Controlled Completion of Dual-Affine Entropy Geometry: Input-Output Born Structures, Compact Holonomy, and Relative-Entropy Generating Functions

1. Introduction

1.1. Scope and Guiding Distinctions

Information geometry equips a statistical model with a metric, dual affine connections, and directed divergences [1,2,3]. Quantum-state geometry additionally uses amplitude complex structures, noncommutative transport, phase forms, and evolution laws. These are not interchangeable: a state manifold is not an amplitude space, an entropy Hessian is not a phase form, and a dimensionless divergence is not a mechanical action. Likewise, statistical symmetric-space Yang-Mills criticality is not spacetime gauge propagation. Directional statistical asymmetry, channel-induced loss of distinguishability, and dynamical dissipation remain distinct; the first two alone supply neither an evolution law nor a clock.

Unless stated otherwise, $n\ge 2$ and all Hilbert spaces are finite-dimensional. On a supplied complex Hermitian carrier $V\simeq {\mathbb{C}}^n$, write $I_{V}v=\mathrm{i}v$ on its underlying real space and consider faithful trace-one states. The reconstruction problem is to recover the logarithmic/Legendre and diagonal-jet data of Umegaki contrast, then compare the structures enabled by each additional channel, metric, or calibration. The roadmap is a typed fork, not a single implication chain:

$$\begin{array}{c}\hfill \begin{array}{cccc}\hfill \left(R\right)& (Herm\left(V\right),Tr,D_{\mathrm{U}})\hfill & \hfill & ⟶\left([log\rho -log\sigma ],g_{\mathrm{BKM}},{\mathcal{C}}^{\left(\mathrm{m}\right)},\mathrm{constructed}\mathrm{dual}\mathrm{Hessian}/\mathrm{Born}\mathrm{data}\right),\hfill \\ \hfill \left(Q\right)& (D_{\mathrm{U}},\mathsf{\Lambda })\hfill & \hfill & ⟶\left({\mathsf{\Delta }}_{\mathsf{\Lambda }},g_{\mathsf{\Lambda }}^{\mathrm{loss}},{\mathcal{C}}_{\mathsf{\Lambda }}^{\mathrm{loss}}\right),\hfill \\ \hfill \left(C\right)& (V,\mathsf{\Phi },-log,g_{\mathrm{AI}})\hfill & \hfill & ⟶\left(A,{\overline{D}}^{\mathrm{AI}},J,{\mathcal{B}}^{\mathrm{rel}}=D^{\mathsf{\Psi }}-{\widehat{D}}^{\mathrm{AI}}\right),\hfill \\ \hfill \left(H\right)& ({\kappa }_A,D_{\mathrm{U}})\hfill & \hfill & ⟶\mathrm{exact}\mathrm{endpoint}\mathrm{canonical}\mathrm{relation}.\hfill \end{array}\end{array}$$

(1)

The labels are mnemonic: (R) denotes relative-entropy reconstruction and regularization, (Q) quotient/channel geometry, (C) compact comparison, and (H) Hamiltonian endpoint reconstruction. In (R), the first variation recovers relative logarithmic cotangent classes modulo scalars on the supplied trace pairing, not the carrier V, $I_V$, Hermitian form, or trace. The constructed Born complex structures act on $T\left(T{\mathcal{S}}_n^{+}\right)$ and are not identified with $I_V$. Determinant normalization and principal functional calculus select the global traceless Gibbs representative (Proposition 5 and Corollary 2). Its state-space Rényi and moment-space Hopf-Lax continuations are exact but inequivalent; interpreting the former as BKM relaxation additionally requires mobility and a clock (Theorems 6 and 7).

In (Q), the output Born geometry is intrinsic to the support-reduced Hessian quotient, not automatic descent of input tensors. One common CPTP left inverse on a smooth embedded faithful model with embedded image makes the restricted channel a contrast-preserving diffeomorphism and preserves its intrinsic metric-through-holonomy stack. Born intertwining additionally requires mixture-affine Hessian models and affinity of that restriction (Remark 7 and Proposition 10).

In (C), the affine-invariant metric is independent input and J is a positive metric normalizer, not a complex structure. The Legendre-exponential and normalized bridges compare compact and BKM transport by defects, not curvature or holonomy conjugacy. The tracial qutrit metric-cubic split is pointwise; its global oriented $\mathrm{PSU}\left(3\right)$ reduction uses a chosen nonzero ${\widehat{D}}^{\mathrm{AI}}$-parallel Cartan form and central alignment, while principal $\mathrm{SU}\left(3\right)$ requires the retained canonical Cartan lift (Theorems 12 and 17). In (H), fixed action conversion ${\kappa }_A>0$ gives the endpoint canonical relation; mechanical interpretation instead supplies inertia and a clock with ${\kappa }_h={\mu }_{\ast }/h$. These calibrations are not identified (Remark 19).

For each fixed $n\ge 2$, only signatures satisfying their dimension and auxiliary-input predicates are indexed. Full BKM tangent holonomy and the general central metric-cubic stabilizer enter for $n\ge 3$; qutrit compact and endpoint packages require $n=3$; AIII requires $n=5$ and an independently supplied orthogonal complex-linear $3+2$ grading. Theorem 26 gives the exact catalogue, including the additional Weyl-field and gauge-action inputs of the indexed AIII anomaly and coupling clauses. Dynamical and flagged-record interpretations retain their stated evolution and apparatus data. The separate massive/null, spacetime-color, and local-horizon applications lie outside that index (Section 9.3).

Definition 1

(Controlled completion). Fix specified base data $\mathfrak{I}$. A finite family of typed constructions

$$F_j:(\mathfrak{I},{\mathfrak{a}}_j)⟶{\mathfrak{O}}_j,j\in J,$$

(2)

is acontrolled completionof $\mathfrak{I}$, relative to its displayed conclusions, if:

• (C1) the carrier, domain, and codomain of every $F_j$ are explicit;
• (C2) every datum used to define ${\mathfrak{O}}_j$ or state a conclusion appears either in $\mathfrak{I}$ or among the declared auxiliary inputs ${\mathfrak{a}}_j$;
• (C3) every asserted relation between branches is represented by an explicit map and proved as a commuting identity, quantitative defect identity, inequality, or obstruction; and
• (C4) no dynamical law, physical representation assignment, or dimensionful normalization absent from the stated inputs is inferred.

This is an input-and-defect completeness condition. It is not a categorical completion, a universal property, or a uniqueness assertion.

The exposition follows R, Q, C, and H before the final dependency audit; detailed calculations and longer proofs appear in the cross-referenced appendices.

• Relation to prior work and theorem-level novelty.

Entropy and channel branches. Finite-dimensional Umegaki relative entropy is used in its standard form [4], with the established BKM/dual-affine geometry [5,6], maximum-entropy families and boundary theory [7,8,9,10], data processing and recovery [11,12,13], Donald’s ensemble identity [14], and Sakamoto’s Hessian Born lift [15]. Moreau/Bregman proximal theory is used from [16,17,18,19,20]. In particular, the weighted Umegaki barycenter, normalized-exponential optimizer, log-Euclidean Rényi value, and sided residual are established [21], Theorem III.6 and Proposition III.8, [22], Equation (I.6) and Proposition V.29, [23]. For $F\left(\omega \right)=Tr(\omega log\omega )$ this is the standard left (variable-first) Bregman-Moreau envelope [20], Section 2; the precise convention mapping and boundary between this established core and the derived BKM extensions are stated before Theorem 6.

The contribution is the compatibility package in Theorems 4, 6, 7 and 10 and Propositions 7 and 11: support-reduced channel/Born geometry with exact loss and response defects; noncommutative moment mirror, information-certificate, and susceptibility identities; the entropic envelope’s BKM endpoint gradients, extended-real scale-Hamiltonian, Hamilton-Jacobi, two-sided action, and calibrated dissipation consequences; and the information-geometric Hopf-Lax specialization with explicit susceptibility shift.

Compact branch. Invariant symmetric-space connections and their Yang-Mills property are background [24,25,26], as are monotone metrics and their curvature calculations [6,27,28]. Cartan-three-form stabilizers and the locally three-symmetric geometry of $\mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right)$ are used from [29,30,31,32]. The complex branching ${\mathsf{\Lambda }}^2\mathbf{8}=\mathbf{8}\oplus \mathbf{10}\oplus \overline{\mathbf{10}}$-equivalently, the real split $\mathfrak{so}\left(8\right)=ad\mathfrak{su}\left(3\right)\oplus {\mathfrak{m}}_{20}$-is standard representation theory [33]; so are the $\mathrm{SU}\left(5\right)$ family branching and the mod-two $\mathrm{SU}\left(2\right)$ anomaly criterion [34,35]. The article-specific claims of Theorems 12, 14, 15, 17, 20 and 21 are their coefficient-explicit realization and comparison on the stated statistical carriers: the normalized compact transporter and curvature defect; the central BKM curvature eigenvalues, plane fractions, and Jordan-normalization reconstruction within the standard $8\oplus 20$ branching; and the pullback realization of the supplied AIII state sector and its associated-representation interfaces.

Endpoint branch. Discrete generating functions and the standard unscaled variational-order mechanism are used from [36,37]. Modular automorphism theory, its thermal-time interpretation, and the Bisognano-Wichmann wedge identification are distinct background inputs [38,39,40]. The claims of Theorems 23 and 24, Corollaries 49 and 52, and Propositions 8 and 35 are the closed-form relative-entropy endpoint generators, their calibrated branch-local Hamiltonian reconstructions and response formulas, the scale obstruction, the scaled-endpoint derivative transfer, sharp cubic defect, and qualified compact-tube phase-map estimates, and the conditional modular clock locking.

2. Geometric Framework and Methods

The core arguments are analytic and finite-dimensional. The methods are spectral and matrix calculus, convex and Legendre-Fenchel duality, Hessian information geometry, reductive homogeneous connections, and variational generating functions. Symbolic and numerical matrix checks were used only to audit signs, ranks, and normalizations; no proof depends on them.

We use the standard affine-connection conventions [1,41]

$$\begin{array}{cc}\hfill T^{\nabla }(X,Y)& ={\nabla }_{X}Y-{\nabla }_{Y}X-[X,Y],\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill R^{\nabla }(X,Y)Z& ={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z.\hfill \end{array}$$

(4)

2.1. Dual Affine Structures

Connections ∇ and ${\nabla }^{\ast }$ are g-dual when $X\left[g(Y,Z)\right]=g({\nabla }_{X}Y,Z)+g(Y,{\nabla }_X^{\ast }Z)$. Set

$$D:=\frac{1}{2}(\nabla +{\nabla }^{\ast }),K:=\frac{1}{2}(\nabla -{\nabla }^{\ast }),$$

(5)

so $\nabla =D+K$ and ${\nabla }^{\ast }=D-K$. Standard duality gives $Dg=0$ and makes $K_X$g-self-adjoint. If both connections are torsion-free, then $D={\nabla }^{\mathrm{LC}}$, $K_{X}Y=K_{Y}X$, and

$$C_K(X,Y,Z):=g(K_{X}Y,Z)$$

(6)

is fully symmetric [1,3]. We reserve $C_K$ for this half-difference cubic, while $\mathcal{C}={\nabla }^{\mathrm{aff}}g$ denotes the Hessian cubic of a chosen flat affine connection; Section 6.2 fixes their sign and factor conventions.

Proposition 1

(Intrinsic curvature adjointness of dual connections). For every g-dual pair $\nabla ,{\nabla }^{\ast }$,

$$g\left(R^{\nabla }(X,Y)Z,W\right)=-g\left(Z,R^{{\nabla }^{\ast }}(X,Y)W\right).$$

(7)

Equivalently, $R^{{\nabla }^{\ast }}(X,Y)=-R^{\nabla }{(X,Y)}^{{†}_g}$. This identity uses only metric duality; it asserts no conjugacy by a Legendre map or by an exponential map. In a dually flat Hessian pair both sides vanish, whereas the generally nonzero curvature considered below is the curvature of the Levi-Civita midpoint or of a different affine-invariant metric.

Proof from standard dual-connection theory. 

Curvature adjointness is the standard curvature identity for metric-dual connections [3]. With the convention (4), applying $XY-YX-[X,Y]$ to $g(Z,W)$ cancels the one-derivative terms and fixes the minus sign in (7). Only metric duality is used, so the stated absence of torsion, flatness, or conjugacy assumptions is essential. □

Supplied amplitude complex structure.

Let $V_{\mathbb{R}}$ be the underlying real space of the supplied Hermitian amplitude carrier $V\simeq {\mathbb{C}}^n$ and define

$$I_V:V_{\mathbb{R}}⟶V_{\mathbb{R}},I_{V}v:=\mathrm{i}v,I_V^2=-\mathrm{id}.$$

(8)

The pair $(I_V,{\langle ·,·\rangle }_V)$, the adjoint, and the trace are kinematic input, not consequences of the entropy Hessian. Ambient scalar multiplication $A\mapsto \mathrm{i}A$ on ${End}_{\mathbb{C}}\left(V\right)$ sends Hermitian operators to skew-Hermitian operators; it therefore does not endow $Herm\left(V\right)$, ${\mathcal{S}}_n^{+}$, or their real tangent spaces with a complex structure. The state manifold ${\mathcal{P}}_n^1$ likewise parametrizes determinant-normalized positive Hermitian forms on V and is a real manifold (of dimension three when $n=2$). Matrix formulas use an orthonormal frame, but the quotient and connection constructions are unitary-frame invariant. We call $I_V$ supplied; an endomorphism produced functorially from the displayed real geometric data is constructed. No identification between differently typed complex structures is implicit.

2.2. The Strongly Integrable Born Lift of a Hessian Manifold

For a Hessian manifold $(M,\nabla ,g)$, write

$${\mathcal{B}}_{\nabla }(M,g):=\left(TM,I_B^{\nabla },J_B^{\nabla },K_B^{\nabla },h_B^{\nabla },{\eta }_B^{\nabla },{\omega }_B^{\nabla }\right)$$

(9)

for Sakamoto’s real tangent-bundle Born tuple. Here $I_B^{\nabla }\in \mathsf{\Gamma }(End\left(T\left(TM\right)\right))$ is the constructed Dombrowski-Sakamoto almost-complex structure determined by the ∇ horizontal-vertical splitting; it acts on the tangent of the doubled real carrier $TM$, not on V. A Hessian structure here means that ∇ is flat and torsion-free and $\nabla g$ is totally symmetric. Throughout, “Born structure” means exactly the tuple in Equation (9); it is distinct both from the supplied scalar multiplication $I_V$ and from generalized-tangent-bundle Born geometry [42] and from the related Künneth/recursion-operator formulation [43]. In particular, generalized torsion in the former setting is not the affine torsion of Equation (3).

The underlying horizontal-vertical almost-Hermitian tangent-bundle construction goes back to Dombrowski [44]; the full Born tuple used here is Sakamoto’s [15], Note 4.14. Using the ∇-horizontal and vertical lifts at $u\in T_{x}M$, its tensors are

$$\begin{array}{cc}\hfill h_B(X^H+U^V,Y^H+V^V)& =g(X,Y)+g(U,V),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\eta }_B(X^H+U^V,Y^H+V^V)& =g(X,V)+g(U,Y),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\omega }_B(X^H+U^V,Y^H+V^V)& =g(X,V)-g(U,Y).\hfill \end{array}$$

(12)

The endomorphisms

$$\begin{array}{cc}\hfill I_B{X}^H=X^V,& I_B{U}^V=-U^H,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill J_B{X}^H=X^V,& J_B{U}^V=U^H,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill K_B{X}^H=X^H,& K_B{U}^V=-U^V\hfill \end{array}$$

(15)

satisfy

$$-I_B^2=J_B^2=K_B^2=\mathrm{id},I_B{J}_B{K}_B=-\mathrm{id},K_B=-I_B{J}_B,$$

(16)

with ${\omega }_B=h_B(I_B·,·)$ and ${\eta }_B=h_B(J_B·,·)$. Thus $h_B$ is positive, ${\eta }_B$ neutral, and ${\omega }_B$ nondegenerate. Only $I_B^2=-\mathrm{id}$ is an almost-complex structure; $J_B$ and $K_B$ square to $+\mathrm{id}$ and are para-complex/product involutions, not additional complex structures.

These formulas define an almost-Born structure for any affine connection and Riemannian metric. Sakamoto proves [15], Proposition 3.11

$$(M,\nabla ,g)\mathrm{is}\mathrm{Hessian}⟺(TM,I_B,J_B,K_B,h_B,{\eta }_B,{\omega }_B)\mathrm{is}\mathrm{strongly}\mathrm{integrable},$$

(17)

Strong integrability includes the para-hypercomplex triple, $\mathrm{d}{\omega }_B=0$, and the ∇-determined bi-Lagrangian splitting. In particular, under the Hessian hypotheses used here $I_B^{\nabla }$ is integrable. For an arbitrary affine connection the same formula defines only an almost-complex structure.

Remark 1

(Born rotation and dual horizontal shear). For $A=X_u^{H,\nabla }+U_u^V$,

$$I_{B}A=-U_u^{H,\nabla }+X_u^V,{\mathrm{e}}^{\theta I_B}A=cos\theta A+sin\theta I_{B}A.$$

(18)

More precisely, the connection-dependent real-linear isomorphism

$${\chi }_u^{\nabla }:T_u\left(TM\right)⟶T_{x}M{\otimes }_{\mathbb{R}}\mathbb{C},{\chi }_u^{\nabla }(X_u^{H,\nabla }+U_u^V)=X+\mathrm{i}U,{\chi }_u^{\nabla }{I}_B^{\nabla }=\mathrm{i}{\chi }_u^{\nabla }$$

(19)

identifies $I_B^{\nabla }$ with multiplication by the formal scalar $\mathrm{i}$ on the complexification of the real statistical tangent fiber. Since $I_B$ is $h_B$-skew and ${\omega }_B=h_B(I_B·,·)$, this pointwise tangent-space rotation at u preserves $h_B$ and ${\omega }_B$.

For the dual torsion-free pair of Equation (5), the horizontal lifts satisfy

$$\begin{array}{cc}\hfill X_u^{H,\nabla }& =X_u^{H,D}-{\left(K_{X}u\right)}_u^V,\hfill \\ \hfill X_u^{H,{\nabla }^{\ast }}& =X_u^{H,D}+{\left(K_{X}u\right)}_u^V.\hfill \end{array}$$

(20)

Hence

$$X_u^{H,{\nabla }^{\ast }}-X_u^{H,\nabla }=2{\left(K_{X}u\right)}_u^V,g(2K_{X}u,Z)=-\mathcal{C}(X,u,Z),$$

(21)

where $\mathcal{C}=\nabla g$. The pointwise endomorphism

$$N_u\left(A\right):={\left(2K_{{\pi }_{\ast }A}u\right)}_u^V.$$

(22)

satisfies $N_u^2=0$. Set $S_u:={\mathrm{e}}^{N_u}=\mathrm{id}+N_u$; then $S_u{X}_u^{H,\nabla }=X_u^{H,{\nabla }^{\ast }}$. Direct substitution into Equations (12) and (15) gives

$$\begin{array}{cccccc}\hfill S_u{I}_B^{\nabla }& =I_B^{{\nabla }^{\ast }}{S}_u,\hfill & \hfill S_u{J}_B^{\nabla }& =J_B^{{\nabla }^{\ast }}{S}_u,\hfill & \hfill S_u{K}_B^{\nabla }& =K_B^{{\nabla }^{\ast }}{S}_u,\hfill \\ \hfill S_u^{\ast }{h}_{B,u}^{{\nabla }^{\ast }}& =h_{B,u}^{\nabla },\hfill & \hfill S_u^{\ast }{\eta }_{B,u}^{{\nabla }^{\ast }}& ={\eta }_{B,u}^{\nabla },\hfill & \hfill S_u^{\ast }{\omega }_{B,u}^{{\nabla }^{\ast }}& ={\omega }_{B,u}^{\nabla }={\omega }_{B,u}.\hfill \end{array}$$

(23)

For example, on $A=X_u^{H,\nabla }+U_u^V$, both sides of the first identity equal $-U_u^{H,{\nabla }^{\ast }}+X_u^V$. The two constructed complex structures are integrable under the dually flat Hessian hypotheses. Full symmetry of $g(K_{X}Y,Z)$ also gives

$${\omega }_B(N_{u}A,B)+{\omega }_B(A,N_{u}B)=0,{\omega }_B(N_{u}A,N_{u}B)=0,$$

(24)

Thus $S_u$ is a symplectic full-tuple intertwiner, but generally not orthogonal for either metric held fixed; for example $S_u^{\ast }{h}_{B,u}^{\nabla }\ne h_{B,u}^{\nabla }$ in general. The Born rotation is $h_B$-orthogonal. Neither pointwise map is asserted to integrate to a diffeomorphism.

More generally, a fiberwise complex identification with a separately supplied physical Hermitian bundle $(E_{\mathbb{R}},I_E,g_E)\to B$, where $g_E=Re{\langle ·,·\rangle }_E$, requires a specified smooth base map $\varphi :TM\to B$ and a rank-matched real bundle isomorphism

$$\mathsf{\Xi }:T\left(TM\right)\stackrel{\sim }{⟶}{\varphi }^{\ast }{E}_{\mathbb{R}},\mathsf{\Xi }\circ I_B^{\nabla }=\left({\varphi }^{\ast }{I}_E\right)\circ \mathsf{\Xi }.$$

(25)

A Hermitian identification further requires ${\mathsf{\Xi }}^{\ast }{g}_E=h_B$. If ${\omega }_E:=g_E(I_E·,·)$, these two conditions imply ${\mathsf{\Xi }}^{\ast }{\omega }_E={\omega }_B$. Even then, transport, quantum superposition, unitary evolution, phase, and measurement interpretations require their own compatible connections, preparations, flows, scales, and instruments. Relative entropy supplies none of $E,I_E,\varphi$, or Ξ. In particular, for $M={\mathcal{S}}_n^{+}$ the complex rank of $(T\left(TM\right),I_B)$ is $n^2-1$, whereas ${dim}_{\mathbb{C}}V=n$; these ranks never agree for an integer $n\ge 2$, since then $n^2-1>n$. This rules out a direct complex-bundle isomorphism with V, but not noninvertible complex-linear morphisms, embeddings into a larger physical bundle, or quotients onto a smaller one. The natural formal target is ${Herm}_0\left(V\right){\otimes }_{\mathbb{R}}\mathbb{C}\simeq \mathfrak{sl}\left(V\right)$, not V.

For the cotangent comparison, set ${\mathsf{\Theta }}_{\mathrm{can}}=p_i\mathrm{d}{x}^i$, ${\mathsf{\Omega }}_{\mathrm{can}}=\mathrm{d}{\mathsf{\Theta }}_{\mathrm{can}}$, and let $b_g:TM\to T^{\ast }M$, $u\mapsto g(u,·)$, be the musical diffeomorphism.

Proposition 2

(One Born carrier, dual-connection polarizations, and midpoint mechanics). For a dually flat Hessian pair, the two Sakamoto constructions satisfy

$${\omega }_B^{\nabla }={\omega }_B^{{\nabla }^{\ast }}=-{\left(b_g\right)}^{\ast }{\mathsf{\Omega }}_{\mathrm{can}}=:{\omega }_B,$$

(26)

Write ${\mathcal{H}}^{\nabla },{\mathcal{H}}^{{\nabla }^{\ast }}\subset T\left(TM\right)$ for the two horizontal distributions. They are integrable Lagrangian polarizations, each complementary to the common integrable vertical Lagrangian polarization, and are related pointwise by the symplectic shear of Equation (22).

Every independently supplied canonical Hamiltonian H on $T^{\ast }M$ also has two exact affine descriptions on this carrier. If ${\iota }_{X_H}{\mathsf{\Omega }}_{\mathrm{can}}=-\mathrm{d}H$, $Z={\left(b_g^{-1}\right)}_{\ast }{X}_H$, and $\widehat{H}=-H\circ b_g$, then

$${\iota }_Z{\omega }_B=-\mathrm{d}\widehat{H},Z=X^{H,\nabla }+U^V=X^{H,{\nabla }^{\ast }}+{\left(U-2K_{X}u\right)}^V,$$

(27)

where $X={\pi }_{\ast }Z$ at $u\in TM$. Thus the two connections give polarization-dependent decompositions of one supplied flow; they do not select that flow.

On any dual-coordinate domain where the ordinary Hessian Legendre map ℓ is a diffeomorphism, there is a complementary exact statement. In dual coordinates ${\eta }_i={\partial }_i\psi \left(\theta \right)$, the contravariant cotangent lift $\widehat{\ell }:=T^{\ast }\left({\ell }^{-1}\right)$ is

$$\widehat{\ell }(\theta ,p)=\left(\eta ,{\pi }^i=g^{ij}\left(\theta \right)p_j\right),{\widehat{\ell }}^{\ast }\left({\pi }^i\mathrm{d}{\eta }_i\right)=p_i\mathrm{d}{\theta }^i.$$

(28)

It therefore conjugates every supplied $H_{\theta }$ to $H_{\eta }=H_{\theta }\circ {\widehat{\ell }}^{-1}$. This is exact canonical equivalence of the mixture and exponential coordinate descriptions, not an identification of their affine geodesic sprays or of the later affine-invariant symmetric-space mechanics.

More specifically, let $S^{\nabla },S^{{\nabla }^{\ast }},S^D$ be the geodesic sprays and put

$$E\left(u\right):=\frac{1}{2}g(u,u),{\beta }_u\left(A\right):=C_K\left(u,u,{\pi }_{\ast }A\right).$$

(29)

Then

$$\begin{array}{cccccc}\hfill S^{\nabla }& =S^D-{\left(K_{u}u\right)}^V,\hfill & \hfill S^{{\nabla }^{\ast }}& =S^D+{\left(K_{u}u\right)}^V,\hfill & \hfill S^D& =\frac{1}{2}(S^{\nabla }+S^{{\nabla }^{\ast }}),\hfill \end{array}$$

(30)

$$\begin{array}{cccccc}\hfill {\iota }_{S^D}{\omega }_B& =\mathrm{d}E,\hfill & \hfill {\iota }_{S^{\nabla }}{\omega }_B& =\mathrm{d}E+\beta ,\hfill & \hfill {\iota }_{S^{{\nabla }^{\ast }}}{\omega }_B& =\mathrm{d}E-\beta .\hfill \end{array}$$

(31)

Accordingly $S^D$ is Hamiltonian for $-E$ on $(TM,{\omega }_B)$-or for E on $(TM,-{\omega }_B)$-under the convention ${\iota }_{X_H}\omega =-\mathrm{d}H$. For every open $\mathcal{O}\subset M$, either affine spray preserves ${\omega }_B$ on $T\mathcal{O}$ if and only if ${K|}_{\mathcal{O}}=0$; then both equal $S^D$ and are Hamiltonian for $-E$. Where $K\ne 0$, neither spray is even locally Hamiltonian. Born geometry therefore constructs a common phase space and a Hamiltonian midpoint, not one common affine geodesic flow.

Lemma 1

(Affine-isometric functoriality of the Sakamoto lift). Let $F:(M,\nabla ,g)\to (N,{\nabla }^{\prime },g^{\prime })$ be an affine isometric diffeomorphism: $F^{\ast }{g}^{\prime }=g$ and $F_{\ast }\left({\nabla }_{X}Y\right)={\nabla }_{F_{\ast }X}^{\prime }\left(F_{\ast }Y\right)$. Then F also intertwines the metric-dual connections. For $u\in T_{x}M$,

$$\begin{array}{cccc}\hfill T\left(TF\right)\left(X_u^{H,\nabla }\right)& ={\left(F_{\ast }X\right)}_{TF\left(u\right)}^{H,{\nabla }^{\prime }},\hfill & \hfill T\left(TF\right)\left(U_u^V\right)& ={\left(F_{\ast }U\right)}_{TF\left(u\right)}^V.\hfill \end{array}$$

(32)

Consequently $TF:TM\to TN$ intertwines the three Sakamoto endomorphisms and is an isometry for all three bilinear tensors:

$$\begin{array}{cccccc}\hfill T\left(TF\right)\circ I_B^{\nabla }& =I_B^{{\nabla }^{\prime }}\circ T\left(TF\right),\hfill & \hfill T\left(TF\right)\circ J_B^{\nabla }& =J_B^{{\nabla }^{\prime }}\circ T\left(TF\right),\hfill & \hfill T\left(TF\right)\circ K_B^{\nabla }& =K_B^{{\nabla }^{\prime }}\circ T\left(TF\right),\hfill \end{array}$$

(33)

$$\begin{array}{cccccc}\hfill {\left(TF\right)}^{\ast }{h}_B^{{\nabla }^{\prime }}& =h_B^{\nabla },\hfill & \hfill {\left(TF\right)}^{\ast }{\eta }_B^{{\nabla }^{\prime }}& ={\eta }_B^{\nabla },\hfill & \hfill {\left(TF\right)}^{\ast }{\omega }_B^{{\nabla }^{\prime }}& ={\omega }_B^{\nabla }.\hfill \end{array}$$

(34)

The midpoint connections are intertwined as well. If $b_g:TM\to T^{\ast }M$ and $b_{g^{\prime }}:TN\to T^{\ast }N$ are the musical maps and $\widehat{F}:=T^{\ast }\left(F^{-1}\right):T^{\ast }M\to T^{\ast }N$ is the exact cotangent lift, then

$$\begin{array}{|c|}\hline \widehat{F}\circ b_g=b_{g^{\prime }}\circ TF.\\ \hline\end{array}$$

(35)

Thus affine isometries induce isomorphisms of the dual Sakamoto Born tuples, their midpoint carrier, and their cotangent realizations.

Proof from Sakamoto functoriality. 

Sakamoto’s affine-lift naturality and affine-isometric Born functoriality apply directly to F [15], Propositions 4.13 and 4.22(2). Metric duality shows that the same F intertwines the dual connections; averaging gives the midpoint assertion. Finally, $F^{\ast }{g}^{\prime }=g$ identifies the musical maps under the exact cotangent lift, yielding (35). Thus the cited construction gives all three tuple and cotangent intertwinings with the conventions above. □

Proposition 3

(Cubic anholonomy and transport memory). Define the raised Hessian cubic by

$$g({\mathsf{C}}_X^{♯}Y,Z):=\mathcal{C}(X,Y,Z).$$

(36)

For a dually flat Hessian pair,

$$\begin{array}{cccc}\hfill K_X& =-\frac{1}{2}{\mathsf{C}}_X^{♯},\hfill & \hfill R^D(X,Y)& =-[K_X,K_Y]=-\frac{1}{4}[{\mathsf{C}}_X^{♯},{\mathsf{C}}_Y^{♯}],\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill [X^{H,D},Y^{H,D}{]}_u-{[X,Y]}_u^{H,D}& =\frac{1}{4}{\left([{\mathsf{C}}_X^{♯},{\mathsf{C}}_Y^{♯}]u\right)}_u^V.\hfill \end{array}$$

(38)

Thus a nonzero cubic is not sufficient for midpoint curvature: its self-adjoint raised endomorphisms must fail to commute. By the Ambrose-Singer theorem [45], the restricted midpoint holonomy algebra is generated by the parallel translates of $-\frac{1}{4}[{\mathsf{C}}_U^{♯},{\mathsf{C}}_V^{♯}]$.

For two piecewise-smooth paths ${\gamma }_0,{\gamma }_1:x\to y$,

$${\left(P_{{\gamma }_1}^D\right)}^{-1}{P}_{{\gamma }_0}^D={Hol}^D\left({\gamma }_0\mathit{followed}\mathit{by}{\overline{\gamma }}_1\right).$$

(39)

For the positively oriented boundary ${\gamma }_{\epsilon }$ of a sufficiently small coordinate rectangle in D-normal coordinates, based at x with initial sides $\epsilon X,\epsilon Y$,

$$log{P}_{{\gamma }_{\epsilon }}^D=-{\epsilon }^2{R}_x^D(X,Y)+O\left({\epsilon }^3\right)=\frac{{\epsilon }^2}{4}[{\mathsf{C}}_X^{♯},{\mathsf{C}}_Y^{♯}]+O\left({\epsilon }^3\right).$$

(40)

Hence, even after an initial vector or frame is fixed, the two endpoint base states do not determine its transport when midpoint holonomy is nontrivial. Holonomy is only a many-to-one, gauge-covariant residue of the path, not a reconstruction of its full history. The flat ∇- and ${\nabla }^{\ast }$-transports themselves have no local anholonomy; the effect belongs to their midpoint and is reversible, so it is not by itself dissipative hysteresis.

Corollary 1

(Midpoint curvature as tangent nonassociativity). At each point of a dually flat Hessian manifold define the difference-tensor product

$$X★Y:=K_{X}Y,{Assoc}_{★}(X,Y,Z):=\left(X★Y\right)★Z-X★\left(Y★Z\right).$$

(41)

It is a commutative metrized product:

$$X★Y=Y★X,g(X★Y,Z)=g(X,Y★Z)=-\frac{1}{2}\mathcal{C}(X,Y,Z).$$

(42)

Its antisymmetrized associator is exactly the Levi-Civita midpoint curvature:

$$\begin{array}{|c|}\hline R^D(X,Y)Z={Assoc}_{★}(X,Y,Z)-{Assoc}_{★}(Y,X,Z).\\ \hline\end{array}$$

(43)

Equivalently, for normalized antisymmetrization, $R^D=2{Alt}_{X,Y}{Assoc}_{★}$. Consequently,

$$R_p^D=0⟺(T_{p}M,{★}_p)\mathit{is}\mathit{associative},$$

(44)

and every nondegenerate two-plane satisfies

$$\begin{array}{|c|}\hline {sec}_D(X,Y)=\frac{{∥X★Y∥}^2-g(X★X,Y★Y)}{g(X,X)g(Y,Y)-g{(X,Y)}^2}.\\ \hline\end{array}$$

(45)

Thus it is midpoint, rather than affine, flatness that is equivalent to fiberwise associativity: the dual affine connections remain flat by hypothesis.

Proof. 

Torsion freedom of the dual pair makes $K_{X}Y$ symmetric in $X,Y$, while metric duality makes $g(K_{X}Y,Z)$ totally symmetric. Commutativity gives

$${Assoc}_{★}(X,Y,Z)-{Assoc}_{★}(Y,X,Z)=-[K_X,K_Y]Z,$$

which is Equation (37). If the product is associative, all $K_X$ commute. Conversely, if all $K_X$ commute, then

$$\left(X★Y\right)★Z=Z★\left(X★Y\right)=X★\left(Z★Y\right)=X★\left(Y★Z\right),$$

so the product is associative. Finally, invariance of g under the product yields $g(R^D(X,Y)Y,X)={∥X★Y∥}^2-g(X★X,Y★Y)$. □

The proofs of Propositions 2 and 3 are given in Appendix C.2.

2.3. Positive Forms, Determinant Gauge, and the Global Log Chart

Let ${\mathcal{P}}_n:=\{P=P^{†}>0\}$ be the positive Hermitian cone on the supplied carrier, and let

$${\mathcal{P}}_n^1:=\{P\in {\mathcal{P}}_n:detP=1\}.$$

(46)

Congruence and polar decomposition identify ${\mathcal{P}}_n\cong \mathrm{GL}(n,\mathbb{C})/\mathrm{U}\left(n\right)$ [26,46].

Theorem 1

(Global state/quotient/log identification). Let

$${\mathcal{S}}_n:=\{\rho ={\rho }^{†}⪰0:Tr\rho =1\},{\mathcal{S}}_n^{+}:=\{\rho \in {\mathcal{S}}_n:\rho >0\}$$

(47)

be the closed and faithful state spaces, respectively. The maps

$$\begin{array}{cccc}\hfill \mathsf{\Phi }:{\mathcal{S}}_n^{+}& ⟶{\mathcal{P}}_n^1,\hfill & \hfill \mathsf{\Phi }\left(\rho \right)& ={(det\rho )}^{-1/n}\rho ,\hfill \end{array}$$

(48)

$$\begin{array}{cccc}\hfill {\mathsf{\Phi }}^{-1}:{\mathcal{P}}_n^1& ⟶{\mathcal{S}}_n^{+},\hfill & \hfill {\mathsf{\Phi }}^{-1}\left(P\right)& =\frac{P}{TrP}\hfill \end{array}$$

(49)

are smooth inverses, and

$${\mathcal{S}}_n^{+}\simeq {\mathcal{P}}_n^1\cong \mathrm{SL}(n,\mathbb{C})/\mathrm{SU}\left(n\right),{dim}_{\mathbb{R}}{\mathcal{P}}_n^1=n^2-1.$$

(50)

With $\mathfrak{p}:={Herm}_0\left(n\right)$, the matrix exponential is a global diffeomorphism

$$exp:\mathfrak{p}⟶{\mathcal{P}}_n^1,Y⟼{\mathrm{e}}^Y$$

(51)

with inverse the principal Hermitian logarithm. Equivalently, every faithful density operator has a unique traceless Hermitian Gibbs coordinate $X=-log\mathsf{\Phi }\left(\rho \right)$ such that

$${\rho }_X=\frac{{\mathrm{e}}^{-X}}{Tr{\mathrm{e}}^{-X}},X\in {Herm}_0\left(n\right).$$

(52)

Proof from positive-cone geometry. 

Polar decomposition and Hermitian functional calculus give the quotient and global exponential/logarithm diffeomorphisms [26,46]. The manuscript-specific normalization is $det(P/TrP)={(TrP)}^{-n}$ together with $TrlogP=logdetP$; these identities give the inverse state map and the unique traceless Gibbs representative. □

The principal log is fixed by positive-form geometry; entropy subsequently selects a Hessian metric in this chart rather than constructing the chart.

2.4. Cartan Decomposition

For later connection formulas, view $\mathfrak{sl}(n,\mathbb{C})$ as a real Lie algebra. The Cartan involution $\mathsf{\Theta }\left(X\right)=-X^{†}$ has the eigenspace decomposition [26]

$$\mathfrak{sl}(n,\mathbb{C})=\mathfrak{k}\oplus \mathfrak{p}=\mathfrak{su}\left(n\right)\oplus \mathrm{i}\mathfrak{su}\left(n\right).$$

(53)

Here $\mathfrak{k}=\mathfrak{su}\left(n\right)$ and $\mathfrak{p}={Herm}_0\left(n\right)=\mathrm{i}\mathfrak{su}\left(n\right)$, and

$$[\mathfrak{k},\mathfrak{k}]\subset \mathfrak{k},[\mathfrak{k},\mathfrak{p}]\subset \mathfrak{p},[\mathfrak{p},\mathfrak{p}]\subset \mathfrak{k}.$$

(54)

Thus two noncompact tangent directions have a compact commutator.

The shifted-connection identities used below are collected in Appendix C.1.

3. Relative Entropy and Lie-Log Geometry

3.1. Global Lie-Log and Bregman Forms of Umegaki Relative Entropy

For faithful states, Umegaki relative entropy is [4]

$$D_{\mathrm{U}}(\rho \parallel \sigma ):=Tr\rho (log\rho -log\sigma )$$

(55)

and, for a differentiable convex potential F, our Bregman orientation is

$$B_F(x,y):=F\left(x\right)-F\left(y\right)-D{F}_y[x-y].$$

(56)

We first establish their global relation and the score-tangent dictionary; regular exponential families will then be affine restrictions.

For faithful $\rho$, define

$${\mathsf{K}}_{\rho }\left(A\right):={\int }_0^1{\rho }^{s}A{\rho }^{1-s}\mathrm{d}s.$$

(57)

This Kubo-Mori operator is strictly positive and Hilbert-Schmidt self-adjoint. Write ${\tilde{A}}_{\rho }:=A-Tr\left(\rho A\right)I$.

Proposition 4

(BKM score-tangent and covariance dictionary). For ρ-centered Hermitian scores $A,B$ and trace-zero density tangents $U,W$, the two realizations of the BKM metric are

$$\begin{array}{cc}\hfill g_{\rho ,\mathrm{sc}}^{\mathrm{BKM}}(A,B)& :=Tr\left[A{\mathsf{K}}_{\rho }\left(B\right)\right],\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill g_{\rho }^{\mathrm{BKM}}(U,W)& :=Tr\left[U{\mathsf{K}}_{\rho }^{-1}\left(W\right)\right].\hfill \end{array}$$

(59)

The map $A\mapsto -{\mathsf{K}}_{\rho }\left(A\right)$ is onto the density-tangent space and

$$g_{\rho }^{\mathrm{BKM}}(-{\mathsf{K}}_{\rho }A,-{\mathsf{K}}_{\rho }B)=g_{\rho ,\mathrm{sc}}^{\mathrm{BKM}}(A,B).$$

(60)

For arbitrary Hermitian $A,B$, the cotangent covariance is

$${Cov}_{\rho }^{\mathrm{KM}}(A,B):={\int }_0^{1}Tr\left({\rho }^s{\tilde{A}}_{\rho }{\rho }^{1-s}{\tilde{B}}_{\rho }\right)\mathrm{d}s=g_{\rho ,\mathrm{sc}}^{\mathrm{BKM}}({\tilde{A}}_{\rho },{\tilde{B}}_{\rho }).$$

(61)

Its kernel consists exactly of scalar representatives. ThusBKMrefers to both realizations only through the displayed isometry. Channels act directly on density tangents; their induced score map is generally state-dependent.

Proof. 

Since ${\mathsf{K}}_{\rho }\left(I\right)=\rho$ and $Tr{\mathsf{K}}_{\rho }\left(A\right)=Tr\left(\rho A\right)$, its invertibility identifies centered scores with trace-zero tangents. Self-adjointness gives the isometry and covariance identity, and positivity gives the scalar kernel. These are the standard BKM identifications [5,6,47]. □

Corollary 2

(Global Gibbs chart and Umegaki-Bregman geometry). Define $\mathsf{\Psi }:\mathfrak{p}\to \mathbb{R}$ and $\mathcal{X}:{\mathcal{S}}_n^{+}\to \mathfrak{p}$ by

$$\mathsf{\Psi }\left(X\right):=logTr{\mathrm{e}}^{-X},\mathcal{X}\left(\rho \right):=-log\mathsf{\Phi }\left(\rho \right)=-log\rho +\frac{1}{n}Tr(log\rho )I.$$

(62)

Then $X\mapsto {\rho }_X={\mathrm{e}}^{-X}/Tr{\mathrm{e}}^{-X}$ and $\rho \mapsto \mathcal{X}\left(\rho \right)$ are smooth inverses between $\mathfrak{p}$ and the full faithful-state manifold. For all $X,Y\in \mathfrak{p}$,

$$D_{\mathrm{U}}({\rho }_X\parallel {\rho }_Y)=\mathsf{\Psi }\left(Y\right)-\mathsf{\Psi }\left(X\right)-D{\mathsf{\Psi }}_X[Y-X]=B_{\mathsf{\Psi }}(Y,X),$$

(63)

For $H,J\in \mathfrak{p}$, the Hessian is the Gibbs pullback of the density-tangent BKM metric:

$$D^2{\mathsf{\Psi }}_X[H,J]={Cov}_{{\rho }_X}^{\mathrm{KM}}(H,J)=g_{{\rho }_X}^{\mathrm{BKM}}(D{\rho }_X\left[H\right],D{\rho }_X\left[J\right]).$$

(64)

It is positive definite on $\mathfrak{p}$. Hence Umegaki relative entropy is globally Bregman in the determinant-normalized Lie-log chart, with no commutativity assumption.

Proof. 

The inverse maps follow from Theorem 1 and $logdet\rho =Trlog\rho$. Duhamel differentiation of the normalized exponential gives, for $H\in \mathfrak{p}$,

$$D{\mathsf{\Psi }}_X\left[H\right]=-Tr\left({\rho }_{X}H\right),D{\rho }_X\left[H\right]=-{\mathsf{K}}_{{\rho }_X}\left({\tilde{H}}_{{\rho }_X}\right).$$

(65)

Substituting $log{\rho }_X=-X-\mathsf{\Psi }\left(X\right)I$ in Equation (55) proves Equation (63). Differentiating the first identity and applying Proposition 4 gives Equation (64); the scalar kernel meets $\mathfrak{p}$ only at zero. □

Regular exponential subfamilies.

Let $T_a\in \mathfrak{p}$ be linearly independent modulo the identity and restrict the global construction to $W=span\left\{T_a\right\}$:

$$X\left(\theta \right)={\theta }^a{T}_a,\psi \left(\theta \right)=\mathsf{\Psi }\left(X\left(\theta \right)\right),{\rho }_{\theta }=exp[-X\left(\theta \right)-\psi \left(\theta \right)I].$$

(66)

The preceding corollary gives ${\partial }_a\psi =-Tr\left({\rho }_{\theta }{T}_a\right)$ and

$$D_{\mathrm{U}}({\rho }_{\theta }\parallel {\rho }_{\eta })=\psi \left(\eta \right)-\psi \left(\theta \right)-({\eta }^a-{\theta }^a){\partial }_a\psi \left(\theta \right)=B_{\psi }(\eta ,\theta ).$$

(67)

Writing ${\tilde{T}}_a:=T_a-Tr\left({\rho }_{\theta }{T}_a\right)I$, its metric is

$$g_{ab}^{\mathrm{BKM}}\left(\theta \right):={\partial }_a{\partial }_b\psi \left(\theta \right)={Cov}_{{\rho }_{\theta }}^{\mathrm{KM}}(T_a,T_b)=g_{{\rho }_{\theta },\mathrm{sc}}^{\mathrm{BKM}}({\tilde{T}}_a,{\tilde{T}}_b).$$

(68)

This positive-definite pullback equips the regular family with its intrinsic Hessian-dual exponential/mixture pair [5], Sections 2-3; restriction alone does not assert ambient mixture autoparallelity. Its fully symmetric cubic

$${\mathcal{C}}_{abc}={\partial }_a{\partial }_b{\partial }_c\psi$$

(69)

controls the leading correction to the quadratic approximation.

For a parameter tangent $\delta ={\delta }^a{\partial }_a$, the same dictionary specializes to

$$\begin{array}{cc}\hfill \delta \tilde{X}& :={\delta }^a{\tilde{T}}_a,\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill V_{\delta }:={\delta }^a{\partial }_a{\rho }_{\theta }& =-{\mathsf{K}}_{{\rho }_{\theta }}\left(\delta \tilde{X}\right),\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill {∥\delta \tilde{X}∥}_{{\rho }_{\theta },\mathrm{BKM}}^2& :=g_{{\rho }_{\theta },\mathrm{sc}}^{\mathrm{BKM}}(\delta \tilde{X},\delta \tilde{X})=g_{ab}^{\mathrm{BKM}}\left(\theta \right){\delta }^a{\delta }^b=g_{{\rho }_{\theta }}^{\mathrm{BKM}}(V_{\delta },V_{\delta }).\hfill \end{array}$$

(72)

Entropy as the dual potential.

Legendre duality identifies the mixture potential with negative von Neumann entropy. Globally on the faithful trace-one affine space, set $\mathcal{F}\left(\rho \right):=-S_{\mathrm{vN}}\left(\rho \right)=Tr(\rho log\rho )$. For trace-zero V,

$$D{\mathcal{F}}_{\sigma }\left[V\right]=Tr\left[V(log\sigma +I)\right],D^2{\mathcal{F}}_{\sigma }[V,W]=g_{\sigma }^{\mathrm{BKM}}(V,W).$$

(73)

Consequently,

$$\begin{array}{|c|}\hline D_{\mathrm{U}}(\rho \parallel \sigma )=B_{\mathcal{F}}(\rho ,\sigma )=B_{\mathsf{\Psi }}(\mathcal{X}\left(\sigma \right),\mathcal{X}\left(\rho \right)),g_{\mathrm{BKM}}={Hess}_{\mathrm{m}}\mathcal{F}={Hess}_{\mathrm{e}}\mathsf{\Psi }.\\ \hline\end{array}$$

(74)

The first equality uses $Tr(\rho -\sigma )=0$ and the second is Equation (63). The definition of relative entropy uses the supplied matrix logarithm and does not construct its functional calculus. Once the two-point contrast is given, however, its first derivative recovers the relative logarithmic cotangent class, as the next proposition makes precise. The Hessian statement is affine-coordinate dependent in the usual dual-affine sense [1,5].

Proposition 5

(Relative-entropy reconstruction and diagonal jets). View ${\mathcal{S}}_n^{+}$ as an open convex subset of the trace-one affine hyperplane, with mixture coordinates $x=\left(x^a\right)$. The contrast $\mathfrak{D}(x,y):=D_{\mathrm{U}}(x\parallel y)$ determines $\mathcal{F}\left(x\right)=Tr(xlogx)$ up to addition of an affine functional. More precisely, for any fixed ${\sigma }_0\in {\mathcal{S}}_n^{+}$,

$${\tilde{\mathcal{F}}}_{{\sigma }_0}\left(\rho \right):=\mathfrak{D}(\rho ,{\sigma }_0)=\mathcal{F}\left(\rho \right)-Tr(\rho log{\sigma }_0)$$

(75)

is such a representative, and

$$D{\tilde{\mathcal{F}}}_{{\sigma }_0,\rho }\left[V\right]=Tr\left[V(log\rho -log{\sigma }_0)\right],TrV=0.$$

(76)

Thus the contrast reconstructs the relative logarithmic cotangent class in $Herm\left(n\right)/\mathbb{R}I$; determinant normalization and the principal logarithm in Corollary 2 select its global traceless matrix representative, namely $\mathcal{X}\left({\sigma }_0\right)-\mathcal{X}\left(\rho \right)$.

Write unprimed indices for derivatives in the first argument of $\mathfrak{D}$ and primed indices for derivatives in the second. On the diagonal $y=x$ its second- and third-order slot identities are

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{ab}& =-{\mathfrak{D}}_{a{b}^{\prime }}={\mathfrak{D}}_{a^{\prime }{b}^{\prime }}=g_{ab}^{\mathrm{BKM}},\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{abc}& =-{\mathfrak{D}}_{a{b}^{\prime }{c}^{\prime }}={\mathcal{C}}_{abc}^{\left(\mathrm{m}\right)},{\mathfrak{D}}_{ab{c}^{\prime }}=0,{\mathfrak{D}}_{a^{\prime }{b}^{\prime }{c}^{\prime }}=2{\mathcal{C}}_{abc}^{\left(\mathrm{m}\right)}.\hfill \end{array}$$

(78)

Consequently the Eguchi connections satisfy

$${\mathsf{\Gamma }}_{abc}^{\left(\mathrm{m}\right)}=-{\mathfrak{D}}_{ab{c}^{\prime }}=0,{\mathsf{\Gamma }}_{abc}^{\left(\mathrm{e}\right)}=-{\mathfrak{D}}_{a^{\prime }{b}^{\prime }c}={\mathcal{C}}_{abc}^{\left(\mathrm{m}\right)},$$

(79)

so the lowered Levi-Civita coefficient is ${\mathsf{\Gamma }}_{abc}^{\mathrm{LC}}={\mathcal{C}}_{abc}^{\left(\mathrm{m}\right)}/2$ and, with $K=({\nabla }^{\left(\mathrm{m}\right)}-{\nabla }^{\left(\mathrm{e}\right)})/2$,

$$g_{\mathrm{BKM}}(K_{X}Y,Z)=-\frac{1}{2}{\mathcal{C}}^{\left(\mathrm{m}\right)}(X,Y,Z).$$

(80)

Proof from contrast-function geometry. 

Because $\mathfrak{D}=B_{\mathcal{F}}$, its affine-gauge and cotangent statements follow directly. Applying the standard diagonal-jet construction for an oriented contrast [1,48] gives Equations (77)–(79) with the displayed slot signs; metric duality then gives (80). Thus the standard theorem supplies the jets, while the proposition records the manuscript’s Bregman orientation and traceless cotangent representative. □

Corollary 3

(Global Gibbs-Born and cotangent bridge). Put $g^{\mathsf{\Psi }}:=Hess\mathsf{\Psi }$ on $\mathfrak{p}$, let ∂ be its flat coordinate connection, and let ${\partial }^{\ast ,\mathsf{\Psi }}$ be the $g^{\mathsf{\Psi }}$-dual connection. The Gibbs map is an affine isometry

$$\mathcal{X}:({\mathcal{S}}_n^{+},{\nabla }^{\left(\mathrm{e}\right)},g_{\mathrm{BKM}})\stackrel{\simeq }{\to }(\mathfrak{p},\partial ,g^{\mathsf{\Psi }}),{\mathcal{X}}_{\ast }{\nabla }^{\left(\mathrm{m}\right)}={\partial }^{\ast ,\mathsf{\Psi }}.$$

(81)

Consequently $T\mathcal{X}$ is an isomorphism of both dual Sakamoto Born tuples and sends their Levi-Civita midpoint $D=({\nabla }^{\left(\mathrm{e}\right)}+{\nabla }^{\left(\mathrm{m}\right)})/2$ to $D^{\mathsf{\Psi }}:={\nabla }_{g^{\mathsf{\Psi }}}^{\mathrm{LC}}$. Moreover, with $\widehat{\mathcal{X}}:=T^{\ast }\left({\mathcal{X}}^{-1}\right)$, the exact diagram

$$\begin{array}{|c|}\hline \widehat{\mathcal{X}}\circ b_{g_{\mathrm{BKM}}}=b_{g^{\mathsf{\Psi }}}\circ T\mathcal{X}\\ \hline\end{array}$$

(82)

identifies their common Born symplectic carrier with the corresponding cotangent carrier in global Gibbs coordinates.

Proof. 

The definition of ${\nabla }^{\left(\mathrm{e}\right)}$, the global inverse in Corollary 2, and Equation (74) prove the affine-isometry statement. Metric duality then gives the asserted image of ${\nabla }^{\left(\mathrm{m}\right)}$, and isometries intertwine Levi-Civita connections. Apply 1 to the exponential connection and its metric dual; its musical identity gives Equation (82). □

Theorem 2

(Quantum log-partition/relative-entropy Legendre duality). Fix $\sigma \in {\mathcal{S}}_n^{+}$ and define

$${\mathcal{Z}}_{\sigma }\left(H\right):=Trexp(log\sigma +H),{\mathsf{\Lambda }}_{\sigma }\left(H\right):=log{\mathcal{Z}}_{\sigma }\left(H\right),H\in Herm\left(n\right).$$

(83)

On ${\mathcal{S}}_n$ we use the continuous extension in the first argument, with $0log0:=0$; because σ is faithful, $D_{\mathrm{U}}(\rho \parallel \sigma )$ is finite and continuous on ${\mathcal{S}}_n$. Then ${\mathsf{\Lambda }}_{\sigma }$ is smooth and convex on $Herm\left(n\right)$, obeys ${\mathsf{\Lambda }}_{\sigma }(H+cI)={\mathsf{\Lambda }}_{\sigma }\left(H\right)+c$, and has precisely the scalar direction as its Hessian kernel. Its restriction to $\mathfrak{p}$ is strictly convex, and

$${\rho }_{\sigma ,H}:=\frac{exp(log\sigma +H)}{Trexp(log\sigma +H)}$$

(84)

satisfies

$$\begin{array}{cc}\hfill D{\mathsf{\Lambda }}_{\sigma }{|}_H\left[A\right]& =Tr\left({\rho }_{\sigma ,H}A\right),\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill D^2{\mathsf{\Lambda }}_{\sigma }{|}_H[A,B]& ={Cov}_{{\rho }_{\sigma ,H}}^{\mathrm{KM}}(A,B),\hfill \end{array}$$

(86)

The restricted map $H\mapsto {\rho }_{\sigma ,H}$ is a diffeomorphism from $\mathfrak{p}$ onto ${\mathcal{S}}_n^{+}$, and the exact Legendre-Fenchel identities are

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_{\sigma }\left(H\right)& ={\displaystyle \underset{\rho \in {\mathcal{S}}_n}{max}}\left\{Tr\left(\rho H\right)-D_{\mathrm{U}}(\rho \parallel \sigma )\right\},\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill D_{\mathrm{U}}(\rho \parallel \sigma )& ={\displaystyle \underset{H\in \mathfrak{p}}{max}}\left\{Tr\left(\rho H\right)-{\mathsf{\Lambda }}_{\sigma }\left(H\right)\right\}.\hfill \end{array}$$

(88)

The second identity and its finite maximizer apply to $\rho \in {\mathcal{S}}_n^{+}$; the closed extension to singular states is the conjugate formula below. More generally, identify the trace-one affine hyperplane with $\mathfrak{p}$ by $Y=\rho -I/n$. The closed extended-real conjugate is

$$\begin{array}{|c|}\hline {\left({\mathsf{\Lambda }}_{\sigma }{|}_{\mathfrak{p}}\right)}^{\ast }\left(Y\right)=\left\{\begin{array}{cc}{D}_{\mathrm{U}}(Y+I/n\parallel \sigma ),\hfill & Y+I/n⪰0,\hfill \\ +\infty ,\hfill & Y+I/n\neg ⪰0.\hfill \end{array}\right.\\ \hline\end{array}$$

(89)

The supremum is attained by a finite source exactly when $Y+I/n$ is faithful; singular states are obtained as limiting maximizers. The unique maximizers are respectively $\rho ={\rho }_{\sigma ,H}$ and

$$H_{\rho \mid \sigma }=log\rho -log\sigma -\frac{1}{n}Tr(log\rho -log\sigma )I=log\mathsf{\Phi }\left(\rho \right)-log\mathsf{\Phi }\left(\sigma \right)=\mathcal{X}\left(\sigma \right)-\mathcal{X}\left(\rho \right).$$

(90)

Thus the dual variable lies in the same traceless Hermitian carrier as the symmetric-space Lie-log coordinate, although at a noncentral reference it is a difference of exponential-affine logs rather than the affine-invariant Riemannian logarithm.

Proof from Gibbs variational and Fenchel duality. 

The Gibbs variational formula gives the optimizer, Fenchel gap, and first conjugacy [8], Corollary 2 and Theorem 9. Equality of normalized Hermitian exponentials determines the source modulo scalars; the traceless gauge therefore gives (90) and the stated source-state diffeomorphism. With $Y=q-I/n$, finite-dimensional Fenchel-Moreau duality gives (89), including the nonattained singular boundary and $+\infty$ outside the state body [49]. For $H_0:=H-Tr\left(H\right)I/n$ and $X_{\sigma }:=\mathcal{X}\left(\sigma \right)$, ${\rho }_{\sigma ,H}={\rho }_{X_{\sigma }-H_0}$. The scalar-shift law and Equation (65) therefore give the positive-source derivative $D{\rho }_{\sigma ,H}\left[A\right]={\mathsf{K}}_{{\rho }_{\sigma ,H}}\left({\tilde{A}}_{{\rho }_{\sigma ,H}}\right)$ and the displayed first derivative and Hessian. Positivity and the scalar kernel are Proposition 4. □

Theorem 3

(Global mirror action, partition cocycle, and accumulated response). For $\sigma \in {\mathcal{S}}_n^{+}$ and $H\in \mathfrak{p}$, write $E_{\sigma }\left(H\right):={\rho }_{\sigma ,H}$ and $X_{\sigma }:=\mathcal{X}\left(\sigma \right)$. Then every reference partition is a translate of the single global Massieu potential Ψ:

$$\begin{array}{|c|}\hline {\mathsf{\Lambda }}_{\sigma }\left(H\right)=\mathsf{\Psi }(X_{\sigma }-H)-\mathsf{\Psi }\left(X_{\sigma }\right), \mathcal{X}\left(E_{\sigma }\left(H\right)\right)=X_{\sigma }-H.\\ \hline\end{array}$$

(91)

Consequently, the additive group $(\mathfrak{p},+)$ acts freely and transitively on ${\mathcal{S}}_n^{+}$ by E, and for $H,K\in \mathfrak{p}$,

$$\begin{array}{cc}\hfill E_{E_{\sigma }\left(H\right)}\left(K\right)& =E_{\sigma }(H+K),\hfill \\ \hfill {\mathsf{\Lambda }}_{\sigma }(H+K)& ={\mathsf{\Lambda }}_{\sigma }\left(H\right)+{\mathsf{\Lambda }}_{E_{\sigma }\left(H\right)}\left(K\right).\hfill \end{array}$$

(92)

The second identity is the additive partition cocycle. Equivalently, the relative scores satisfy $H_{\rho \mid \sigma }=H_{\rho \mid \tau }+H_{\tau \mid \sigma }$.

For every $\rho \in {\mathcal{S}}_n^{+}$, the exact Fenchel gap and Bregman reciprocity identities are

$$\begin{array}{cc}\hfill D_{\mathrm{U}}(\rho \parallel \sigma )+{\mathsf{\Lambda }}_{\sigma }\left(H\right)-Tr\left(\rho H\right)& =D_{\mathrm{U}}(\rho \parallel E_{\sigma }\left(H\right)),\hfill \end{array}$$

(93)

$$\begin{array}{cc}\hfill B_{{\mathsf{\Lambda }}_{\sigma }}(H,K)& =D_{\mathrm{U}}(E_{\sigma }\left(K\right)\parallel E_{\sigma }\left(H\right)).\hfill \end{array}$$

(94)

Finally, fix nonzero $H\in \mathfrak{p}$, put $\lambda \left(t\right):={\mathsf{\Lambda }}_{\sigma }\left(tH\right)$ and ${\rho }_t:=E_{\sigma }\left(tH\right)$. Then

$${\lambda }^{\prime \prime }\left(t\right)={Cov}_{{\rho }_t}^{\mathrm{KM}}(H,H)>0,$$

(95)

and the two orientations of relative entropy are complementary moments of this same noncommutative susceptibility:

$$\begin{array}{cc}\hfill D_{\mathrm{U}}({\rho }_1\parallel \sigma )& =B_{{\mathsf{\Lambda }}_{\sigma }}(0,H)={\int }_0^{1}t{Cov}_{{\rho }_t}^{\mathrm{KM}}(H,H)\mathrm{d}t,\hfill \\ \hfill D_{\mathrm{U}}(\sigma \parallel {\rho }_1)& =B_{{\mathsf{\Lambda }}_{\sigma }}(H,0)={\int }_0^1(1-t){Cov}_{{\rho }_t}^{\mathrm{KM}}(H,H)\mathrm{d}t,\hfill \\ \hfill D_{\mathrm{U}}({\rho }_1\parallel \sigma )+D_{\mathrm{U}}(\sigma \parallel {\rho }_1)& =Tr\left[({\rho }_1-\sigma )H\right]={\int }_0^1{Cov}_{{\rho }_t}^{\mathrm{KM}}(H,H)\mathrm{d}t.\hfill \end{array}$$

(96)

Proof by reduction to global Gibbs geometry. 

Since $log\sigma =-X_{\sigma }-\mathsf{\Psi }\left(X_{\sigma }\right)I$, normalization gives (91). Iterating these translations gives the action and cocycle, and relative logarithmic scores add. The identity $log{E}_{\sigma }\left(H\right)=log\sigma +H-{\mathsf{\Lambda }}_{\sigma }\left(H\right)I$ gives (93) by substitution; the translation and Equation (63) give (94). Finally, Equation (86) gives the ray susceptibility. Integrating ${\lambda }^{\prime \prime }$ by parts with weights t and $1-t$, using $\lambda \left(0\right)=0$, gives the two directed entropies in Equation (96); their sum gives the last identity. □

Corollary 4

(Global Hilbert-Schmidt stability of mirror response). Fix $\sigma \in {\mathcal{S}}_n^{+}$. For all $H,K\in \mathfrak{p}$, the mirror response obeys

$$\begin{array}{cc}\hfill {∥E_{\sigma }\left(H\right)-E_{\sigma }\left(K\right)∥}_2& \le \frac{1}{2}{∥H-K∥}_2,\hfill \end{array}$$

(97)

$$\begin{array}{cc}\hfill Tr\left[(E_{\sigma }\left(H\right)-E_{\sigma }\left(K\right))(H-K)\right]& \ge 2{∥E_{\sigma }\left(H\right)-E_{\sigma }\left(K\right)∥}_2^2,\hfill \end{array}$$

(98)

and

$$max\left\{D_{\mathrm{U}}(E_{\sigma }\left(H\right)\parallel E_{\sigma }\left(K\right)),D_{\mathrm{U}}(E_{\sigma }\left(K\right)\parallel E_{\sigma }\left(H\right))\right\}\le \frac{1}{4}{∥H-K∥}_2^2.$$

(99)

Writing $osc\left(A\right):={\lambda }_{max}\left(A\right)-{\lambda }_{min}\left(A\right)$, the sharper gauge-invariant entropy bounds are

$$\begin{array}{cc}\hfill max\left\{D_{\mathrm{U}}(E_{\sigma }\left(H\right)\parallel E_{\sigma }\left(K\right)),D_{\mathrm{U}}(E_{\sigma }\left(K\right)\parallel E_{\sigma }\left(H\right))\right\}& \le \frac{1}{8}osc{(H-K)}^2,\hfill \end{array}$$

(100)

$$\begin{array}{cc}\hfill D_{\mathrm{U}}(E_{\sigma }\left(H\right)\parallel E_{\sigma }\left(K\right))+D_{\mathrm{U}}(E_{\sigma }\left(K\right)\parallel E_{\sigma }\left(H\right))& \le \frac{1}{4}osc{(H-K)}^2.\hfill \end{array}$$

(101)

Here ${∥·∥}_2$ is the Hilbert-Schmidt norm. The constants are dimension-independently sharp, already infinitesimally on the central qubit family. Even for $\sigma =I/2$, the inverse score map has no global Lipschitz constant as the state-space boundary is approached.

Proof. 

For $A\in \mathfrak{p}$ and $\rho =E_{\sigma }\left(L\right)$, the logarithmic-mean inequality gives

$${Cov}_{\rho }^{\mathrm{KM}}(A,A)\le Tr\left[\rho {(A-Tr\left(\rho A\right)I)}^2\right]\le \frac{{({\lambda }_{max}\left(A\right)-{\lambda }_{min}\left(A\right))}^2}{4}\le \frac{1}{2}Tr{A}^2.$$

(102)

The first inequality follows by diagonalizing $\rho$ and using that the logarithmic mean is at most the arithmetic mean; the last uses $TrA=0$. Thus, on $\mathfrak{p}$ with its Hilbert-Schmidt metric, $0⪯D^2{\mathsf{\Lambda }}_{\sigma }⪯\frac{1}{2}I$, so $\nabla {\mathsf{\Lambda }}_{\sigma }=E_{\sigma }-I/n$ is $1/2$-Lipschitz. The Baillon-Haddad theorem makes it 2-cocoercive, proving Equations (97) and (98) [50], Corollary 18.17. The response-integral formulas in Equation (96), applied at reference $E_{\sigma }\left(K\right)$ with source $H-K$, and Equation (102) prove Equation (99). Using instead the middle variance bound in Equation (102) and integrating the weights t and $1-t$ proves Equations (100) and (101). Equality constants are approached with $\sigma =I/2$, $K=0$, and $H=\epsilon {\sigma }_z$ as $\epsilon \to 0$. Finally, $E_{I/2}\left(t{\sigma }_z\right)$ and $E_{I/2}\left((t+1){\sigma }_z\right)$ coalesce as $t\to \infty$ although their sources remain a fixed distance apart, which excludes a global Lipschitz inverse. □

Remark 2

(Literal Laplace transform and noncommutative qualification). If $[H,\sigma ]=0$, a common eigenbasis gives

$$\sigma =diag(q_1,\dots ,q_n),H=diag(h_1,\dots ,h_n),{\mathsf{\Lambda }}_{\sigma }\left(tH\right)=log{\displaystyle \sum _{j=1}^n}{q}_j{\mathrm{e}}^{t{h}_j},t\in \mathbb{R}.$$

(103)

This is literally the log-Laplace transform of ${\sum }_j{q}_j{\delta }_{h_j}$; the original value is obtained at $t=1$. For $[H,\sigma ]\ne 0$, ${\mathsf{\Lambda }}_{\sigma }$ is instead the noncommutative log-partition functional. The entropy tilt $E_{\sigma }\left(H\right)$ must be distinguished from the normalized congruence orbit

$$C_{\sigma }\left(H\right):=\frac{{\mathrm{e}}^{H/2}\sigma {\mathrm{e}}^{H/2}}{Tr\left(\sigma {\mathrm{e}}^H\right)}.$$

(104)

They agree when $[H,\sigma ]=0$, in particular for $\sigma =I/n$, but not generally. Golden-Thompson gives ${\mathsf{\Lambda }}_{\sigma }\left(H\right)\le logTr\left(\sigma {\mathrm{e}}^H\right)$ [51,52], while the symmetric Trotter formula provides the exact interleaving bridge

$${\mathrm{e}}^{log\sigma +H}=\underset{N\to \infty }{lim}{\left({\sigma }^{1/\left(2N\right)}{\mathrm{e}}^{H/N}{\sigma }^{1/\left(2N\right)}\right)}^N.$$

(105)

The phrase Laplace transform is therefore literal in commuting reductions; the general exact statement is quantum log-partition plus Legendre-Fenchel duality.

3.2. Sakamoto Born Lift and Local BKM Tangent Kinematics

Corollary 5

(Sakamoto tangent-bundle Born lift of a regular quantum exponential family). Let Θ be the parameter manifold of the regular exponential family in Equation (66), and let ${\nabla }^{\left(\mathrm{e}\right)}$ be the flat connection for which the natural coordinates ${\theta }^a$ are affine. Then

$$g_{\mathrm{BKM}}={Hess}_{{\nabla }^{\left(\mathrm{e}\right)}}\psi ,$$

(106)

so $(\mathsf{\Theta },{\nabla }^{\left(\mathrm{e}\right)},g_{\mathrm{BKM}})$ is Hessian and $T\mathsf{\Theta }$ carries Sakamoto’s strongly integrable Born structure. If ${\nabla }^{\left(\mathrm{m}\right)}$ is the $g_{\mathrm{BKM}}$-dual connection and $K=({\nabla }^{\left(\mathrm{e}\right)}-{\nabla }^{\left(\mathrm{m}\right)})/2$, then

$$g_{\mathrm{BKM}}\left(2K_{{\partial }_a}u,{\partial }_c\right)=-{\psi }_{,abc}{u}^b.$$

(107)

Thus the quantum cubic is precisely the vertical shear between the exponential and mixture Born horizontal polarizations.

Proof. 

In the coordinates ${\theta }^a$, ${\nabla }^{\left(\mathrm{e}\right)}$ is flat and torsion-free, while $g_{ab}^{\mathrm{BKM}}={\partial }_a{\partial }_b\psi$; hence ${\nabla }^{\left(\mathrm{e}\right)}{g}_{\mathrm{BKM}}$ is fully symmetric. The Born assertion follows from Equation (17), and the shear formula is Equation (21). □

Proposition 6

(Exact weighted metric energy along the exponential segment). Let ${\theta }_s=\theta +s\delta$, $0\le s\le 1$, and

$$\delta {\tilde{X}}_s={\delta }^a\left(T_a-{\langle T_a\rangle }_{{\theta }_s}I\right),$$

(108)

Then

$$D_{\mathrm{U}}({\rho }_{\theta }\parallel {\rho }_{\theta +\delta })={\int }_0^1(1-s){∥\delta {\tilde{X}}_s∥}_{{\rho }_{{\theta }_s},\mathrm{BKM}}^2\mathrm{d}s.$$

(109)

Thus relative entropy is exactly a weighted integrated squared speed of the logarithmic displacement along the exponential-affine path, not merely a quadratic approximation at its initial point.

Proof. 

Apply the second orientation in Equation (96) with reference ${\rho }_{\theta }$ and source $H=-{\delta }^a{T}_a$. Then $E_{{\rho }_{\theta }}\left(sH\right)={\rho }_{{\theta }_s}$ and its susceptibility is the squared norm in Equation (72). The zero displacement is immediate. □

Corollary 6

(Local Lie-log expansion).

$$D_{\mathrm{U}}({\rho }_{\theta }\parallel {\rho }_{\theta +\delta })=\frac{1}{2}{∥\delta \tilde{X}∥}_{{\rho }_{\theta },\mathrm{BKM}}^2+\frac{1}{6}{\mathcal{C}}_{abc}\left(\theta \right){\delta }^a{\delta }^b{\delta }^c+O\left({∥\delta ∥}^4\right).$$

(110)

This is the standard third-order contrast expansion of a Bregman divergence [48]; applying it to (67) fixes the displayed cubic coefficient. Thus, to quadratic order, relative entropy is one half of the squared infinitesimal BKM norm; it is not the squared affine-invariant positive-cone distance [1,46].

Variational limitation.

The exact quantum formula in Equation (109) is the reversed-orientation specialization of the standard Hessian identity $B_F(x,y)={\int }_0^{1}t{g}_{x+t(y-x)}(y-x,y-x)\mathrm{d}t$; its short derivation is given in Appendix G.6. Evaluation on a prescribed affine geodesic is not an on-shell variational theorem and does not make the divergence a continuous Hamilton principal function. The flat one-dimensional obstruction and the positive autonomous Hamilton-Jacobi obstruction are recorded in Appendix G.7; the regular two-endpoint completion is developed in Section 8.2. The complete integral in Theorem 22 instead uses an explicitly scale-dependent Rényi-Moreau Hamiltonian and therefore does not evade that obstruction or supply physical time.

3.3. Noncommutative Moment Effective Potentials and the Mirror Bridge

The full-state conjugacy in Theorem 2 has a useful finite-dimensional reduction that does not require commuting observables. It is the precise static effective-potential content of the partition construction [7,8,9,10].

Theorem 4

(Noncommutative moment effective potential). Fix $\sigma \in {\mathcal{S}}_n^{+}$, and let $O_1,\dots ,O_d\in Herm\left(n\right)$ be minimal modulo the identity:

$$v^a{O}_a\in \mathbb{R}I⟹v=0.$$

(111)

No commutativity is assumed. Put

$$\begin{array}{c}O\left(\theta \right):={\theta }^a{O}_a,M{\left(\rho \right)}_a:=Tr\left(\rho O_a\right),{\mathcal{C}}_O:=M\left({\mathcal{S}}_n\right),\end{array}$$

(112)

$$\begin{array}{c}{\lambda }_{\sigma ,O}\left(\theta \right):={\mathsf{\Lambda }}_{\sigma }\left(O\left(\theta \right)\right),{\rho }_{\theta }:=E_{\sigma }\left(O\left(\theta \right)\right),\theta \in {\mathbb{R}}^d.\end{array}$$

(113)

Then ${\mathcal{C}}_O$ is a compact full-dimensional convex body, ${\lambda }_{\sigma ,O}$ is real analytic and strictly convex, and

$$\begin{array}{cc}\hfill {\partial }_a{\lambda }_{\sigma ,O}\left(\theta \right)& =m_a\left(\theta \right):=M{\left({\rho }_{\theta }\right)}_a,\hfill \\ \hfill {\chi }_{ab}\left(\theta \right):={\partial }_a{\partial }_b{\lambda }_{\sigma ,O}\left(\theta \right)& ={Cov}_{{\rho }_{\theta }}^{\mathrm{KM}}(O_a,O_b)\succ 0.\hfill \end{array}$$

(114)

The moment map is a real-analytic diffeomorphism

$$\nabla {\lambda }_{\sigma ,O}:{\mathbb{R}}^d\stackrel{\simeq }{\to }int{\mathcal{C}}_O.$$

(115)

Its Legendre-Fenchel conjugate is exactly constrained Umegaki information:

$$\begin{array}{|c|}\hline {\mathsf{\Gamma }}_{\sigma ,O}\left(m\right):={\lambda }_{\sigma ,O}^{\ast }\left(m\right)=\left\{\begin{array}{cc}{\displaystyle \underset{\omega \in {\mathcal{S}}_n,M\left(\omega \right)=m}{min}}{D}_{\mathrm{U}}(\omega \parallel \sigma ),\hfill & m\in {\mathcal{C}}_O,\hfill \\ +\infty ,\hfill & m\notin {\mathcal{C}}_O.\hfill \end{array}\right.\\ \hline\end{array}$$

(116)

The minimizer ${\rho }^m$ is unique. If $m\in int{\mathcal{C}}_O$, then

$${\rho }^m={\rho }_{\theta \left(m\right)},\theta \left(m\right)=\nabla {\mathsf{\Gamma }}_{\sigma ,O}\left(m\right),{\nabla }^2{\mathsf{\Gamma }}_{\sigma ,O}\left(m\right)=\chi {\left(\theta \left(m\right)\right)}^{-1}.$$

(117)

Every ω with $M\left(\omega \right)=m\in int{\mathcal{C}}_O$ obeys the exact information projection

$$\begin{array}{|c|}\hline D_{\mathrm{U}}(\omega \parallel \sigma )=D_{\mathrm{U}}(\omega \parallel {\rho }^m)+{\mathsf{\Gamma }}_{\sigma ,O}\left(m\right).\\ \hline\end{array}$$

(118)

Thus the first term is information unresolved by the retained moments, whereas ${\mathsf{\Gamma }}_{\sigma ,O}$ is their irreducible macroscopic cost.

There is also a reference-independent global stiffness bound. Define

$${\overline{O}}_a:=O_a-\frac{Tr{O}_a}{n}I,G_{ab}:=Tr\left({\overline{O}}_a{\overline{O}}_b\right).$$

(119)

Then $G\succ 0$,

$$0\prec \chi \left(\theta \right)⪯\frac{1}{2}G,{\nabla }^2{\mathsf{\Gamma }}_{\sigma ,O}\left(m\right)⪰2G^{-1},$$

(120)

and, for $m\in {\mathcal{C}}_O$ and $q\in int{\mathcal{C}}_O$,

$$\begin{array}{|c|}\hline D_{\mathrm{U}}({\rho }^m\parallel {\rho }^q)={\mathsf{\Gamma }}_{\sigma ,O}\left(m\right)-{\mathsf{\Gamma }}_{\sigma ,O}\left(q\right)-\theta \left(q\right)·(m-q)\ge {(m-q)}^{\mathsf{T}}{G}^{-1}(m-q).\\ \hline\end{array}$$

(121)

In particular, with $m_0=M\left(\sigma \right)$,

$${\mathsf{\Gamma }}_{\sigma ,O}\left(m\right)\ge {(m-m_0)}^{\mathsf{T}}{G}^{-1}(m-m_0).$$

(122)

The constant is approached infinitesimally by a central qubit family.

Proof from quantum maximum-entropy duality. 

Compactness and minimality give a compact full-dimensional moment body. The Gibbs variational principle and finite-dimensional convex duality identify the constrained minimum with ${\lambda }_{\sigma ,O}^{\ast }$ [8,49]; standard quantum maximum-entropy theory identifies every interior optimizer with a unique faithful exponential-family state and makes the moment map a real-analytic diffeomorphism [7,9,10]. Legendre equality then gives Equations (117) and (118).

The manuscript-specific uniform estimate follows by pulling (102) back through $v\mapsto v^a{\overline{O}}_a$, giving $\chi ⪯G/2$. Legendre inversion and strong convexity prove Equations (120)–(122); the central-qubit family in Corollary 4 proves sharpness. □

Observable coarse-graining, exposed-face source limits, and an explicit noncommuting Pauli example are given in Appendix B. In particular, finite sources cover exactly $int{\mathcal{C}}_O$; boundary inference may require support reduction and need not be continuous.

The third derivative now has a direct response meaning. With

$$C_{abc}\left(\theta \right):={\partial }_c{\chi }_{ab}\left(\theta \right)={\partial }_a{\partial }_b{\partial }_c{\lambda }_{\sigma ,O}\left(\theta \right),{\chi }^{ab}:={\left({\chi }^{-1}\right)}^{ab},$$

(123)

Legendre differentiation gives

$$\begin{array}{|c|}\hline {\partial }_{m_a}{\partial }_{m_b}{\partial }_{m_c}{\mathsf{\Gamma }}_{\sigma ,O}=-{\chi }^{ai}{\chi }^{bj}{\chi }^{ck}{C}_{ijk}.\\ \hline\end{array}$$

(124)

Thus C is at once nonlinear static susceptibility, the Hessian cubic, and-through Remark 1-the tensor driving the dual-horizontal Born shear. This is an equality of roles on the same statistical parameter manifold, not an identification with a real-time response kernel or with the complex amplitude carrier.

Corollary 7

(Thermal effective potential and exact static response). Let $\beta >0$, ${\sigma }_{\beta }={\mathrm{e}}^{-\beta H_0}/Z_0$, and couple real fields by

$$H\left(f\right):=H_0-f^a{O}_a,{\rho }_f:=\frac{{\mathrm{e}}^{-\beta H\left(f\right)}}{Z\left(f\right)}.$$

(125)

Then ${\rho }_f={\rho }_{\beta f}$ and

$${\lambda }_{{\sigma }_{\beta },O}\left(\beta f\right)=logZ\left(f\right)-log{Z}_0=\beta [F_0-F_{\mathrm{eq}}\left(f\right)],$$

(126)

where $F_0=-{\beta }^{-1}log{Z}_0$ and $F_{\mathrm{eq}}\left(f\right):=-{\beta }^{-1}logZ\left(f\right)$. The constrained Helmholtz potential

$${\mathcal{F}}_{\beta ,c}\left(m\right):=\underset{M\left(\omega \right)=m}{min}\{Tr\left(\omega H_0\right)-{\beta }^{-1}{S}_{\mathrm{vN}}\left(\omega \right)\},$$

(127)

satisfies, for every $m\in {\mathcal{C}}_O$,

$$\begin{array}{|c|}\hline {\mathcal{F}}_{\beta ,c}\left(m\right)=F_0+{\beta }^{-1}{\mathsf{\Gamma }}_{{\sigma }_{\beta },O}\left(m\right)={\displaystyle \underset{f\in {\mathbb{R}}^d}{sup}}\{F_{\mathrm{eq}}\left(f\right)+f·m\}.\\ \hline\end{array}$$

(128)

For $m\in int{\mathcal{C}}_O$, the supremum is attained at the unique $f=f\left(m\right)$ and

$$\begin{array}{|c|}\hline {\nabla }_m{\mathcal{F}}_{\beta ,c}\left(m\right)=f\left(m\right),{\nabla }_m^2{\mathcal{F}}_{\beta ,c}\left(m\right)={\left.{\left(\frac{\partial m}{\partial f}\right)}^{-1}\right|}_{f=f\left(m\right)}.\\ \hline\end{array}$$

(129)

At a boundary moment the value formula remains valid, but a finite maximizing field need not exist. More explicitly,

$$\begin{array}{cc}\hfill \frac{\partial m_a}{\partial f^b}& =\beta {\chi }_{ab}\left(\beta f\right)={\int }_0^{\beta }{〈{\tilde{O}}_a(-\mathrm{i}\tau ){\tilde{O}}_b〉}_{{\rho }_f}\mathrm{d}\tau ,\hfill \end{array}$$

(130)

$$\begin{array}{cc}\hfill m_a\left(f_1\right)-m_a\left(f_0\right)& =\beta {\int }_0^1{\chi }_{ab}\left(\beta (f_0+s\delta f)\right)\delta f^b\mathrm{d}s,\delta f:=f_1-f_0.\hfill \end{array}$$

(131)

Here $O_a(-\mathrm{i}\tau ):={\mathrm{e}}^{\tau H\left(f\right)}{O}_a{\mathrm{e}}^{-\tau H\left(f\right)}$ and ${\tilde{O}}_a:=O_a-m_a\left(f\right)I$. For every state with interior moment m,

$$\begin{array}{|c|}\hline F_{\beta }^{\left(0\right)}\left(\omega \right)-F_0={\mathcal{F}}_{\beta ,c}\left(m\right)-F_0+{\beta }^{-1}{D}_{\mathrm{U}}(\omega \parallel {\rho }^m),\\ \hline\end{array}$$

(132)

where $F_{\beta }^{\left(0\right)}\left(\omega \right)=Tr\left(\omega H_0\right)-{\beta }^{-1}{S}_{\mathrm{vN}}\left(\omega \right)$ and $m_0:=M\left({\sigma }_{\beta }\right)$. Moreover,

$${\mathcal{F}}_{\beta ,c}\left(m\right)-F_0\ge {\beta }^{-1}{(m-m_0)}^{\mathsf{T}}{G}^{-1}(m-m_0).$$

(133)

Equations (130) and (131) are static isothermal (imaginary-time/Matsubara) Kubo-Mori response identities [53]. Identifying them with an $\omega \to 0$ retarded susceptibility requires dynamical and order-of-limits assumptions; causal real-time response additionally requires a propagator and time ordering.

Proof. 

Since $log{\sigma }_{\beta }+\beta f^a{O}_a=-\beta H\left(f\right)-log{Z}_{0}I$, normalization gives the partition and state identities. Equation (199) and Equations (116) and (117) give Equation (128). Applying Equation (86) to the source $\beta f^a{O}_a$ gives $\partial m_a/\partial f^b=\beta {\chi }_{ab}$. Insert ${\rho }_f={\mathrm{e}}^{-\beta H\left(f\right)}/Z\left(f\right)$ in the covariance and set $\tau =\beta (1-s)$ to obtain the imaginary-time integral; integration along the field segment gives the finite response. Finally, Equations (118) and (122) give Equations (132) and (133). □

Theorem 5

(Mirror bridge, reverse-information barycenter, and BKM energy). For $\rho ,\sigma \in {\mathcal{S}}_n^{+}$ define

$$L_{\rho \mid \sigma }:=log\rho -log\sigma ,{\phi }_{\rho \mid \sigma }\left(t\right):=logTrexp[(1-t)log\sigma +tlog\rho ],{\gamma }_t:=\frac{exp\left[\right(1-t)log\sigma +tlog\rho ]}{exp\left[{\phi }_{\rho \mid \sigma }\left(t\right)\right]},$$

(134)

for $0\le t\le 1$, and put $C_t^{\mathrm{b}}(\rho ,\sigma ):=-{\phi }_{\rho \mid \sigma }\left(t\right)$. Then for every $\omega \in {\mathcal{S}}_n$,

$$\begin{array}{|c|}\hline (1-t)D_{\mathrm{U}}(\omega \parallel \sigma )+t{D}_{\mathrm{U}}(\omega \parallel \rho )=D_{\mathrm{U}}(\omega \parallel {\gamma }_t)+C_t^{\mathrm{b}}(\rho ,\sigma ).\\ \hline\end{array}$$

(135)

Consequently, ${\gamma }_t$ is the unique reverse-Umegaki barycenter and $C_t^{\mathrm{b}}$ is its minimum cost. For $0<t<1$,

$$D_t^{\mathrm{b}}(\rho \parallel \sigma ):=\frac{1}{t-1}{\phi }_{\rho \mid \sigma }\left(t\right)=\frac{C_t^{\mathrm{b}}(\rho ,\sigma )}{1-t},$$

(136)

is the log-Euclidean Rényi divergence. Its Umegaki variational formula is [21], Theorem III.6, and it is the Umegaki-generated barycentric Rényi divergence of [22], Equation (I.6) and Proposition V.29. We retain the superscript $\mathrm{b}$ for this barycentric construction; it does not denote symmetrization. In the global log chart,

$$\begin{array}{cc}\hfill C_t^{\mathrm{b}}(\rho ,\sigma )& =(1-t)\mathsf{\Psi }\left(\mathcal{X}\right(\sigma \left)\right)+t\mathsf{\Psi }\left(\mathcal{X}\right(\rho \left)\right)-\mathsf{\Psi }\left(\right(1-t\left)\mathcal{X}\right(\sigma )+t\mathcal{X}(\rho \left)\right),\hfill \end{array}$$

(137)

$$\begin{array}{cc}\hfill \mathsf{\Phi }\left({\gamma }_t\right)& =exp\left[(1-t)log\mathsf{\Phi }\left(\sigma \right)+tlog\mathsf{\Phi }\left(\rho \right)\right].\hfill \end{array}$$

(138)

Thus the same curve is the normalized complex-Hermitian analogue of the log-Euclidean geodesic introduced for real positive-definite matrices [54]; it is generally not the affine-invariant geodesic.

The bridge obeys

$$\begin{array}{cccc}\hfill {\phi }^{\prime }\left(t\right)& =Tr\left({\gamma }_t{L}_{\rho \mid \sigma }\right),\hfill & \hfill {\phi }^{″}\left(t\right)& ={Cov}_{{\gamma }_t}^{\mathrm{KM}}(L_{\rho \mid \sigma },L_{\rho \mid \sigma })=g_{{\gamma }_t}^{\mathrm{BKM}}({\dot{\gamma }}_t,{\dot{\gamma }}_t).\hfill \end{array}$$

(139)

Hence its exact unweighted BKM energy is

$$\begin{array}{|c|}\hline {\int }_0^1{g}_{{\gamma }_t}^{\mathrm{BKM}}({\dot{\gamma }}_t,{\dot{\gamma }}_t)\mathrm{d}t=D_{\mathrm{U}}(\rho \parallel \sigma )+D_{\mathrm{U}}(\sigma \parallel \rho ),\\ \hline\end{array}$$

(140)

Define the BKM length of a piecewise smooth path ζ and the induced Riemannian distance by

$$L_{\mathrm{BKM}}\left(\zeta \right):={\int }_0^1\sqrt{g_{{\zeta }_t}^{\mathrm{BKM}}({\dot{\zeta }}_t,{\dot{\zeta }}_t)}\mathrm{d}t,d_{\mathrm{BKM}}(\rho ,\sigma ):=\underset{\zeta :\rho \to \sigma }{inf}{L}_{\mathrm{BKM}}\left(\zeta \right).$$

(141)

Then

$$d_{\mathrm{BKM}}{(\rho ,\sigma )}^2\le L_{\mathrm{BKM}}{\left(\gamma \right)}^2\le D_{\mathrm{U}}(\rho \parallel \sigma )+D_{\mathrm{U}}(\sigma \parallel \rho ).$$

(142)

This is a path-energy statement, not a claim that γ is a BKM geodesic or a universal dissipation-minimizing protocol.

For distinct endpoints, φ has a unique minimizer $t_{\ast }\in (0,1)$, and

$$\begin{array}{|c|}\hline C_{\mathrm{b}}(\rho ,\sigma ):=-{\displaystyle \underset{0\le t\le 1}{min}}\phi \left(t\right)=D_{\mathrm{U}}({\gamma }_{t_{\ast }}\parallel \rho )=D_{\mathrm{U}}({\gamma }_{t_{\ast }}\parallel \sigma )={\displaystyle \underset{\omega \in {\mathcal{S}}_n}{min}}max\{D_{\mathrm{U}}(\omega \parallel \rho ),D_{\mathrm{U}}(\omega \parallel \sigma )\}.\\ \hline\end{array}$$

(143)

Finally, every Hermitian observable A satisfies the finite-response bound

$$\begin{array}{|c|}\hline \left|Tr\left[\right(\rho -\sigma \left)A\right]\right|\le \frac{osc\left(A\right)}{2}\sqrt{D_{\mathrm{U}}(\rho \parallel \sigma )+D_{\mathrm{U}}(\sigma \parallel \rho )}.\\ \hline\end{array}$$

(144)

Proof from barycentric and response theory. 

Put $H=H_{\rho \mid \sigma }$ and $c=Tr\left(L_{\rho \mid \sigma }\right)/n$. Then ${\gamma }_t=E_{\sigma }\left(tH\right)$ and $\phi \left(t\right)={\mathsf{\Lambda }}_{\sigma }\left(tH\right)+tc$. The Fenchel gap (93), extended to singular competitors by continuity, gives Equation (135), its unique minimizer ${\gamma }_t$, and minimum $C_t^{\mathrm{b}}$. This recovers the cited barycentric formula in the variable-first orientation. Rewriting the endpoint logarithms through $\mathcal{X}$ gives the Jensen-gap and determinant-normalized shape identities. The covariance dictionary and accumulated response in Theorem 3 give Equations (139) and (140); the zero-score case is immediate. Riemannian distance is bounded by path length, and Cauchy-Schwarz on the unit interval gives the stated length bound.

For distinct endpoints, strict positivity of the BKM variance and the endpoint signs ${\phi }^{\prime }\left(0\right)=-D_{\mathrm{U}}(\sigma \parallel \rho )<0$ and ${\phi }^{\prime }\left(1\right)=D_{\mathrm{U}}(\rho \parallel \sigma )>0$ give the unique $t_{\ast }$. At ${\phi }^{\prime }\left(t_{\ast }\right)=0$, substitution gives the two equal divergences in Equation (143). Every competitor’s maximum endpoint divergence dominates its $t_{\ast }$-weighted mean; the cited residual identity bounds that mean below by $C_{\mathrm{b}}$, and ${\gamma }_{t_{\ast }}$ attains equality. Finally,

$$Tr\left[(\rho -\sigma )A\right]={\int }_0^1{Cov}_{{\gamma }_t}^{\mathrm{KM}}(A,L_{\rho \mid \sigma })\mathrm{d}t.$$

(145)

Cauchy-Schwarz in the varying BKM metrics, the Jeffreys-energy identity, and ${Cov}_{\tau }^{\mathrm{KM}}(A,A)\le osc{\left(A\right)}^2/4$ prove Equation (144). □

The quantity $C_{\mathrm{b}}$ becomes the operational Chernoff exponent on commuting families. In general Golden-Thompson gives $C_{\mathrm{b}}\ge C_{\mathrm{Q}}$, where $C_{\mathrm{Q}}$ is the quantum Chernoff exponent [55]; neither ${\phi }_{\rho \mid \sigma }$ nor its bridge is, without a specified protocol, a two-point-measurement (TPM) work cumulant generator.

3.4. Two Exact Proximal Regularizations

The mirror bridge supports a variable-first entropic Bregman penalty on state space [17,18,19], whereas the moment effective potential uses a quadratic Moreau-Yosida penalty in the $G^{-1}$ metric [16,49,50]. Their spaces and scales are distinct; the partition identities below relate, but do not identify, these regularizations.

3.4.1. Entropic Bregman-Moreau Relaxation on State Space

The weighted Umegaki minimization, normalized-exponential optimizer, log-Euclidean Rényi minimum, and exact objective residual used below are established barycentric and sided-centroid facts [21], Theorem III.6 and Proposition III.8, [22], Equation (I.6) and Proposition V.29, [23]. In the standard terminology of [20], Section 2, the first minimization is the left Bregman-Moreau envelope of $\omega \mapsto D_{\mathrm{U}}(\omega \parallel \sigma )$ at $\rho$, generated by negative entropy; $r=(1-t)/t$ is only its change of parameter. No novelty is claimed for this variational core or for the basic uniqueness, faithful-state analyticity, endpoint, and relative-log composition properties that follow from it. We retain the theorem and a convention-mapping proof sketch to fix matrix orientation, sidedness, support, and normalization. What is derived here from this standard core is its integration with both BKM endpoint gradients, the extended-real scale Hamiltonian and Hamilton-Jacobi equation, the scale-response and two-sided action identities, and the subsequent calibrated BKM relaxation and dissipation package.

Theorem 6

(Umegaki barycentric envelope and derived BKM scale identities). Fix faithful $\rho ,\sigma \in {\mathcal{S}}_n^{+}$ and $r>0$, put $t_r:={(1+r)}^{-1}$, and define

$${\mathcal{M}}_r^{\sigma }\left(\rho \right):=\underset{\omega \in {\mathcal{S}}_n}{min}\left\{D_{\mathrm{U}}(\omega \parallel \sigma )+\frac{1}{r}{D}_{\mathrm{U}}(\omega \parallel \rho )\right\}.$$

(146)

Denote its unique minimizer by ${prox}_{r,\sigma }^{\mathrm{U}}\left(\rho \right)$. The minimization is over the full compact state space, including singular competitors. Since $\mathcal{F}\left(\omega \right):=Tr(\omega log\omega )$ generates $B_{\mathcal{F}}(\omega ,\rho )=D_{\mathrm{U}}(\omega \parallel \rho )$ on the trace-one affine space, the displayed variable-first orientation is the entropic Bregman-Moreau envelope. Equivalently, for every $0<t<1$,

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill (1-t)D_t^{\mathrm{b}}(\rho \parallel \sigma )& ={\displaystyle \underset{\omega \in {\mathcal{S}}_n}{min}}\left\{(1-t)D_{\mathrm{U}}(\omega \parallel \sigma )+t{D}_{\mathrm{U}}(\omega \parallel \rho )\right\},\hfill \\ \hfill D_t^{\mathrm{b}}(\rho \parallel \sigma )& ={\mathcal{M}}_{(1-t)/t}^{\sigma }\left(\rho \right),\mathrm{argmin}={\gamma }_t.\hfill \end{array}\\ \hline\end{array}$$

(147)

Then the minimizer is faithful and

$$\begin{array}{cc}\hfill {prox}_{r,\sigma }^{\mathrm{U}}\left(\rho \right)& ={\gamma }_{t_r}=\frac{exp[(1-t_r)log\sigma +t_{r}log\rho ]}{Trexp[(1-t_r)log\sigma +t_{r}log\rho ]},\hfill \end{array}$$

(148)

$$\begin{array}{cc}\hfill {\mathcal{M}}_r^{\sigma }\left(\rho \right)& =D_{t_r}^{\mathrm{b}}(\rho \parallel \sigma ).\hfill \end{array}$$

(149)

Thus the Moreau scale and Rényi order are related bijectively by $r=(1-t)/t$. Writing $p_r:={prox}_{r,\sigma }^{\mathrm{U}}\left(\rho \right)$, the maps $(\rho ,\sigma ,r)\mapsto p_r$ and $(\rho ,\sigma ,r)\mapsto {\mathcal{M}}_r^{\sigma }\left(\rho \right)$ are real analytic on ${\mathcal{S}}_n^{+}\times {\mathcal{S}}_n^{+}\times (0,\infty )$. With endpoint values understood by continuity,

$$\begin{array}{c}{p}_0=\rho ,{\mathcal{M}}_0^{\sigma }\left(\rho \right)=D_1^{\mathrm{b}}(\rho \parallel \sigma )=D_{\mathrm{U}}(\rho \parallel \sigma ),\\ p_{\infty }=\sigma ,{\mathcal{M}}_{\infty }^{\sigma }\left(\rho \right)=D_0^{\mathrm{b}}(\rho \parallel \sigma )=0,\underset{r\to \infty }{lim}r{\mathcal{M}}_r^{\sigma }\left(\rho \right)=D_{\mathrm{U}}(\sigma \parallel \rho ).\end{array}$$

(150)

More strongly, every $\omega \in {\mathcal{S}}_n$ obeys the exact residual identity

$$\begin{array}{|c|}\hline D_{\mathrm{U}}(\omega \parallel \sigma )+\frac{1}{r}{D}_{\mathrm{U}}(\omega \parallel \rho )={\mathcal{M}}_r^{\sigma }\left(\rho \right)+\left(1+\frac{1}{r}\right)D_{\mathrm{U}}(\omega \parallel p_r).\\ \hline\end{array}$$

(151)

In the traceless relative-log coordinate ${\ell }_{\sigma }\left(\rho \right):=H_{\rho \mid \sigma }$ of Equation (90),

$$\begin{array}{|c|}\hline {\ell }_{\sigma }\left(p_r\right)=\frac{1}{1+r}{\ell }_{\sigma }\left(\rho \right),\frac{{\ell }_{\sigma }\left(\rho \right)-{\ell }_{\sigma }\left(p_r\right)}{r}={\ell }_{\sigma }\left(p_r\right).\\ \hline\end{array}$$

(152)

Thus this is the Bregman resolvent of $\omega \mapsto D_{\mathrm{U}}(\omega \parallel \sigma )$ in negative-entropy dual coordinates. Using the Kubo-Mori operator and density-tangent BKM metric fixed in Equations (57) and (59), every trace-zero Hermitian density tangent V satisfies

$$\begin{array}{cc}\hfill D{\mathcal{M}}_r^{\sigma }\left(\rho \right)\left[V\right]& =\frac{1}{r}{g}_{\rho }^{\mathrm{BKM}}(\rho -p_r,V),\hfill \end{array}$$

(153)

$$\begin{array}{cc}\hfill {grad}_{\mathrm{BKM}}{\mathcal{M}}_r^{\sigma }\left(\rho \right)& =\frac{\rho -p_r}{r},{\partial }_r{\mathcal{M}}_r^{\sigma }\left(\rho \right)=-\frac{1}{r^2}{D}_{\mathrm{U}}(p_r\parallel \rho ).\hfill \end{array}$$

(154)

For every trace-zero tangent W at the reference endpoint, the same envelope calculation gives the complementary differential

$$D_{\sigma }{\mathcal{M}}_r^{\sigma }\left(\rho \right)\left[W\right]=g_{\sigma }^{\mathrm{BKM}}(\sigma -p_r,W),{grad}_{\mathrm{BKM}}^{\left(\sigma \right)}{\mathcal{M}}_r^{\sigma }\left(\rho \right)=\sigma -p_r.$$

(155)

For a cotangent vector α at ρ, let ${\alpha }^{{♯}_{\rho }}$ be its BKM metric dual and define the scale-dependent, extended-real Bregman Hamiltonian

$${\mathcal{H}}_r^{\mathrm{U}}(\rho ,\alpha ):=\left\{\begin{array}{cc}{r}^{-2}{D}_{\mathrm{U}}(\rho -r{\alpha }^{{♯}_{\rho }}\parallel \rho ),\hfill & \rho -r{\alpha }^{{♯}_{\rho }}\in {\mathcal{S}}_n,\hfill \\ +\infty ,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(156)

This is the proper lower-semicontinuous convex extension in the cotangent variable; it is smooth wherever the displaced state is faithful. The boundary extension is needed for Legendre-Fenchel duality, while the solution below remains faithful. Combining the state differential and scale derivative gives the exact Bregman Hamilton-Jacobi value equation

$$\begin{array}{|c|}\hline {\partial }_r{\mathcal{M}}_r^{\sigma }\left(\rho \right)+{\mathcal{H}}_r^{\mathrm{U}}\left(\rho ,D_{\rho }{\mathcal{M}}_r^{\sigma }\left(\rho \right)\right)=0.\\ \hline\end{array}$$

(157)

The displaced state is $\rho -r{grad}_{\mathrm{BKM}}{\mathcal{M}}_r^{\sigma }\left(\rho \right)=p_r\in {\mathcal{S}}_n^{+}$. For fixed faithful ρ, if $V_r:={grad}_{\mathrm{BKM}}{\mathcal{M}}_r^{\sigma }\left(\rho \right)$, then

$$\frac{1}{r^2}{D}_{\mathrm{U}}(\rho -r{V}_r\parallel \rho )=\frac{1}{2}{g}_{\rho }^{\mathrm{BKM}}(V_r,V_r)+O\left(r\right),r\downarrow 0.$$

(158)

Thus the state-space equation has an exact nonquadratic Bregman Hamiltonian in regularization scale, not physical time. Its local quadratic limit is BKM, whereas Equation (189) is exactly quadratic in the moment-space gradient at every smoothing scale. Writing $L_{\rho \mid \sigma }=log\rho -log\sigma$, its endpoint asymptotics are

$$\begin{array}{cccc}\hfill {\mathcal{M}}_r^{\sigma }\left(\rho \right)& =D_{\mathrm{U}}(\rho \parallel \sigma )-\frac{r}{2}{Cov}_{\rho }^{\mathrm{KM}}(L_{\rho \mid \sigma },L_{\rho \mid \sigma })+O\left(r^2\right),\hfill & \hfill & r\downarrow 0,\hfill \end{array}$$

(159)

$$\begin{array}{cccc}\hfill {\mathcal{M}}_r^{\sigma }\left(\rho \right)& =\frac{1}{r}{D}_{\mathrm{U}}(\sigma \parallel \rho )-\frac{1}{2r^2}{Cov}_{\sigma }^{\mathrm{KM}}(L_{\rho \mid \sigma },L_{\rho \mid \sigma })+O\left(r^{-3}\right),\hfill & \hfill & r\to \infty .\hfill \end{array}$$

(160)

Consequently, for $0<r_1<r_2$,

$$\begin{array}{cc}\hfill {\mathcal{M}}_{r_1}^{\sigma }\left(\rho \right)-{\mathcal{M}}_{r_2}^{\sigma }\left(\rho \right)& ={\int }_{r_1}^{r_2}\frac{1}{r^2}{D}_{\mathrm{U}}(p_r\parallel \rho )\mathrm{d}r,\hfill \end{array}$$

(161)

$$\begin{array}{cc}\hfill D_{\mathrm{U}}(\rho \parallel \sigma )& ={\int }_0^{\infty }\frac{1}{r^2}{D}_{\mathrm{U}}(p_r\parallel \rho )\mathrm{d}r.\hfill \end{array}$$

(162)

Corollary 8

(Order reversal and two-sided information action). Write $p_r^{\sigma }\left(\rho \right):={prox}_{r,\sigma }^{\mathrm{U}}\left(\rho \right)$. For $0<t<1$ and $r=(1-t)/t$,

$$\begin{array}{cc}\hfill (1-t)D_t^{\mathrm{b}}(\rho \parallel \sigma )& =t{D}_{1-t}^{\mathrm{b}}(\sigma \parallel \rho )=-{\phi }_{\rho \mid \sigma }\left(t\right),\hfill \end{array}$$

(163)

$$\begin{array}{cc}\hfill p_r^{\sigma }\left(\rho \right)& =p_{1/r}^{\rho }\left(\sigma \right),r{\mathcal{M}}_r^{\sigma }\left(\rho \right)={\mathcal{M}}_{1/r}^{\rho }\left(\sigma \right).\hfill \end{array}$$

(164)

Although $D_t^{\mathrm{b}}$ is directed, its two endpoint orientations are therefore exchanged exactly by $t\leftrightarrow 1-t$, or $r\leftrightarrow r^{-1}$. Its order derivative is

$$\begin{array}{|c|}\hline \frac{\mathrm{d}}{\mathrm{d}t}{D}_t^{\mathrm{b}}(\rho \parallel \sigma )=\frac{D_{\mathrm{U}}({\gamma }_t\parallel \rho )}{{(1-t)}^2}\ge 0,\\ \hline\end{array}0<t<1,$$

(165)

strictly for distinct faithful endpoints. Hence

$$\underset{t\downarrow 0}{lim}\frac{D_t^{\mathrm{b}}(\rho \parallel \sigma )}{t}=D_{\mathrm{U}}(\sigma \parallel \rho ),\underset{t\uparrow 1}{lim}{D}_t^{\mathrm{b}}(\rho \parallel \sigma )=D_{\mathrm{U}}(\rho \parallel \sigma ).$$

(166)

In addition,

$$\begin{array}{|c|}\hline {\partial }_r\left[r{\mathcal{M}}_r^{\sigma }\left(\rho \right)\right]=D_{\mathrm{U}}(p_r^{\sigma }\left(\rho \right)\parallel \sigma ).\\ \hline\end{array}$$

(167)

Consequently,

$$\begin{array}{cc}\hfill D_{\mathrm{U}}(\sigma \parallel \rho )& ={\int }_0^{\infty }{D}_{\mathrm{U}}(p_r^{\sigma }\left(\rho \right)\parallel \sigma )\mathrm{d}r,\hfill \\ \hfill D_{\mathrm{U}}(\rho \parallel \sigma )+D_{\mathrm{U}}(\sigma \parallel \rho )& ={\int }_0^{\infty }\left[\frac{1}{r^2}{D}_{\mathrm{U}}(p_r^{\sigma }\left(\rho \right)\parallel \rho )+D_{\mathrm{U}}(p_r^{\sigma }\left(\rho \right)\parallel \sigma )\right]\mathrm{d}r\hfill \end{array}$$

(168)

$$\begin{array}{cc}& ={\int }_0^1{g}_{{\gamma }_t}^{\mathrm{BKM}}({\dot{\gamma }}_t,{\dot{\gamma }}_t)\mathrm{d}t.\hfill \end{array}$$

(169)

Thus the Jeffreys BKM path energy is also the total two-sided proximal-scale action.

Corollary 9

(Partition dual and Rényi response of the scale Hamiltonian). Fix $\rho \in {\mathcal{S}}_n^{+}$ and $r>0$. For a Hermitian cotangent representative P, modulo scalar matrices, define the centered negative-source cumulant

$$\begin{array}{cc}\hfill {\mathcal{K}}_{r,\rho }^{-}\left(\left[P\right]\right)& :=\frac{{\mathsf{\Lambda }}_{\rho }(-rP)+rTr\left(\rho P\right)}{r^2}\hfill \\ & =\frac{1}{r^2}{D}_{\mathrm{U}}\left(\rho \parallel E_{\rho }(-rP)\right)={\int }_0^1(1-s){Cov}_{E_{\rho }(-srP)}^{\mathrm{KM}}(P,P)\mathrm{d}s.\hfill \end{array}$$

(170)

It is gauge invariant and strictly convex on $Herm\left(n\right)/\mathbb{R}I$. The scale Hamiltonian is its exact Legendre-Fenchel dual after the BKM musical identification:

$$\begin{array}{|c|}\hline {\mathcal{H}}_r^{\mathrm{U}}(\rho ,\alpha )={\displaystyle \underset{\left[P\right]\in Herm\left(n\right)/\mathbb{R}I}{sup}}\left\{Tr\left({\alpha }^{{♯}_{\rho }}P\right)-{\mathcal{K}}_{r,\rho }^{-}\left(\left[P\right]\right)\right\}.\\ \hline\end{array}$$

(171)

If $q:=\rho -r{\alpha }^{{♯}_{\rho }}$ is faithful, the supremum is attained uniquely at

$$\left[P^{★}\right]=\frac{1}{r}[log\rho -logq].$$

(172)

On the envelope solution $q=p_r={\gamma }_t$, where $t={(1+r)}^{-1}$, this optimizer and the two oriented residuals are

$$\begin{array}{cc}\hfill \left[P_r^{★}\right]& =[log{p}_r-log\sigma ]=t[log\rho -log\sigma ],\hfill \end{array}$$

(173)

$$\begin{array}{cc}\hfill {\mathcal{H}}_r^{\mathrm{U}}\left(\rho ,D_{\rho }{\mathcal{M}}_r^{\sigma }\right)& =t^2\frac{\mathrm{d}}{\mathrm{d}t}{D}_t^{\mathrm{b}}(\rho \parallel \sigma ),\hfill \end{array}$$

(174)

$$\begin{array}{cc}\hfill D_{\mathrm{U}}({\gamma }_t\parallel \sigma )& =D_t^{\mathrm{b}}(\rho \parallel \sigma )-t(1-t)\frac{\mathrm{d}}{\mathrm{d}t}{D}_t^{\mathrm{b}}(\rho \parallel \sigma ).\hfill \end{array}$$

(175)

Thus the Rényi value and its order derivative determine both barycentric legs; the derivative alone determines the exact nonquadratic Hamiltonian density and the ρ-oriented leg.

The additive semigroup clock gives an equivalent Hamilton-Jacobi form. Put

$$u:=log(1+r),{\mathcal{V}}_u^{\sigma }\left(\rho \right):={\mathcal{M}}_{{\mathrm{e}}^u-1}^{\sigma }\left(\rho \right),{\widehat{\mathcal{H}}}_u^{\mathrm{U}}:={\mathrm{e}}^u{\mathcal{H}}_{{\mathrm{e}}^u-1}^{\mathrm{U}}.$$

(176)

Then

$$\begin{array}{|c|}\hline {\partial }_u{\mathcal{V}}_u^{\sigma }+{\widehat{\mathcal{H}}}_u^{\mathrm{U}}\left(\rho ,D_{\rho }{\mathcal{V}}_u^{\sigma }\right)=0,\\ \hline\end{array}{\widehat{\mathcal{H}}}_u^{\mathrm{U}}{|}_{\mathrm{solution}}=t{\partial }_t{D}_t^{\mathrm{b}}{|}_{t={\mathrm{e}}^{-u}}.$$

(177)

This is regularization time. It is not the physical clock introduced only after a mobility and τ are supplied.

The proof is given in Appendix A.1.

Remark 3

(Orientation and Rényi-family comparison). The superscript $\mathrm{b}$ denotes the barycentric/log-Euclidean construction and not a symmetrization; only at $t=1/2$ is the displayed order invariant under exchange of the two states. The variable-first orientation in Equation (147) is essential. The opposite centroid problem has

$$\begin{array}{cc}\hfill {\overline{\rho }}_t& :=\underset{\omega }{\mathrm{argmin}}\left\{(1-t)D_{\mathrm{U}}(\sigma \parallel \omega )+t{D}_{\mathrm{U}}(\rho \parallel \omega )\right\}=(1-t)\sigma +t\rho ,\hfill \\ \hfill \underset{\omega }{min}\left\{(1-t)D_{\mathrm{U}}(\sigma \parallel \omega )+t{D}_{\mathrm{U}}(\rho \parallel \omega )\right\}& =S_{\mathrm{vN}}\left({\overline{\rho }}_t\right)-(1-t)S_{\mathrm{vN}}\left(\sigma \right)-t{S}_{\mathrm{vN}}\left(\rho \right).\hfill \end{array}$$

(178)

Its value is the weighted quantum Jensen-Shannon divergence, not a log-Euclidean Rényi divergence.

If $[\rho ,\sigma ]=0$, with common-basis eigenvalues ${\rho }_i,{\sigma }_i>0$, then

$${\left({\gamma }_t\right)}_i=\frac{{\sigma }_i^{1-t}{\rho }_i^t}{{\sum }_j{\sigma }_j^{1-t}{\rho }_j^t},D_t^{\mathrm{b}}(\rho \parallel \sigma )=\frac{1}{t-1}log\sum _i{\rho }_i^t{\sigma }_i^{1-t},$$

(179)

so the construction reduces to the classical Rényi divergence. For noncommuting endpoints, however, its defining quantity is $Trexp[tlog\rho +(1-t)log\sigma ]$, not $Tr\left({\rho }^t{\sigma }^{1-t}\right)$. In particular, for $0<t<1$,

$${\tilde{D}}_t(\rho \parallel \sigma )\le D_t^{\mathrm{P}}(\rho \parallel \sigma )\le D_t^{\mathrm{b}}(\rho \parallel \sigma ),$$

(180)

where the first two terms are the sandwiched and Petz Rényi divergences; all three coincide in the commuting case [56]. Thus the envelope theorem neither replaces the exponential by an ordered product nor identifies the log-Euclidean member with the Petz or sandwiched member. The variational formula, optimizer, and sided-centroid residual are established inputs to the BKM extensions of Theorem 6.

The differential, BKM-gradient, and Hamilton-Jacobi assertions require faithful states. The variational identity extends facewise to singular endpoints but has no smooth continuation across rank-changing strata. The exact support-reduced optimizer and residual, rank obstruction, and large-scale face limit are recorded in Remark A1.

3.4.2. Quadratic Moreau-Yosida Continuation on Moment Space

Theorem 7

(Metric Moreau-Yosida regularization of the moment potential). Assume the hypotheses of Theorem 4, abbreviate $\lambda :={\lambda }_{\sigma ,O}$, $\mathsf{\Gamma }:={\mathsf{\Gamma }}_{\sigma ,O}$, and let G be the Hilbert-Schmidt Gram matrix in Equation (119). For every $m\in {\mathcal{C}}_O$ and $\theta \in {\mathbb{R}}^d$, the Fenchel mismatch has the exact information certificate

$$\begin{array}{|c|}\hline \mathsf{\Gamma }\left(m\right)+\lambda \left(\theta \right)-\theta ·m=D_{\mathrm{U}}({\rho }^m\parallel {\rho }_{\theta }).\\ \hline\end{array}$$

(181)

For $\epsilon >0$, define on all of ${\mathbb{R}}^d$

$${\mathsf{\Gamma }}_{\epsilon }^G\left(m\right):=\underset{q\in {\mathcal{C}}_O}{min}\left\{\mathsf{\Gamma }\left(q\right)+\frac{1}{2\epsilon }{(m-q)}^{\mathsf{T}}{G}^{-1}(m-q)\right\}.$$

(182)

Then

$$\begin{array}{|c|}\hline {\mathsf{\Gamma }}_{\epsilon }^G={\left(\lambda +\frac{\epsilon }{2}\langle ·,G·\rangle \right)}^{\ast }.\\ \hline\end{array}$$

(183)

For each $m\in {\mathbb{R}}^d$ there is a unique ${\theta }_{\epsilon }\left(m\right)\in {\mathbb{R}}^d$ satisfying

$$m=\nabla \lambda \left({\theta }_{\epsilon }\right)+\epsilon G{\theta }_{\epsilon },q_{\epsilon }\left(m\right):=\nabla \lambda \left({\theta }_{\epsilon }\right)=m-\epsilon G{\theta }_{\epsilon }\in int{\mathcal{C}}_O.$$

(184)

The envelope is real analytic and

$$\begin{array}{cc}\hfill \nabla {\mathsf{\Gamma }}_{\epsilon }^G\left(m\right)& ={\theta }_{\epsilon }\left(m\right),\hfill \end{array}$$

(185)

$$\begin{array}{cc}\hfill {\nabla }^2{\mathsf{\Gamma }}_{\epsilon }^G\left(m\right)& ={\left[\chi \left({\theta }_{\epsilon }\left(m\right)\right)+\epsilon G\right]}^{-1},\hfill \end{array}$$

(186)

$$\begin{array}{cc}\hfill \frac{2}{1+2\epsilon }{G}^{-1}& ⪯{\nabla }^2{\mathsf{\Gamma }}_{\epsilon }^G\left(m\right)\prec \frac{1}{\epsilon }{G}^{-1}.\hfill \end{array}$$

(187)

If $\mathsf{A}:=G\partial \mathsf{\Gamma }$ is viewed as a maximal monotone operator for the inner product ${\langle x,y\rangle }_{G^{-1}}=x^{\mathsf{T}}{G}^{-1}y$, then

$$\begin{array}{|c|}\hline {(\mathrm{id}+\epsilon \mathsf{A})}^{-1}m=q_{\epsilon }\left(m\right),{\mathsf{A}}_{\epsilon }\left(m\right):=\frac{m-q_{\epsilon }\left(m\right)}{\epsilon }=G\nabla {\mathsf{\Gamma }}_{\epsilon }^G\left(m\right).\\ \hline\end{array}$$

(188)

Finally,

$$\begin{array}{cc}\hfill {\partial }_{\epsilon }{\mathsf{\Gamma }}_{\epsilon }^G+\frac{1}{2}{(\nabla {\mathsf{\Gamma }}_{\epsilon }^G)}^{\mathsf{T}}G(\nabla {\mathsf{\Gamma }}_{\epsilon }^G)& =0,\hfill \end{array}$$

(189)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\theta }_{\epsilon }}{\mathrm{d}\epsilon }& =-{[\chi \left({\theta }_{\epsilon }\right)+\epsilon G]}^{-1}G{\theta }_{\epsilon },\hfill \end{array}$$

(190)

where the second equation holds at fixed m. As $\epsilon \downarrow 0$, ${\mathsf{\Gamma }}_{\epsilon }^G\to \mathsf{\Gamma }$ pointwise in the extended-real sense and epigraphically. The first equation is a Hamilton-Jacobi equation in regularization scale, not the physical-time Hamilton-Jacobi equation excluded in Appendix G.7. The proof, exact primal residual, boundary behavior, and a possibly noninjective noisy-record extension are given in Appendix B.

For every fixed $\epsilon >0$, the proximal moment is interior and its selected state ${\rho }^{q_{\epsilon }}$ is faithful even when the imposed m lies on or outside the moment body. This regularizes finite-source and stiffness blow-up only; its remaining scope limitations are collected in Theorem 26 and Section 10.

Proposition 7

(Hopf-Lax semigroup, characteristics, and scale action). Extend Γ by $+\infty$ outside ${\mathcal{C}}_O$ and, for every proper closed convex $f:{\mathbb{R}}^d\to (-\infty ,+\infty ]$, define

$$\left({\mathsf{Q}}_{\epsilon }^{G}f\right)\left(m\right):=\underset{q\in {\mathbb{R}}^d}{inf}\left\{f\left(q\right)+\frac{1}{2\epsilon }{(m-q)}^{\mathsf{T}}{G}^{-1}(m-q)\right\}.$$

(191)

Then, for $\epsilon ,\delta >0$,

$$\begin{array}{|c|}\hline {\mathsf{\Gamma }}_{\epsilon }^G={\mathsf{Q}}_{\epsilon }^G\mathsf{\Gamma },{\mathsf{Q}}_{\delta }^G{\mathsf{Q}}_{\epsilon }^{G}f={\mathsf{Q}}_{\epsilon +\delta }^{G}f.\\ \hline\end{array}$$

(192)

By Fenchel-Moreau duality, Equation (183) uniquely determines $u(\epsilon ,·)={\mathsf{\Gamma }}_{\epsilon }^G$ within the proper closed convex class. It is classical for $\epsilon >0$, satisfies Equation (189), and has the extended-real epigraphical initial trace Γ . This qualified statement does not assert unrestricted viscosity uniqueness for a finite continuous datum on all of ${\mathbb{R}}^d$.

Write ${\theta }_{\epsilon }\left(m\right):=\nabla {\mathsf{\Gamma }}_{\epsilon }^G\left(m\right)$ and $m_{\sigma }:=\nabla \lambda \left(0\right)=M\left(\sigma \right)$. For $0<{\epsilon }_1<{\epsilon }_2$,

$$\begin{array}{cc}\hfill {\mathsf{\Gamma }}_{{\epsilon }_1}^G\left(m\right)-{\mathsf{\Gamma }}_{{\epsilon }_2}^G\left(m\right)& =\frac{1}{2}{\int }_{{\epsilon }_1}^{{\epsilon }_2}{\theta }_s{\left(m\right)}^{\mathsf{T}}G{\theta }_s\left(m\right)\mathrm{d}s,\hfill \end{array}$$

(193)

$$\begin{array}{cc}\hfill {\mathsf{\Gamma }}_{\epsilon }^G\left(m\right)& =\frac{1}{2}{\int }_{\epsilon }^{\infty }{\theta }_s{\left(m\right)}^{\mathsf{T}}G{\theta }_s\left(m\right)\mathrm{d}s.\hfill \end{array}$$

(194)

For $m\in {\mathcal{C}}_O$, the improper endpoint gives

$$\begin{array}{|c|}\hline \mathsf{\Gamma }\left(m\right)=\frac{1}{2}{\int }_0^{\infty }{\theta }_s{\left(m\right)}^{\mathsf{T}}G{\theta }_s\left(m\right)\mathrm{d}s.\\ \hline\end{array}$$

(195)

The Hamilton-Jacobi characteristics are also exact: if $q\in int{\mathcal{C}}_O$, $\theta =\nabla \mathsf{\Gamma }\left(q\right)$, and $m_{\epsilon }=q+\epsilon G\theta$, then

$$q_{\epsilon }\left(m_{\epsilon }\right)=q,{\theta }_{\epsilon }\left(m_{\epsilon }\right)=\theta ,{\mathsf{\Gamma }}_{\epsilon }^G\left(m_{\epsilon }\right)=\mathsf{\Gamma }\left(q\right)+\frac{\epsilon }{2}{\theta }^{\mathsf{T}}G\theta .$$

(196)

For fixed $m\in {\mathbb{R}}^d$, as $\epsilon \to \infty$,

$$\begin{array}{cc}\hfill {\theta }_{\epsilon }\left(m\right)& =\frac{1}{\epsilon }{G}^{-1}(m-m_{\sigma })+O\left({\epsilon }^{-2}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill q_{\epsilon }\left(m\right)& =m_{\sigma }+\frac{1}{\epsilon }\chi \left(0\right)G^{-1}(m-m_{\sigma })+O\left({\epsilon }^{-2}\right),\hfill \end{array}$$

(197)

$$\begin{array}{cc}\hfill {\mathsf{\Gamma }}_{\epsilon }^G\left(m\right)& =\frac{1}{2\epsilon }{(m-m_{\sigma })}^{\mathsf{T}}{G}^{-1}(m-m_{\sigma })+O\left({\epsilon }^{-2}\right).\hfill \end{array}$$

(198)

Hence the selected state ${\rho }^{q_{\epsilon }}={\rho }_{{\theta }_{\epsilon }}$ returns to σ as $\epsilon \to \infty$. The Legendre-Fenchel transform remains exact as an extended-real duality: ε is an inference/noise continuation scale, not physical time. The proof is in Appendix B.

3.5. Operational and Thermodynamic Selection of Relative Entropy

Quantum Stein’s lemma identifies $D_{\mathrm{U}}(\rho \parallel \sigma )$ as the optimal asymptotic type-II error exponent at fixed type-I tolerance [57,58]. For a Gibbs reference ${\tau }_{\beta }={\mathrm{e}}^{-\beta H_{\mathrm{sys}}}/Z_{\beta }$ and $F_{\beta }\left(\rho \right)=Tr\left(\rho H_{\mathrm{sys}}\right)-{\beta }^{-1}{S}_{\mathrm{vN}}\left(\rho \right)$,

$${\beta }^{-1}{D}_{\mathrm{U}}(\rho \parallel {\tau }_{\beta })=F_{\beta }\left(\rho \right)-F_{\beta }\left({\tau }_{\beta }\right).$$

(199)

These operational and thermodynamic identifications select neither a mechanical action nor dynamics.

Corollary 10

(Modular Clausius identity and BKM remainder). For $\rho ,\sigma \in {\mathcal{S}}_n^{+}$, define the modular Hamiltonian $K_{\sigma }^{\mathrm{mod}}:=-log\sigma$. Then

$$\begin{array}{|c|}\hline D_{\mathrm{U}}(\rho \parallel \sigma )=Tr\left[(\rho -\sigma )K_{\sigma }^{\mathrm{mod}}\right]-[S_{\mathrm{vN}}\left(\rho \right)-S_{\mathrm{vN}}\left(\sigma \right)].\\ \hline\end{array}$$

(200)

Along the mixture chord ${\rho }_s:=\sigma +s(\rho -\sigma )$,

$$\begin{array}{|c|}\hline D_{\mathrm{U}}(\rho \parallel \sigma )={\int }_0^1(1-s)g_{{\rho }_s}^{\mathrm{BKM}}(\rho -\sigma ,\rho -\sigma )\mathrm{d}s.\\ \hline\end{array}$$

(201)

Consequently, if ${\rho }_{ϵ}$ is a $C^3$ faithful curve with ${\rho }_0=\sigma$ and $V={\dot{\rho }}_0$, then

$$\begin{array}{cc}\hfill {\left.\frac{\mathrm{d}}{\mathrm{d}ϵ}{S}_{\mathrm{vN}}\left({\rho }_{ϵ}\right)\right|}_0& =Tr\left(V{K}_{\sigma }^{\mathrm{mod}}\right),\hfill \end{array}$$

(202)

$$\begin{array}{cc}\hfill D_{\mathrm{U}}({\rho }_{ϵ}\parallel \sigma )& =\frac{{ϵ}^2}{2}{g}_{\sigma }^{\mathrm{BKM}}(V,V)+O\left({ϵ}^3\right).\hfill \end{array}$$

(203)

Thus the modular first law is first-order saturation of the exact relative-entropy inequality [59,60]; by the finite-dimensional expansion above, the BKM metric is its positive second-order defect [5,47]. For arbitrary finite density matrices this is the entropy or modular first law; it is an entanglement first law only when the states are specified as reduced states of a subsystem or regulated local algebra. Moreover,

$$\begin{array}{|c|}\hline \mathcal{X}\left(\sigma \right)=K_{\sigma }^{\mathrm{mod}}-\frac{1}{n}Tr\left(K_{\sigma }^{\mathrm{mod}}\right)I,\\ \hline\end{array}$$

(204)

so the global Gibbs coordinate is precisely the traceless modular Hamiltonian.

Proof from relative-entropy contrast theory. 

Substitution of $K_{\sigma }^{\mathrm{mod}}=-log\sigma$ gives (200), the standard relative-entropy form of the modular first law [59,60]. Since $D_{\mathrm{U}}=B_{-S_{\mathrm{vN}}}$, the standard integral remainder for a Hessian contrast along the mixture chord gives (201); its first two jets give Equations (202) and (203). Equation (62) supplies the final traceless-gauge identity. □

Remark 4

(Conditional Rindler specialization and action normalization). This specialization uses physical data not supplied by the state geometry. Assume a finite-dimensional cutoff model (for example, a split inclusion followed by a finite-mode or energy truncation) and an action-valued boost generator ${\widehat{\mathcal{K}}}_{\chi }$ such that, in units $k_{\mathrm{B}}=1$,

$$H_{\chi }:=\kappa {\widehat{\mathcal{K}}}_{\chi },{\beta }_{\mathrm{U}}:=\frac{2\pi }{\hslash \kappa },{\sigma }_{\mathrm{U}}:=\frac{exp[-(2\pi /\hslash ){\widehat{\mathcal{K}}}_{\chi }]}{Trexp[-(2\pi /\hslash ){\widehat{\mathcal{K}}}_{\chi }]}=\frac{{\mathrm{e}}^{-{\beta }_{\mathrm{U}}{H}_{\chi }}}{Z_{\mathrm{U}}}.$$

(205)

The Unruh normalization and the local-boost use below are external field-theoretic inputs [61,62]. If $\mathsf{\Delta }Q:=Tr\left[(\rho -{\sigma }_{\mathrm{U}})H_{\chi }\right]$ is the supplied boost heat flux, then

$$\begin{array}{|c|}\hline D_{\mathrm{U}}(\rho \parallel {\sigma }_{\mathrm{U}})={\beta }_{\mathrm{U}}\mathsf{\Delta }Q-[S_{\mathrm{vN}}\left(\rho \right)-S_{\mathrm{vN}}\left({\sigma }_{\mathrm{U}}\right)].\\ \hline\end{array}$$

(206)

Hence positivity gives the finite modular-energy (Bekenstein-Casini) bound $\mathsf{\Delta }{S}_{\mathrm{vN}}\le {\beta }_{\mathrm{U}}\mathsf{\Delta }Q$, while Equation (202) gives equality to first order [59].

The resulting clock-action locking is proved in Proposition 8. It fixes only ${\kappa }_A/\hslash$ and does not identify the boost generator with the scalar endpoint functional.

For sharp quantum-field-theoretic wedges, reduced density matrices and the separate von Neumann entropies above need not exist. The appropriate object is Araki relative entropy on the local von Neumann algebra [63]. Under the Bisognano-Wichmann hypotheses, wedge modular flow is boost flow [40]; related relative-entropy formulations are given in [59,64]. These are external operator-algebraic results, not consequences of the finite-dimensional controlled completion.

Proposition 8

(Reference modular flow and conditional clock-action locking). Let $\tau \in {\mathcal{S}}_n^{+}$ be distinguished, put $X_{\tau }:=\mathcal{X}\left(\tau \right)$, and use the standard modular orientation

$${\sigma }_s^{\tau }\left(A\right):={\tau }^{\mathrm{i}s}A{\tau }^{-\mathrm{i}s}={\mathrm{e}}^{-\mathrm{i}s{X}_{\tau }}A{\mathrm{e}}^{\mathrm{i}s{X}_{\tau }},s\in \mathbb{R}.$$

(207)

Then ${\sigma }^{\tau }$ is a canonical dimensionless one-parameter group of ∗-automorphisms with generator

$${\delta }_{\tau }\left(A\right)=-\mathrm{i}[X_{\tau },A].$$

(208)

Its Schrödinger dual ${\rho }_s={\tau }^{-\mathrm{i}s}\rho {\tau }^{\mathrm{i}s}$ preserves the spectrum, $S_{\mathrm{vN}}\left(\rho \right)$, and $D_{\mathrm{U}}(\rho \parallel \tau )$. It is therefore reversible modular transport, not the dissipative proximal family associated with Theorem 6.

If

$$\tau =\frac{{\mathrm{e}}^{-\beta H}}{Tr{\mathrm{e}}^{-\beta H}},{\alpha }_t^H\left(A\right):={\mathrm{e}}^{\mathrm{i}tH/\hslash }A{\mathrm{e}}^{-\mathrm{i}tH/\hslash },$$

(209)

then

$$X_{\tau }=\beta \left(H-\frac{1}{n}Tr\left(H\right)I\right),\begin{array}{|c|}\hline {\sigma }_s^{\tau }={\alpha }_{-\beta \hslash s}^H.\\ \hline\end{array}$$

(210)

Thus the reference fixes a dimensionless modular ordering and the product $\beta H$, while conversion to physical time requires $\beta \hslash$. The tracial reference has $X_{\tau }=0$ and hence gives the trivial modular flow.

Under the regulated Rindler hypotheses of Remark 4, write

$${\widehat{\mathcal{K}}}_{\chi }^0:={\widehat{\mathcal{K}}}_{\chi }-\frac{1}{n}Tr\left({\widehat{\mathcal{K}}}_{\chi }\right)I.$$

(211)

Then

$$X_{{\sigma }_{\mathrm{U}}}=\frac{2\pi }{\hslash }{\widehat{\mathcal{K}}}_{\chi }^0,{\sigma }_s^{{\sigma }_{\mathrm{U}}}={\alpha }_{-2\pi s/\kappa }^{H_{\chi }}.$$

(212)

If ${\widehat{\mathcal{K}}}_{\chi }^0\ne 0$, the physical-generator intertwining condition

$${\kappa }_A{X}_{{\sigma }_{\mathrm{U}}}={\widehat{\mathcal{K}}}_{\chi }^0$$

(213)

holds if and only if

$$\begin{array}{|c|}\hline {\kappa }_A=\frac{\hslash }{2\pi },t_A={\beta }_{\mathrm{U}}{\kappa }_A={\kappa }^{-1}.\\ \hline\end{array}$$

(214)

Consequently the regulated Bisognano-Wichmann identification removes an additional Killing-time calibration once κ is supplied and locks ${\kappa }_A/\hslash$; it does not determine the absolute action scale. Interpreting the modular parameter as physical time is the additional thermal-time hypothesis [39]. Reversing the modular-flow convention reverses the displayed time orientations.

Proof. 

Finite-dimensional modular theory gives the group, generator, and dual unitary-conjugation invariances [38]. Substituting the Gibbs form into the traceless logarithm proves Equation (210); inserting Equation (205) then proves Equation (212). Finally, Equation (213) reduces to $(2\pi {\kappa }_A/\hslash ){\widehat{\mathcal{K}}}_{\chi }^0={\widehat{\mathcal{K}}}_{\chi }^0$; nonvanishing of the generator proves the equivalence and the formula for $t_A$. □

Corollary 11

(Exact proximal resolution of the regulated modular Clausius defect). Under the cutoff hypotheses of Remark 4, define

$$\mathsf{\Delta }{Q}_{\chi }\left(\omega \right):=Tr\left[(\omega -{\sigma }_{\mathrm{U}})H_{\chi }\right],\mathsf{\Delta }{S}_{\mathrm{U}}\left(\omega \right):=S_{\mathrm{vN}}\left(\omega \right)-S_{\mathrm{vN}}\left({\sigma }_{\mathrm{U}}\right),$$

(215)

and

$${\mathfrak{C}}_{\mathrm{U}}\left(\omega \right):={\beta }_{\mathrm{U}}\mathsf{\Delta }{Q}_{\chi }\left(\omega \right)-\mathsf{\Delta }{S}_{\mathrm{U}}\left(\omega \right)=D_{\mathrm{U}}(\omega \parallel {\sigma }_{\mathrm{U}})\ge 0.$$

(216)

This is a distinguishability/free-energy residual in the chosen past-horizon flux orientation, not the internal entropy-production term in the irreversible balance $\mathrm{d}S=\delta Q/T+{\mathrm{d}}_{i}S$ [65]. For faithful ρ, $r>0$, and $p_r:={prox}_{r,{\sigma }_{\mathrm{U}}}^{\mathrm{U}}\left(\rho \right)$,

$$\begin{array}{cc}\hfill {\mathfrak{C}}_{\mathrm{U}}\left(\rho \right)& ={\mathcal{M}}_r^{{\sigma }_{\mathrm{U}}}\left(\rho \right)+\left(1+\frac{1}{r}\right)D_{\mathrm{U}}(\rho \parallel p_r),\hfill \end{array}$$

(217)

$$\begin{array}{cc}\hfill {\mathfrak{C}}_{\mathrm{U}}\left(p_r\right)& =D_{\mathrm{U}}(p_r\parallel {\sigma }_{\mathrm{U}})={\partial }_r\left[r{\mathcal{M}}_r^{{\sigma }_{\mathrm{U}}}\left(\rho \right)\right],\hfill \end{array}$$

(218)

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{\mathfrak{C}}_{\mathrm{U}}\left(p_r\right)\mathrm{d}r& =D_{\mathrm{U}}({\sigma }_{\mathrm{U}}\parallel \rho ).\hfill \end{array}$$

(219)

Equivalently, for $0<t<1$,

$$\begin{array}{|c|}\hline D_t^{\mathrm{b}}(\rho \parallel {\sigma }_{\mathrm{U}})={\displaystyle \underset{\omega \in {\mathcal{S}}_n}{min}}\left\{{\mathfrak{C}}_{\mathrm{U}}\left(\omega \right)+\frac{t}{1-t}{D}_{\mathrm{U}}(\omega \parallel \rho )\right\},\\ \hline\end{array}$$

(220)

with $r=(1-t)/t$ and minimizer $p_r={\gamma }_t$. Thus the log-Euclidean/Umegaki-barycentric Rényi divergence is exactly the entropic envelope of the regulated modular defect; it is not merely the defect evaluated at the relaxed state, because its value also includes the movement penalty $r^{-1}{D}_{\mathrm{U}}(p_r\parallel \rho )$.

For fixed r and ${\rho }_{ϵ}={\sigma }_{\mathrm{U}}+ϵV+O\left({ϵ}^2\right)$ with $TrV=0$,

$$\begin{array}{cc}\hfill p_r\left({\rho }_{ϵ}\right)& ={\sigma }_{\mathrm{U}}+\frac{ϵ}{1+r}V+O\left({ϵ}^2\right),\hfill \end{array}$$

(221)

$$\begin{array}{cc}\hfill {\mathcal{M}}_r^{{\sigma }_{\mathrm{U}}}\left({\rho }_{ϵ}\right)& =\frac{{ϵ}^2}{2(1+r)}{g}_{{\sigma }_{\mathrm{U}}}^{\mathrm{BKM}}(V,V)+O\left({ϵ}^3\right),\hfill \end{array}$$

(222)

$$\begin{array}{cc}\hfill \left(1+\frac{1}{r}\right)D_{\mathrm{U}}({\rho }_{ϵ}\parallel p_r\left({\rho }_{ϵ}\right))& =\frac{r{ϵ}^2}{2(1+r)}{g}_{{\sigma }_{\mathrm{U}}}^{\mathrm{BKM}}(V,V)+O\left({ϵ}^3\right),\hfill \end{array}$$

(223)

$$\begin{array}{cc}\hfill {\mathfrak{C}}_{\mathrm{U}}\left(p_r\left({\rho }_{ϵ}\right)\right)& =\frac{{ϵ}^2}{2{(1+r)}^2}{g}_{{\sigma }_{\mathrm{U}}}^{\mathrm{BKM}}(V,V)+O\left({ϵ}^3\right).\hfill \end{array}$$

(224)

The envelope therefore preserves the first-order modular Clausius law, splits its BKM second-order defect in the fractions $1/(1+r)$ and $r/(1+r)$, and contracts the residual defect at the proximal state by ${(1+r)}^{-2}$.

Proof. 

The displayed identities are the corresponding specializations of Equations (147), (151), (167) and (168). The linear and quadratic formulas follow from Equations (152) and (203). □

For a non-scalar $H_{\chi }$, the one-observable specialization of Theorem 4 is the minimum modular disequilibrium at fixed boost energy and obeys

$${\mathsf{\Gamma }}_{{\sigma }_{\mathrm{U}},H_{\chi }}^{\prime \prime }\left(q\right)={Cov}_{{\rho }^q}^{\mathrm{KM}}{(H_{\chi },H_{\chi })}^{-1}.$$

(225)

Its distinct moment-space Moreau regularization replaces this inverse response by ${[{Cov}_{{\rho }^{q_{\epsilon }}}^{\mathrm{KM}}(H_{\chi },H_{\chi })+\epsilon G_{\chi }]}^{-1}$, where $G_{\chi }:=Tr\left[{(H_{\chi }-Tr\left(H_{\chi }\right)I/n)}^2\right]$. This is a regularized static constitutive law, not by itself horizon relaxation or a gravitational equation of state.

The exact Gibbs-Fenchel, BKM, shifted-cone, response, and global-bound identities for forward and reverse thermal quenches are proved in Corollary A6 and Equation (A302). Their sum is a bidirectional aggregate for two separately prepared quenches, not the work of a closed hysteresis cycle.

4. Channel Defects, Recovery, and Operational Response

4.1. Data Processing and Channel-Accessible BKM Geometry

Let $\mathsf{\Lambda }:{\mathrm{M}}_n\left(\mathbb{C}\right)\to {\mathrm{M}}_m\left(\mathbb{C}\right)$ be a finite-dimensional completely positive, trace-preserving (CPTP) map. Umegaki relative entropy obeys [11]

$$D_{\mathrm{U}}\left(\mathsf{\Lambda }\left(\rho \right)\parallel \mathsf{\Lambda }\left(\sigma \right)\right)\le D_{\mathrm{U}}(\rho \parallel \sigma ).$$

(226)

Define its contraction defect by

$${\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma ):=D_{\mathrm{U}}(\rho \parallel \sigma )-D_{\mathrm{U}}\left(\mathsf{\Lambda }\left(\rho \right)\parallel \mathsf{\Lambda }\left(\sigma \right)\right)\ge 0.$$

(227)

The defect is the endpoint distinguishability lost under $\mathsf{\Lambda }$. For faithful endpoint pairs, $D_{\mathrm{U}}$ also has the weighted BKM representation above, linking the exact endpoint contraction to its local metric Hessian. DPI and ${\mathsf{\Delta }}_{\mathsf{\Lambda }}$ remain well defined for nonfaithful outputs under the usual support convention.

Remark 5

(Exterior accessibility under a supplied split regulator). If an independently supplied finite-dimensional split identifies $\mathcal{H}={\mathcal{H}}_{\mathrm{ext}}\otimes {\mathcal{H}}_{\mathrm{hid}}$ and $\mathsf{\Lambda }={Tr}_{\mathrm{hid}}$, then

$${\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma )=D_{\mathrm{U}}(\rho \parallel \sigma )-D_{\mathrm{U}}({\rho }_{\mathrm{ext}}\parallel {\sigma }_{\mathrm{ext}})$$

(228)

is, in the iid Stein setting with arbitrary collective measurements on ${\mathcal{H}}_{\mathrm{ext}}^{\otimes N}$, exactly the reduction of the optimal asymptotic type-II error exponent available to the exterior observer. Vanishing means sufficiency for this pair and, for faithful states, exact Petz recovery; it does not mean that the hidden factor is absent. The interpretation depends on the supplied split: a generic CPTP map is not thereby a causal horizon. Sharp relativistic local algebras need not admit density matrices or a literal partial trace, and their corresponding statement uses restriction of normal states and monotonicity of Araki relative entropy [59,63]. No algebraic-QFT extension is asserted here.

Taking the second variation of DPI and using the diagonal metric identity Equation (77) gives the metric contraction in the following theorem.

Theorem 8

(CPTP contraction and intrinsic channel-accessible Hessian geometry). Let $P_{\mathsf{\Lambda }}$ be the support projection of $\mathsf{\Lambda }\left(I\right)$, regard outputs as states on $P_{\mathsf{\Lambda }}{\mathbb{C}}^m$, and define

$${L:=\mathsf{\Lambda }|}_{{Herm}_0\left(n\right)},{\mathcal{Q}}_{\mathsf{\Lambda }}:=\mathsf{\Lambda }\left({\mathcal{S}}_n^{+}\right).$$

(229)

For $\rho ,\sigma \in {\mathcal{S}}_n^{+}$, put

$$\rho {\sim }_{\mathsf{\Lambda }}\sigma \iff \mathsf{\Lambda }\left(\rho \right)=\mathsf{\Lambda }\left(\sigma \right),{\overline{D}}_{\mathsf{\Lambda }}(\left[\rho \right]\parallel \left[\sigma \right]):=D_{\mathrm{U}}\left(\mathsf{\Lambda }\left(\rho \right)\parallel \mathsf{\Lambda }\left(\sigma \right)\right).$$

(230)

Then $\left[\rho \right]\mapsto \mathsf{\Lambda }\left(\rho \right)$ canonically identifies the quotient with the relatively open convex set ${\mathcal{Q}}_{\mathsf{\Lambda }}$ in its affine hull, and

$$T_q{\mathcal{Q}}_{\mathsf{\Lambda }}=imL\cong {Herm}_0\left(n\right)/kerL,q\in {\mathcal{Q}}_{\mathsf{\Lambda }},dim{\mathcal{Q}}_{\mathsf{\Lambda }}=rankL.$$

(231)

Let ${\overline{\nabla }}^{\left(\mathrm{m}\right)}$ be the flat connection inherited from this affine hull. On ${\mathcal{Q}}_{\mathsf{\Lambda }}$, the mixture-affine potential

$${\mathcal{F}}_{\mathsf{\Lambda }}\left(q\right):={Tr}_{P_{\mathsf{\Lambda }}}(qlogq)=-S_{\mathrm{vN}}\left(q\right)$$

(232)

satisfies

$${\overline{D}}_{\mathsf{\Lambda }}(\left[\rho \right]\parallel \left[\sigma \right])=B_{{\mathcal{F}}_{\mathsf{\Lambda }}}(\mathsf{\Lambda }\left(\rho \right),\mathsf{\Lambda }\left(\sigma \right)),{\overline{g}}_{\mathsf{\Lambda }}={Hess}_{{\overline{\nabla }}^{\left(\mathrm{m}\right)}}{\mathcal{F}}_{\mathsf{\Lambda }}.$$

(233)

Thus $({\mathcal{S}}_n^{+}/{\sim }_{\mathsf{\Lambda }},{\overline{\nabla }}^{\left(\mathrm{m}\right)},{\overline{g}}_{\mathsf{\Lambda }})$ is intrinsically Hessian and dually flat, and its tangent bundle carries the strongly integrable Sakamoto structure ${\mathcal{B}}_{\mathsf{\Lambda },\mathrm{m}}={\mathcal{B}}_{{\overline{\nabla }}^{\left(\mathrm{m}\right)}}({\mathcal{Q}}_{\mathsf{\Lambda }},{\overline{g}}_{\mathsf{\Lambda }})$. Before quotienting, the metric pulls back to the semimetric

$$g_{\mathsf{\Lambda },\rho }^{\mathrm{obs}}(V,W):=g_{\mathsf{\Lambda }\left(\rho \right)}^{\mathrm{BKM}}\left(\mathsf{\Lambda }\left(V\right),\mathsf{\Lambda }\left(W\right)\right),$$

(234)

with

$$\begin{array}{c}0\le {\overline{D}}_{\mathsf{\Lambda }}(\left[\rho \right]\parallel \left[\sigma \right])\le D_{\mathrm{U}}(\rho \parallel \sigma ),\\ rad{g}_{\mathsf{\Lambda },\rho }^{\mathrm{obs}}=kerL,g_{\mathsf{\Lambda },\rho }^{\mathrm{obs}}(V,V)\le g_{\rho }^{\mathrm{BKM}}(V,V).\end{array}$$

(235)

The carrier, quotient-Hessian, and second-variation arguments are given in Appendix F.5; the last step is the Hessian form of DPI.

Corollary 12

(Exact global-to-local contraction-defect bridge). Define the entropy-gain potential

$${\mathcal{H}}_{\mathsf{\Lambda }}\left(\rho \right):=Tr(\rho log\rho )-{Tr}_{P_{\mathsf{\Lambda }}}[\mathsf{\Lambda }\left(\rho \right)log\mathsf{\Lambda }\left(\rho \right)]=S_{\mathrm{vN}}\left(\mathsf{\Lambda }\left(\rho \right)\right)-S_{\mathrm{vN}}\left(\rho \right).$$

(236)

Then, in input mixture-affine coordinates, define the local metric-loss tensor

$${\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma )=B_{{\mathcal{H}}_{\mathsf{\Lambda }}}(\rho ,\sigma ),g_{\mathsf{\Lambda },\rho }^{\mathrm{loss}}:={\left.Hess{\mathcal{H}}_{\mathsf{\Lambda }}\right|}_{\rho }=g_{\rho }^{\mathrm{BKM}}-g_{\mathsf{\Lambda },\rho }^{\mathrm{obs}}⪰0.$$

(237)

Write ${\mathcal{C}}^{\mathrm{in},\left(\mathrm{m}\right)}:={\nabla }^{\left(\mathrm{m}\right)}{g}_{\mathrm{BKM}}={\left({\nabla }^{\left(\mathrm{m}\right)}\right)}^3\mathcal{F}$ and ${\mathcal{C}}^{\mathrm{out},\left(\mathrm{m}\right)}:={\overline{\nabla }}^{\left(\mathrm{m}\right)}{\overline{g}}_{\mathsf{\Lambda }}={\left({\overline{\nabla }}^{\left(\mathrm{m}\right)}\right)}^3{\mathcal{F}}_{\mathsf{\Lambda }}$ on ${\mathcal{Q}}_{\mathsf{\Lambda }}$. The pure first-argument third diagonal jet then gives the symmetric nonlinear-response loss tensor

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill {\mathcal{C}}_{\mathsf{\Lambda },\rho }^{\mathrm{loss}}& :={\left({\nabla }^{\left(\mathrm{m}\right)}\right)}^3{\mathcal{H}}_{\mathsf{\Lambda },\rho }={\mathcal{C}}_{\rho }^{\mathrm{in},\left(\mathrm{m}\right)}-{\left({\mathsf{\Lambda }}^{\ast }{\mathcal{C}}^{\mathrm{out},\left(\mathrm{m}\right)}\right)}_{\rho },\hfill \\ \hfill {\left({\mathsf{\Lambda }}^{\ast }{\mathcal{C}}^{\mathrm{out},\left(\mathrm{m}\right)}\right)}_{\rho }(V,W,Z)& :={\mathcal{C}}_{\mathsf{\Lambda }\left(\rho \right)}^{\mathrm{out},\left(\mathrm{m}\right)}(\mathsf{\Lambda }V,\mathsf{\Lambda }W,\mathsf{\Lambda }Z),\hfill \\ \hfill {\left({\nabla }^{\left(\mathrm{m}\right)}{g}_{\mathsf{\Lambda }}^{\mathrm{loss}}\right)}_{\rho }(V,W,Z)& ={\mathcal{C}}_{\mathsf{\Lambda },\rho }^{\mathrm{loss}}(V,W,Z).\hfill \end{array}\\ \hline\end{array}$$

(238)

For faithful $\rho ,\sigma$, put $Z:=\rho -\sigma$ and ${\rho }_t:=\sigma +tZ$. The finite endpoint loss is exactly the weighted accumulation of local metric loss along this mixture-affine chord:

$$\begin{array}{|c|}\hline {\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma )={\int }_0^1(1-t)g_{\mathsf{\Lambda },{\rho }_t}^{\mathrm{loss}}(Z,Z)\mathrm{d}t.\\ \hline\end{array}$$

(239)

For $V,W\in T_{\sigma }{\mathcal{S}}_n^{+}$,

$$\begin{array}{cc}\hfill g_{\mathsf{\Lambda },\sigma }^{\mathrm{loss}}(V,W)& ={\left.\frac{{\partial }^2}{\partial a\partial b}{\mathsf{\Delta }}_{\mathsf{\Lambda }}(\sigma +aV+bW,\sigma )\right|}_{a=b=0},\hfill \end{array}$$

(240)

$$\begin{array}{cc}\hfill {\mathsf{\Delta }}_{\mathsf{\Lambda }}(\sigma +ϵV,\sigma )& =\frac{{ϵ}^2}{2}{g}_{\mathsf{\Lambda },\sigma }^{\mathrm{loss}}(V,V)+\frac{{ϵ}^3}{6}{\mathcal{C}}_{\mathsf{\Lambda },\sigma }^{\mathrm{loss}}(V,V,V)+O\left({ϵ}^4\right),\hfill \end{array}$$

(241)

locally uniformly on compact faithful sets for bounded tangents. Since the integrand in Equation (239) is continuous and nonnegative,

$${\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma )=0⟺g_{\mathsf{\Lambda },{\rho }_t}^{\mathrm{loss}}(Z,Z)=0\mathit{for}\mathit{every}t\in [0,1].$$

(242)

In particular, ${\mathcal{H}}_{\mathsf{\Lambda }}$ is convex.

Although ${\mathcal{C}}_{\mathsf{\Lambda }}^{\mathrm{loss}}$ is symmetric, it has no positivity property, and $g_{\mathsf{\Lambda }}^{\mathrm{loss}}$ may be degenerate. Thus the full input manifold has no loss Levi-Civita or Born geometry in general. On the input manifold, such a geometry requires a declared open restriction on which $g_{\mathsf{\Lambda }}^{\mathrm{loss}}>0$. Alternatively, descent through its null distribution requires separate constant-rank, integrability, and projectability hypotheses and a declared quotient carrier. Either route defines a new Hessian geometry with cubic ${\mathcal{C}}_{\mathsf{\Lambda }}^{\mathrm{loss}}$; even then, no bare input-minus-output curvature identity follows.

The proof is included with the channel-Hessian calculation in Appendix F.5.

Although ${\mathcal{H}}_{\mathsf{\Lambda }}$ is convex, it need not be nonnegative for a nonunital channel; the invariant statement is $B_{{\mathcal{H}}_{\mathsf{\Lambda }}}={\mathsf{\Delta }}_{\mathsf{\Lambda }}\ge 0$. The weighted identity upgrades diagonal metric contraction to the exact global loss and, together with Equation (A261), characterizes pairwise recovery by vanishing chordwise loss. Related entropy-gain convexity and recoverability bounds appear in [66,67]; the exact Bregman/chord form is the one used here.

Proposition 9

(BKM loss amplitude and fiberwise isometric completion). Fix $\rho \in {\mathcal{S}}_n^{+}$, put $\overline{\rho }:=\mathsf{\Lambda }\left(\rho \right)$ on the fixed output support $P_{\mathsf{\Lambda }}$, and write

$$L_{\rho }:=\mathrm{d}{\mathsf{\Lambda }}_{\rho }:T_{\rho }{\mathcal{S}}_n^{+}⟶T_{\overline{\rho }}{\mathcal{S}}_{P_{\mathsf{\Lambda }}}^{+}.$$

(243)

Let $L_{\rho }^{♯}$ be its BKM adjoint, defined by

$$g_{\rho }^{\mathrm{BKM}}(V,L_{\rho }^{♯}W)=g_{\overline{\rho }}^{\mathrm{BKM}}(L_{\rho }V,W).$$

(244)

BKM monotonicity is equivalent to $0⪯L_{\rho }^{♯}{L}_{\rho }⪯I$. Define the positive loss amplitude

$${\mathsf{E}}_{\mathsf{\Lambda },\rho }:={\left(I-L_{\rho }^{♯}{L}_{\rho }\right)}^{1/2}.$$

(245)

Then

$$\begin{array}{|c|}\hline g_{\mathsf{\Lambda },\rho }^{\mathrm{loss}}(V,W)=g_{\rho }^{\mathrm{BKM}}({\mathsf{E}}_{\mathsf{\Lambda },\rho }V,{\mathsf{E}}_{\mathsf{\Lambda },\rho }W).\\ \hline\end{array}$$

(246)

Consequently

$${\mathcal{J}}_{\mathsf{\Lambda },\rho }V:=(L_{\rho }V,{\mathsf{E}}_{\mathsf{\Lambda },\rho }V)$$

(247)

is a fiberwise isometry from the input BKM tangent space into the output-plus-defect direct sum:

$${\mathcal{J}}_{\mathsf{\Lambda },\rho }^{♯}{\mathcal{J}}_{\mathsf{\Lambda },\rho }=I.$$

(248)

This is a tangent-Hilbert-space completion of metric contraction, not a Stinespring dilation of the CPTP map.

For the mixture chord in 12, the global loss therefore becomes

$$\begin{array}{|c|}\hline {\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma )={\int }_0^1(1-t){∥{\mathsf{E}}_{\mathsf{\Lambda },{\rho }_t}(\rho -\sigma )∥}_{{\rho }_t,\mathrm{BKM}}^2\mathrm{d}t.\\ \hline\end{array}$$

(249)

For faithful endpoints, continuity and Petz equality give the exact chain

$$\begin{array}{|c|}\hline \mathit{Petz}\mathit{recovery}\mathit{of}(\rho ,\sigma )⟺{\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma )=0⟺{\mathsf{E}}_{\mathsf{\Lambda },{\rho }_t}(\rho -\sigma )=0(0\le t\le 1).\\ \hline\end{array}$$

(250)

The same Petz map recovers the mixture chord by linearity, but the local statement here concerns only its chord direction.

Proof. 

The operator inequality is the adjoint form of Equation (235); subtracting from the identity and taking the positive square root gives the Gram factorization and the isometry. Substitute it into Equation (239). A continuous nonnegative integrand has zero weighted integral precisely when it vanishes throughout the chord, and Equation (A261) supplies the first equivalence. □

Remark 6

(Regularity and geometry of the completion). The positive square root in Equation (245) is canonical pointwise. Smoothness as a bundle map is not automatic across rank changes of $I-L_{\rho }^{♯}{L}_{\rho }$; constant rank is a sufficient local hypothesis. If a smooth completion is placed in a fixed metric bundle with a chosen ambient connection, the covariant derivative of its image projector measures variation of the completed subbundle, not the static loss amplitude itself; no identity equating its curvature with a bare input-minus-output BKM curvature is asserted.

Corollary 13

(Sequential composition of channel defects). Let ${\mathsf{\Lambda }}_1:{\mathrm{M}}_n\left(\mathbb{C}\right)\to {\mathrm{M}}_k\left(\mathbb{C}\right)$ and ${\mathsf{\Lambda }}_2:{\mathrm{M}}_k\left(\mathbb{C}\right)\to {\mathrm{M}}_{\ell }\left(\mathbb{C}\right)$ be CPTP, and put

$$P_1:=supp{\mathsf{\Lambda }}_1\left(I_n\right),P_{21}:=supp({\mathsf{\Lambda }}_2\circ {\mathsf{\Lambda }}_1)\left(I_n\right)=supp{\mathsf{\Lambda }}_2\left(P_1\right).$$

(251)

In every second-stage term, restrict ${\mathsf{\Lambda }}_2$ to the CPTP map

$$P_1{\mathrm{M}}_k\left(\mathbb{C}\right)P_1⟶P_{21}{\mathrm{M}}_{\ell }\left(\mathbb{C}\right)P_{21},X⟼{\mathsf{\Lambda }}_2\left(X\right),$$

(252)

and form all entropy potentials, BKM tensors, adjoints, and tangent maps on these stagewise reduced carriers. Let ${\nabla }_0^{\left(\mathrm{m}\right)}$ and ${\nabla }_1^{\left(\mathrm{m}\right)}$ denote the flat mixture connections on the first and intermediate carriers. Then the endpoint defects telescope exactly. At potential level,

$${\mathcal{H}}_{{\mathsf{\Lambda }}_2\circ {\mathsf{\Lambda }}_1}={\mathcal{H}}_{{\mathsf{\Lambda }}_1}+{\mathcal{H}}_{{\mathsf{\Lambda }}_2}\circ {\mathsf{\Lambda }}_1,$$

(253)

and hence

$$\begin{array}{|c|}\hline {\mathsf{\Delta }}_{{\mathsf{\Lambda }}_2\circ {\mathsf{\Lambda }}_1}(\rho ,\sigma )={\mathsf{\Delta }}_{{\mathsf{\Lambda }}_1}(\rho ,\sigma )+{\mathsf{\Delta }}_{{\mathsf{\Lambda }}_2}({\mathsf{\Lambda }}_1\rho ,{\mathsf{\Lambda }}_1\sigma ).\\ \hline\end{array}$$

(254)

Put ${\rho }_1:={\mathsf{\Lambda }}_1\left(\rho \right)$ and let $L_i$ denote the corresponding tangent maps. Locally,

$$\begin{array}{|c|}\hline g_{{\mathsf{\Lambda }}_2\circ {\mathsf{\Lambda }}_1,\rho }^{\mathrm{loss}}(V,W)=g_{{\mathsf{\Lambda }}_1,\rho }^{\mathrm{loss}}(V,W)+g_{{\mathsf{\Lambda }}_2,{\rho }_1}^{\mathrm{loss}}(L_{1}V,L_{1}W).\\ \hline\end{array}$$

(255)

and the cubic-loss tensors obey

$$\begin{array}{|c|}\hline {\mathcal{C}}_{{\mathsf{\Lambda }}_2\circ {\mathsf{\Lambda }}_1,\rho }^{\mathrm{loss}}={\mathcal{C}}_{{\mathsf{\Lambda }}_1,\rho }^{\mathrm{loss}}+{\left({\mathsf{\Lambda }}_1^{\ast }{\mathcal{C}}_{{\mathsf{\Lambda }}_2}^{\mathrm{loss}}\right)}_{\rho }.\\ \hline\end{array}$$

(256)

Equivalently, evaluation on $(V,W,Z)$ replaces the final term by ${\mathcal{C}}_{{\mathsf{\Lambda }}_2,{\rho }_1}^{\mathrm{loss}}(L_{1}V,L_{1}W,L_{1}Z)$. More generally, for every fixed derivative order $r\ge 2$,

$${\left({\nabla }_0^{\left(\mathrm{m}\right)}\right)}^r{\mathcal{H}}_{{\mathsf{\Lambda }}_2\circ {\mathsf{\Lambda }}_1}={\left({\nabla }_0^{\left(\mathrm{m}\right)}\right)}^r{\mathcal{H}}_{{\mathsf{\Lambda }}_1}+{\mathsf{\Lambda }}_1^{\ast }{\left({\nabla }_1^{\left(\mathrm{m}\right)}\right)}^r{\mathcal{H}}_{{\mathsf{\Lambda }}_2}.$$

(257)

Thus the diagonal defect jets form an additive contravariant cocycle for fixed affine channels. This statement does not extend without additional Faà di Bruno terms to nonlinear or state-dependent coarse grainings, nor smoothly across parameter values at which a stagewise support changes. If ${\mathsf{E}}_i^2=I-L_i^{♯}{L}_i$ and ${\mathsf{E}}_{21}^2=I-{\left(L_2{L}_1\right)}^{♯}\left(L_2{L}_1\right)$, then

$$\begin{array}{|c|}\hline {\mathsf{E}}_{21}^2={\mathsf{E}}_1^2+L_1^{♯}{\mathsf{E}}_2^2{L}_1.\\ \hline\end{array}$$

(258)

Thus $V\mapsto ({\mathsf{E}}_{1}V,{\mathsf{E}}_2{L}_{1}V)$ is a Gram amplitude for the composite loss, and

$$V⟼(L_2{L}_{1}V,{\mathsf{E}}_2{L}_{1}V,{\mathsf{E}}_{1}V)$$

(259)

is a fiberwise isometry. There is no additional mixed term.

Proof. 

Because ${\mathsf{\Lambda }}_1\left(I_n\right)$ is positive definite on $P_1$, constants $0<c\le C$ satisfy $c{P}_1⪯{\mathsf{\Lambda }}_1\left(I_n\right)⪯C{P}_1$. Positivity of ${\mathsf{\Lambda }}_2$ gives

$$c{\mathsf{\Lambda }}_2\left(P_1\right)⪯({\mathsf{\Lambda }}_2\circ {\mathsf{\Lambda }}_1)\left(I_n\right)⪯C{\mathsf{\Lambda }}_2\left(P_1\right),$$

which proves the support equality in Equation (251); the same comparison shows that every second-stage image of a faithful input is faithful on $P_{21}$. Insert and subtract the intermediate entropy potential to obtain Equation (253), or insert and subtract $D_{\mathrm{U}}({\mathsf{\Lambda }}_1\rho \parallel {\mathsf{\Lambda }}_1\sigma )$ for the endpoint identity. Because ${\mathsf{\Lambda }}_1$ is affine, differentiating the potential identity proves the metric and cubic laws (255) and (256). Repeated flat covariant differentiation gives (257) without mixed chain-rule terms. The amplitude identity follows by expanding $I-L_1^{♯}{L}_2^{♯}{L}_2{L}_1$. □

Proposition 10

(Common recovery preserves intrinsic contrast geometry). Let $\mathcal{S}\subset {\mathcal{S}}_n^{+}$ be a smooth embedded faithful statistical model, and assume $\mathcal{N}:=\mathsf{\Lambda }\left(\mathcal{S}\right)$ is an embedded submanifold of the reduced faithful output-state space. Suppose one CPTP recovery map

$$\mathcal{R}:P_{\mathsf{\Lambda }}{\mathrm{M}}_m\left(\mathbb{C}\right)P_{\mathsf{\Lambda }}⟶{\mathrm{M}}_n\left(\mathbb{C}\right)$$

satisfies $\mathcal{R}\mathsf{\Lambda }\left(\rho \right)=\rho$ for every $\rho \in \mathcal{S}$. Then ${f:=\mathsf{\Lambda }|}_{\mathcal{S}}:\mathcal{S}\to \mathcal{N}$ is a diffeomorphism, with

$${\mathrm{d}\mathcal{R}\mathrm{d}\mathsf{\Lambda }|}_{T\mathcal{S}}=I,D_{\mathrm{U}}(\rho \parallel \sigma )=D_{\mathrm{U}}(\mathsf{\Lambda }\rho \parallel \mathsf{\Lambda }\sigma )(\rho ,\sigma \in \mathcal{S}).$$

(260)

Differentiating this two-point contrast identity shows that ${\mathsf{\Lambda }|}_{\mathcal{S}}$ preserves the intrinsic metric, cubic, and dual connections induced on $\mathcal{S}$ and $\mathsf{\Lambda }\left(\mathcal{S}\right)$. In particular, for either induced connection,

$$\mathrm{d}\mathsf{\Lambda }\left(R^{\mathcal{S}}(X,Y)Z\right)=R^{\mathsf{\Lambda }\left(\mathcal{S}\right)}(\mathrm{d}\mathsf{\Lambda }X,\mathrm{d}\mathsf{\Lambda }Y)\mathrm{d}\mathsf{\Lambda }Z,$$

(261)

and intrinsic parallel transports are intertwined. At each ${\rho }_0\in \mathcal{S}$ the restricted and full based holonomy groups satisfy

$$\begin{array}{cc}\hfill {Hol}_{\mathsf{\Lambda }\left({\rho }_0\right)}^0\left(\mathcal{N}\right)& =\mathrm{d}{\mathsf{\Lambda }}_{{\rho }_0}{Hol}_{{\rho }_0}^0\left(\mathcal{S}\right)\mathrm{d}{\mathsf{\Lambda }}_{{\rho }_0}^{-1},\hfill \\ \hfill {Hol}_{\mathsf{\Lambda }\left({\rho }_0\right)}\left(\mathcal{N}\right)& =\mathrm{d}{\mathsf{\Lambda }}_{{\rho }_0}{Hol}_{{\rho }_0}\left(\mathcal{S}\right)\mathrm{d}{\mathsf{\Lambda }}_{{\rho }_0}^{-1}.\hfill \end{array}$$

(262)

This is an intrinsic statement about the two models, not an identification with the ambient output-state curvature.

If, in addition, $\mathcal{S}$ and $\mathcal{N}$ are mixture-affine Hessian models and f is affine for their restricted mixture structures, then f is a Hessian isometry. Its tangent lift intertwines their intrinsic mixture Born tuples and, by metric duality, their exponential Born tuples. This extra conclusion is an affine-model statement, not tensorial descent of the ambient input Born tuple through a general channel.

Proof from contrast naturality. 

Data processing for $\mathsf{\Lambda }$ and its common recovery map gives the two-point equality; differentiating ${\mathcal{R}\mathsf{\Lambda }|}_{\mathcal{S}}=\mathrm{id}$ gives the tangent inverse. A contrast-preserving diffeomorphism preserves the metric and dual connections obtained from its diagonal jets [1,48]; curvature, transport, and holonomy then follow by naturality. Under the additional affine-Hessian hypothesis, Lemma 1 gives both Born-tuple intertwinings. □

Corollary 14

(One-reference criterion for model-wide recovery). Under the embedded-model hypotheses above, fix ${\sigma }_0\in \mathcal{S}$. If

$${\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,{\sigma }_0)=0\mathit{for}\mathit{every}\rho \in \mathcal{S},$$

(263)

then the single Petz map ${\mathcal{R}}_{{\sigma }_0,\mathsf{\Lambda }}$ recovers every state in $\mathcal{S}$, and all conclusions of Proposition 10 follow. Pairwise vanishing with reference states allowed to depend on ρ is insufficient: the common recovery map is the rigidity hypothesis.

Proof from Petz equality. 

On the stated faithful supports, equality in data processing for each pair $(\rho ,{\sigma }_0)$ is equivalent to recovery of both states by the Petz map constructed from the fixed pair $({\sigma }_0,\mathsf{\Lambda })$ [12,13]. The reference is common, hence so is the recovery map, and Proposition 10 gives the claimed model geometry and transport. □

Common recovery strengthens the pairwise chord-amplitude criterion in Equation (250) to the model-wide intrinsic transport statement of Proposition 10.

Theorem 9

(Dual partition data processing and response decomposition). To avoid collision with the partition symbol, let $\mathcal{N}:{\mathrm{M}}_n\left(\mathbb{C}\right)\to {\mathrm{M}}_m\left(\mathbb{C}\right)$ be a CPTP map, fix $\sigma \in {\mathcal{S}}_n^{+}$, and work on $P_{\mathcal{N}}:=supp\mathcal{N}\left(I\right)$, where $\overline{\sigma }:=\mathcal{N}\left(\sigma \right)$ is faithful. Let ${\mathcal{N}}^{†}$ denote the trace adjoint. For every Hermitian source K on $P_{\mathcal{N}}{\mathbb{C}}^m$, define

$${\rho }_K:=E_{\sigma }\left({\mathcal{N}}^{†}K\right),{\omega }_K:=E_{\overline{\sigma }}\left(K\right).$$

(264)

Here both $E_{\tau }\left(A\right)$ and ${\mathsf{\Lambda }}_{\tau }\left(A\right)$ denote their Hermitian-source extensions, so $E_{\tau }(A+cI)=E_{\tau }\left(A\right)$ and ${\mathsf{\Lambda }}_{\tau }(A+cI)={\mathsf{\Lambda }}_{\tau }\left(A\right)+c$. On the reduced output the identity is $P_{\mathcal{N}}$ and ${\mathcal{N}}^{†}\left(P_{\mathcal{N}}\right)=I$, so the partition defect below is gauge invariant. Write ${\mathsf{\Delta }}_{\mathcal{N}}$ for the contraction defect in Equation (227) with $\mathsf{\Lambda }=\mathcal{N}$. The ambient partition defect has the exact nonlinear decomposition

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill {\delta }_{\mathcal{N},\sigma }^{\mathrm{part}}\left(K\right)& :={\mathsf{\Lambda }}_{\overline{\sigma }}\left(K\right)-{\mathsf{\Lambda }}_{\sigma }\left({\mathcal{N}}^{†}K\right)\hfill \\ \hfill & =D_{\mathrm{U}}(\mathcal{N}\left({\rho }_K\right)\parallel {\omega }_K)+{\mathsf{\Delta }}_{\mathcal{N}}({\rho }_K,\sigma )\ge 0.\hfill \end{array}\\ \hline\end{array}$$

(265)

Thus output partition response dominates the pulled-back input response. Equality holds precisely when $\mathcal{N}\left({\rho }_K\right)={\omega }_K$ and DPI is saturated for $({\rho }_K,\sigma )$; for faithful pairs, the latter condition is equivalent to Petz recovery [12,13].

Let ${\mathcal{Q}}_{\mathcal{N}}:=\mathcal{N}\left({\mathcal{S}}_n^{+}\right)$ and define its image-restricted ambient partition by

$${\mathsf{\Lambda }}_{\overline{\sigma }}^{{\mathcal{Q}}_{\mathcal{N}}}\left(K\right):=\underset{q\in {\mathcal{Q}}_{\mathcal{N}}}{sup}\{Tr\left(qK\right)-D_{\mathrm{U}}(q\parallel \overline{\sigma })\}.$$

(266)

Then

$${\mathsf{\Lambda }}_{\overline{\sigma }}\left(K\right)\ge {\mathsf{\Lambda }}_{\overline{\sigma }}^{{\mathcal{Q}}_{\mathcal{N}}}\left(K\right)\ge {\mathsf{\Lambda }}_{\sigma }\left({\mathcal{N}}^{†}K\right),{\mathsf{\Lambda }}_{\overline{\sigma }}\left(K\right)-{\mathsf{\Lambda }}_{\overline{\sigma }}^{{\mathcal{Q}}_{\mathcal{N}}}\left(K\right)=\underset{q\in {\mathcal{Q}}_{\mathcal{N}}}{inf}{D}_{\mathrm{U}}(q\parallel {\omega }_K).$$

(267)

The infimum need not be attained on the relatively open image; on the natural restricted Legendre domain it is the corresponding entropic projection.

The Hessian of Equation (265) at $K=0$ gives an exact local dual decomposition. Define the source-to-density response

$${\mathsf{R}}_{\tau }\left(A\right):={\mathsf{K}}_{\tau }(A-Tr\left(\tau A\right)I),V_K:={\mathsf{R}}_{\sigma }\left({\mathcal{N}}^{†}K\right),W_K:={\mathsf{R}}_{\overline{\sigma }}\left(K\right).$$

(268)

Then

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill & {Cov}_{\overline{\sigma }}^{\mathrm{KM}}(K,K)-{Cov}_{\sigma }^{\mathrm{KM}}({\mathcal{N}}^{†}K,{\mathcal{N}}^{†}K)\hfill \\ \hfill & =g_{\overline{\sigma }}^{\mathrm{BKM}}(W_K-\mathcal{N}\left(V_K\right),W_K-\mathcal{N}\left(V_K\right))+g_{\sigma }^{\mathrm{BKM}}(V_K,V_K)-g_{\overline{\sigma }}^{\mathrm{BKM}}(\mathcal{N}\left(V_K\right),\mathcal{N}\left(V_K\right))\ge 0.\hfill \end{array}\\ \hline\end{array}$$

(269)

The first term is failure to intertwine infinitesimal mirror response; the remaining difference is exactly the tangent BKM metric-loss tensor of Equation (237).

Proof. 

The input variational equality and the output Fenchel gap give

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_{\sigma }\left({\mathcal{N}}^{†}K\right)& =Tr\left[\mathcal{N}\left({\rho }_K\right)K\right]-D_{\mathrm{U}}({\rho }_K\parallel \sigma ),\hfill \end{array}$$

(270)

$$\begin{array}{cc}\hfill D_{\mathrm{U}}(\mathcal{N}\left({\rho }_K\right)\parallel \overline{\sigma })+{\mathsf{\Lambda }}_{\overline{\sigma }}\left(K\right)-Tr\left[\mathcal{N}\left({\rho }_K\right)K\right]& =D_{\mathrm{U}}(\mathcal{N}\left({\rho }_K\right)\parallel {\omega }_K).\hfill \end{array}$$

(271)

Adding them proves Equation (265); DPI makes its second summand nonnegative. Applying the output Fenchel gap to each $q\in {\mathcal{Q}}_{\mathcal{N}}$ proves the first gap in Equation (267), while DPI applied before taking the input supremum proves the second inequality.

For the local formula, $g_{\overline{\sigma }}^{\mathrm{BKM}}(W_K,W_K)={Cov}_{\overline{\sigma }}^{\mathrm{KM}}(K,K)$ and $g_{\sigma }^{\mathrm{BKM}}(V_K,V_K)={Cov}_{\sigma }^{\mathrm{KM}}({\mathcal{N}}^{†}K,{\mathcal{N}}^{†}K)$. Since ${\mathsf{K}}_{\overline{\sigma }}^{-1}\left(W_K\right)=K-Tr\left(\overline{\sigma }K\right)I$ and $Tr\mathcal{N}\left(V_K\right)=0$, trace adjointness gives

$$g_{\overline{\sigma }}^{\mathrm{BKM}}(W_K,\mathcal{N}\left(V_K\right))=Tr\left[K\mathcal{N}\left(V_K\right)\right]=Tr\left[\left({\mathcal{N}}^{†}K\right)V_K\right]=g_{\sigma }^{\mathrm{BKM}}(V_K,V_K).$$

(272)

Expanding the squared mismatch proves Equation (269). □

Corollary 15

(Dual response as a direct-sum defect amplitude). With the notation of Theorem 9, define

$${\mathfrak{d}}_{\mathcal{N},\sigma }^{\mathrm{dual}}\left(K\right):=\left(W_K-\mathcal{N}\left(V_K\right),{\mathsf{E}}_{\mathcal{N},\sigma }{V}_K\right).$$

(273)

Equip the direct sum with the output BKM metric at $\overline{\sigma }=\mathcal{N}\left(\sigma \right)$ and the input BKM metric at σ. Then

$$\begin{array}{|c|}\hline {Cov}_{\overline{\sigma }}^{\mathrm{KM}}(K,K)-{Cov}_{\sigma }^{\mathrm{KM}}({\mathcal{N}}^{†}K,{\mathcal{N}}^{†}K)={∥{\mathfrak{d}}_{\mathcal{N},\sigma }^{\mathrm{dual}}\left(K\right)∥}^2.\\ \hline\end{array}$$

(274)

Thus mirror-response mismatch and ordinary tangent BKM loss are orthogonal components in an auxiliary direct sum. This orthogonality does not assert statistical independence of two random contributions.

Proof. 

Apply Equation (246) to the final two terms of Equation (269). □

Corollary 16

(Two-endpoint mirror-bridge channel defect). Let $\rho ,\sigma \in {\mathcal{S}}_n^{+}$, let $0<t<1$, and let ${\gamma }_t$ be their bridge from Equation (134). On the reduced output support, let ${\overline{\gamma }}_t$ be the corresponding bridge between $\mathcal{N}\left(\rho \right)$ and $\mathcal{N}\left(\sigma \right)$. Then

$$\begin{array}{|c|}\hline \begin{array}{c}{C}_t^{\mathrm{b}}(\rho ,\sigma )-C_t^{\mathrm{b}}(\mathcal{N}\left(\rho \right),\mathcal{N}\left(\sigma \right))\hfill \\ =(1-t){\mathsf{\Delta }}_{\mathcal{N}}({\gamma }_t,\sigma )+t{\mathsf{\Delta }}_{\mathcal{N}}({\gamma }_t,\rho )+D_{\mathrm{U}}(\mathcal{N}\left({\gamma }_t\right)\parallel {\overline{\gamma }}_t)\ge 0.\hfill \end{array}\\ \hline\end{array}$$

(275)

Thus the contraction of the scaled log-Euclidean/Umegaki-barycentric Rényi cost $C_t^{\mathrm{b}}=(1-t)D_t^{\mathrm{b}}$ for $0<t<1$ separates exactly into two endpoint DPI losses and failure of the channel to intertwine the finite mirror bridge. Dividing the identity by $1-t$ gives the corresponding decomposition for $D_t^{\mathrm{b}}$. In proximal variables, put

$$p_r:=P_r^{\sigma }\left(\rho \right),{\overline{p}}_r:=P_r^{\mathcal{N}\left(\sigma \right)}\left(\mathcal{N}\left(\rho \right)\right),r=\frac{1-t}{t}.$$

(276)

Then the same identity becomes the exact deformation defect

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill {\mathcal{M}}_r^{\sigma }\left(\rho \right)-{\mathcal{M}}_r^{\mathcal{N}\left(\sigma \right)}\left(\mathcal{N}\left(\rho \right)\right)=& {\mathsf{\Delta }}_{\mathcal{N}}(p_r,\sigma )+\frac{1}{r}{\mathsf{\Delta }}_{\mathcal{N}}(p_r,\rho )\hfill \\ \hfill & +\frac{1+r}{r}{D}_{\mathrm{U}}(\mathcal{N}\left(p_r\right)\parallel {\overline{p}}_r).\hfill \end{array}\\ \hline\end{array}$$

(277)

The last term is precisely the failure of the channel to intertwine the finite proximal deformation. Pinsker’s inequality gives the quantitative consequence

$${∥\mathcal{N}\left(p_r\right)-{\overline{p}}_r∥}_1^2\le \frac{2r}{1+r}\left[{\mathcal{M}}_r^{\sigma }\left(\rho \right)-{\mathcal{M}}_r^{\mathcal{N}\left(\sigma \right)}\left(\mathcal{N}\left(\rho \right)\right)\right].$$

(278)

Equality in the value defect holds precisely when all three displayed nonnegative terms vanish. This finite defect is the two-endpoint counterpart of Equation (265). The data-processing result used here is restricted to $0<t<1$; for every $t>1$ the log-Euclidean Rényi divergence fails CPTP monotonicity in general, including under suitable pinching maps [21], Lemma III.17.

The intertwining term can be strictly positive even for a faithful qubit pair and a noninjective channel. For $b>0$, take

$$\sigma =\frac{{\mathrm{e}}^{b{\sigma }_z}}{2coshb},\rho =\frac{{\mathrm{e}}^{b{\sigma }_x}}{2coshb},\mathcal{N}\left(A\right)=\frac{1}{2}(A+{\sigma }_{z}A{\sigma }_z).$$

(279)

At $t=1/2$,

$$\mathcal{N}\left({\gamma }_{1/2}\right)=\frac{1}{2}\left[I+\frac{tanh(b/\sqrt{2})}{\sqrt{2}}{\sigma }_z\right],{\overline{\gamma }}_{1/2}=\frac{1}{2}[I+tanh(b/2){\sigma }_z],$$

(280)

and these states differ for every $b>0$.

Proof. 

At the input minimizer, Equation (135) gives

$$C_t^{\mathrm{b}}(\rho ,\sigma )=(1-t)D_{\mathrm{U}}({\gamma }_t\parallel \sigma )+t{D}_{\mathrm{U}}({\gamma }_t\parallel \rho ).$$

(281)

Insert the two definitions of ${\mathsf{\Delta }}_{\mathcal{N}}$ and apply the output Pythagorean identity to $\mathcal{N}\left({\gamma }_t\right)$; this gives Equation (275). The qubit formulas follow by exponentiating the two Pauli-vector sources. Their Bloch coefficients are unequal because $x\mapsto tanhx/x$ is strictly decreasing on $(0,\infty )$. □

The global traceless log chart on the reduced support embeds ${\mathcal{Q}}_{\mathsf{\Lambda }}$ in its ambient output state space. The accessible centered score and norm are

$$\delta {\tilde{X}}_{\mathsf{\Lambda },\rho }\left(V\right)=-{\mathsf{K}}_{\mathsf{\Lambda }\left(\rho \right)}^{-1}\left(\mathsf{\Lambda }\left(V\right)\right),g_{\mathsf{\Lambda },\rho }^{\mathrm{obs}}(V,V)={\parallel \delta {\tilde{X}}_{\mathsf{\Lambda },\rho }\left(V\right)\parallel }_{\mathsf{\Lambda }\left(\rho \right),\mathrm{BKM}}^2.$$

(282)

It is centered with respect to $q=\mathsf{\Lambda }\left(\rho \right)$: $Tr\left[q\delta {\tilde{X}}_{\mathsf{\Lambda },\rho }\left(V\right)\right]=-Tr\mathsf{\Lambda }\left(V\right)=0$; its ordinary operator trace need not vanish. Thus precisely $kerL$ disappears. Unless ${\mathcal{Q}}_{\mathsf{\Lambda }}$ is exponential-autoparallel, the ambient exponential connection does not restrict to it. Intrinsically, however, Theorem 8 makes the quotient mixture-Hessian and dually flat, so $T{\mathcal{Q}}_{\mathsf{\Lambda }}$ carries ${\mathcal{B}}_{\mathsf{\Lambda },\mathrm{m}}$. The derivative and reduced-carrier formulas are in Appendix F.5.

Remark 7

(Intrinsic output lift versus tensorial descent). The tuple ${\mathcal{B}}_{\mathsf{\Lambda },\mathrm{m}}$ is constructed intrinsically from the output mixture-Hessian manifold. For a general channel it need not descend tensorially from either input tuple, and no such descent is asserted. If ${f:=\mathsf{\Lambda }|}_{{\mathcal{S}}_n^{+}}:{\mathcal{S}}_n^{+}\to {\mathcal{Q}}_{\mathsf{\Lambda }}$ is a Hessian isomorphism, meaning that it is a diffeomorphism satisfying

$$f^{\ast }{\overline{g}}_{\mathsf{\Lambda }}=g_{\mathrm{BKM}},Tf\left({\nabla }_X^{\left(\mathrm{m}\right)}Y\right)={\overline{\nabla }}_{Tf\left(X\right)}^{\left(\mathrm{m}\right)}Tf\left(Y\right),$$

(283)

then $Tf:T{\mathcal{S}}_n^{+}\to T{\mathcal{Q}}_{\mathsf{\Lambda }}$ intertwines ${\mathcal{B}}_{\mathrm{m}}$ and ${\mathcal{B}}_{\mathsf{\Lambda },\mathrm{m}}$ [15], Proposition 4.22. Identity and unitary-conjugation channels are examples. For a noninjective channel, descent of an input tensor requires the corresponding projectability conditions.

Noninjective and recovery benchmarks.

Complete qubit dephasing has $ker{L}_{{\mathsf{\Pi }}_z}={span}_{\mathbb{R}}\{{\sigma }_x,{\sigma }_y\}$ and ${\mathcal{Q}}_{{\mathsf{\Pi }}_z}\cong (-1,1)$, so its fibers are genuinely positive-dimensional; the exact defect and output geometry are computed in Appendix F.5.1. For faithful pairs, vanishing defect is equivalent to Petz recovery [12,13]. Quantitative small-defect recovery and stationary-semigroup decay require the separate additional hypotheses reviewed in Appendix F.5.3 [68,69,70]. Coset projection alone is not a channel; a CPTP reduction needs a supplied realization such as an ancillary partial trace.

5. Synthesis of the Unconditional Entropy-Channel Core

Theorem 10

(Unconditional input-output entropy and Born core). Fix integers $n\ge 2$ and $m\ge 1$ and supplied complex Hermitian carriers $(V_{\mathbb{R}},I_V,{\langle ·,·\rangle }_V)$ and $(W_{\mathbb{R}},I_W,{\langle ·,·\rangle }_W)$, with $V\simeq {\mathbb{C}}^n$, $W\simeq {\mathbb{C}}^m$, and $I_{V}v=\mathrm{i}v$, $I_{W}w=\mathrm{i}w$. Let $\mathsf{\Lambda }:{End}_{\mathbb{C}}\left(V\right)\to {End}_{\mathbb{C}}\left(W\right)$ be CPTP, and set $P_{\mathsf{\Lambda }}:=supp\mathsf{\Lambda }\left({\mathrm{id}}_V\right)$, $W_{\mathsf{\Lambda }}:=P_{\mathsf{\Lambda }}W$, and $I_{W_{\mathsf{\Lambda }}}:=I_W{|}_{{\left(W_{\mathsf{\Lambda }}\right)}_{\mathbb{R}}}$. Use the state, cone, Gibbs-chart, Umegaki, BKM, and Sakamoto conventions of Theorem 1, Corollary 2, and Equations (9) and (57)–(59). Then the following conclusions hold simultaneously.

1. 

The maps $\mathsf{\Phi }:{\mathcal{S}}_n^{+}\to {\mathcal{P}}_n^1$ and $\mathcal{X}=-log\circ \mathsf{\Phi }:{\mathcal{S}}_n^{+}\to \mathfrak{p}$ are global diffeomorphisms, and

$$D_{\mathrm{U}}({\rho }_X\parallel {\rho }_Y)=B_{\mathcal{F}}({\rho }_X,{\rho }_Y)=B_{\mathsf{\Psi }}(Y,X),g_{\mathrm{BKM}}={Hess}_{{\nabla }^{\left(\mathrm{m}\right)}}\mathcal{F}={Hess}_{{\nabla }^{\left(\mathrm{e}\right)}}(\mathsf{\Psi }\circ \mathcal{X}).$$

(284)

The mixture and exponential connections are BKM-dual and construct two strongly integrable input Born tuples on $T{\mathcal{S}}_n^{+}$, whose complex structures act on $T\left(T{\mathcal{S}}_n^{+}\right)$. They have the common cotangent carrier, horizontal shear, and midpoint mechanics of Remark 1 and Proposition 2.

2. 

On the reduced carrier $W_{\mathsf{\Lambda }}=P_{\mathsf{\Lambda }}W$, the image ${\mathcal{Q}}_{\mathsf{\Lambda }}=\mathsf{\Lambda }\left({\mathcal{S}}_n^{+}\right)$ is the smooth quotient of ${\mathcal{S}}_n^{+}$ by equality of channel outputs. With ${L=\mathsf{\Lambda }|}_{{Herm}_0\left(V\right)}$ and $E_{\mathsf{\Lambda }}=imL$,

$$T_q{\mathcal{Q}}_{\mathsf{\Lambda }}\simeq {Herm}_0\left(V\right)/kerL\simeq E_{\mathsf{\Lambda }},{\overline{g}}_{\mathsf{\Lambda },q}=g_q^{\mathrm{BKM}}{|}_{E_{\mathsf{\Lambda }}}={Hess}_{{\overline{\nabla }}^{\left(\mathrm{m}\right)}}{\mathcal{F}}_{\mathsf{\Lambda }}.$$

(285)

Thus $T{\mathcal{Q}}_{\mathsf{\Lambda }}$ carries its intrinsic mixture-Hessian Born tuple, whose constructed complex endomorphism $I_{B,\mathsf{\Lambda }}^{\mathrm{out}}\in \mathsf{\Gamma }(End\left(T\left(T{\mathcal{Q}}_{\mathsf{\Lambda }}\right)\right))$ acts on $T\left(T{\mathcal{Q}}_{\mathsf{\Lambda }}\right)$, together with the dual flat output connection. This includes the zero-dimensional case.

3. 

With ${\mathcal{H}}_{\mathsf{\Lambda }}=\mathcal{F}-{\mathcal{F}}_{\mathsf{\Lambda }}\circ \mathsf{\Lambda }$, the exact loss identities are

$${\mathsf{\Delta }}_{\mathsf{\Lambda }}=B_{{\mathcal{H}}_{\mathsf{\Lambda }}},{Hess}_{{\nabla }^{\left(\mathrm{m}\right)}}{\mathcal{H}}_{\mathsf{\Lambda }}=g_{\mathrm{BKM}}^{\mathrm{in}}-{\mathsf{\Lambda }}^{\ast }{g}_{\mathrm{BKM}}^{\mathrm{out}}⪰0,{\left({\nabla }^{\left(\mathrm{m}\right)}\right)}^3{\mathcal{H}}_{\mathsf{\Lambda }}={\mathcal{C}}^{\mathrm{in},\left(\mathrm{m}\right)}-{\mathsf{\Lambda }}^{\ast }{\mathcal{C}}^{\mathrm{out},\left(\mathrm{m}\right)}.$$

(286)

Their global-to-local accumulation and sequential cocycles are Corollaries 12 and 13.

The two input Born tuples and the selected mixture-polarized output tuple are separately typed. When $W=V$ and $\mathsf{\Lambda }={\mathrm{id}}_{{End}_{\mathbb{C}}\left(V\right)}$, the corresponding mixture tuples coincide; more generally, an affine Hessian isomorphism intertwines them as in Remark 7; a general noninjective channel supplies no such identification or tensorial descent. None of their complex endomorphisms is identified with the supplied $I_V$ or $I_{W_{\mathsf{\Lambda }}}$, and this core supplies no dynamical, mechanical, or measurement data.

Proof. 

The input statement is Propositions 2 and 5, Theorem 1, and Corollaries 2 and 3. The quotient and intrinsic output geometry are Theorem 8, and the loss statements are Theorem 8 and Corollaries 12 and 13. □

5.1. Measurement Channels and a Covariant Reference Readout

Corollary 17

(Finite POVMs as classical operational quotients). Let $E={\left\{E_a\right\}}_{a=1}^k$ be a finite POVM on V, with zero effects discarded, and define the quantum-to-classical channel

$${\mathcal{M}}_E\left(\rho \right):={\displaystyle \sum _{a=1}^k}{p}_a^E\left(\rho \right)|a\rangle \langle a|,p_a^E\left(\rho \right):=Tr\left(\rho E_a\right).$$

(287)

Then ${\mathcal{M}}_E$ is CPTP and maps faithful states into the interior of the classical probability simplex. Its tangent map and operational quotient are

$$L_E{\left(V\right)}_a=Tr\left(V{E}_a\right),ker{L}_E=\{V\in {Herm}_0\left(n\right):Tr\left(V{E}_a\right)=0\mathit{for}\mathit{every}a\},$$

(288)

$${\mathcal{Q}}_E:=p^E\left({\mathcal{S}}_n^{+}\right),dim{\mathcal{Q}}_E={dim}_{\mathbb{R}}span{\left\{E_a\right\}}_{a=1}^k-1.$$

(289)

For $p=p^E\left(\rho \right)$, $q=p^E\left(\sigma \right)$ and tangent vectors $\dot{p}=L_E\left(V\right)$, $\dot{q}=L_E\left(W\right)$, the output divergence and metric are

$$\begin{array}{cc}\hfill {\overline{D}}_E(\left[\rho \right]\parallel \left[\sigma \right])& ={\displaystyle \sum _a}{p}_{a}log\frac{p_a}{q_a},\hfill \end{array}$$

(290)

$$\begin{array}{cc}\hfill {\overline{g}}_{E,p}(\dot{p},\dot{q})& ={\displaystyle \sum _a}\frac{{\dot{p}}_a{\dot{q}}_a}{p_a},\hfill \end{array}$$

(291)

$$\begin{array}{cc}\hfill g_{E,\rho }^{\mathrm{obs}}(V,W)& ={\displaystyle \sum _a}\frac{Tr\left(V{E}_a\right)Tr\left(W{E}_a\right)}{Tr\left(\rho E_a\right)},g_E^{\mathrm{obs}}⪯g_{\mathrm{BKM}}.\hfill \end{array}$$

(292)

Thus the output BKM metric is precisely the Fisher metric restricted to ${\mathcal{Q}}_E$, and $T{\mathcal{Q}}_E$ has its intrinsic strongly integrable Sakamoto Born lift. The measurement-specific loss potential is

$${\mathcal{H}}_E\left(\rho \right):=Tr(\rho log\rho )-\sum _a{p}_a^E\left(\rho \right)log{p}_a^E\left(\rho \right).$$

(293)

By Corollary 12, ${\mathsf{\Delta }}_E:={\mathsf{\Delta }}_{{\mathcal{M}}_E}=B_{{\mathcal{H}}_E}$ and ${Hess}_{{\nabla }^{\left(\mathrm{m}\right)}}{\mathcal{H}}_E=g_{\mathrm{BKM}}-g_E^{\mathrm{obs}}⪰0$. The POVM is informationally complete exactly when ${span}_{\mathbb{R}}\left\{E_a\right\}=Herm\left(n\right)$, equivalently $ker{L}_E=0$. Informational completeness makes the probability record state-separating; it does not make ${\mathcal{M}}_E$ a reversible quantum channel or a BKM isometry.

For $n\ge 2$ the obstruction is necessarily strict: for every faithful σ there is a faithful ρ such that

$${\mathsf{\Delta }}_E(\rho ,\sigma )>0.$$

(294)

Consequently no finite POVM is globally entropy-lossless or preserves the BKM metric at every state in every tangent direction. For a separately supplied fixed maximal commutative unital ∗-subalgebra, its spectral PVM is lossless on its faithful simplex and identifies the restricted BKM metric with Fisher; see 18.

The proof, including the Petz-recovery obstruction to global losslessness, is given in Appendix F.1.

Corollary 18

(Totally geodesic classical sectors). Fix a maximal commutative unital ∗-subalgebra $\mathcal{A}\subset {\mathrm{M}}_n\left(\mathbb{C}\right)$ with rank-one minimal projections $P_1,\dots ,P_n$. Its full faithful simplex is

$${\mathcal{C}}_{\mathcal{A}}:=\left\{\rho \left(p\right)={\displaystyle \sum _{a=1}^n}{p}_a{P}_a:p_a>0,\sum _a{p}_a=1\right\}.$$

(295)

The associated spectral PVM is lossless on this simplex:

$$D_{\mathrm{U}}\left(\sum _a{p}_a{P}_a\parallel \sum _a{q}_a{P}_a\right)=\sum _a{p}_{a}log\frac{p_a}{q_a}.$$

(296)

For $U={\sum }_a{u}_a{P}_a$, $V={\sum }_a{v}_a{P}_a$, and $W={\sum }_a{w}_a{P}_a$, with ${\sum }_a{u}_a={\sum }_a{v}_a={\sum }_a{w}_a=0$, its restricted metric and mixture-affine cubic are

$$g_{\rho \left(p\right)}^{\mathrm{BKM}}(U,V)=\sum _a\frac{u_a{v}_a}{p_a},{\mathcal{C}}_{\rho \left(p\right)}^{\left(\mathrm{m}\right)}(U,V,W)=-\sum _a\frac{u_a{v}_a{w}_a}{p_a^2}.$$

(297)

The embedding is totally geodesic for the BKM Levi-Civita connection. Consequently, for $n\ge 3$ every tangent two-plane of this full simplex has both intrinsic and ambient BKM sectional curvature $1/4$.

Proof. 

On the fixed algebra, $\mathcal{F}\left(\rho \left(p\right)\right)={\sum }_a{p}_{a}log{p}_a$; its Bregman divergence and second and third derivatives give the displayed restrictions. The common fixed set of conjugations by ${\sum }_a{\mathrm{e}}^{\mathrm{i}{\theta }_a}{P}_a$ is precisely ${\mathcal{C}}_{\mathcal{A}}$. These conjugations are BKM isometries, so uniqueness of a geodesic with fixed initial point and tangent makes this set totally geodesic. The square-root map $p\mapsto 2(\sqrt{p_1},\dots ,\sqrt{p_n})$ is an isometry onto the positive part of the radius-2 sphere [1,6]; its curvature is $1/4$ when $n-1\ge 2$. For $n=2$ the simplex has no tangent two-plane. □

The algebra is fixed across the model: pointwise diagonalization in state-dependent bases is not a classical restriction. Total geodesy and constant curvature refer to the full simplex, not to arbitrary curved statistical submodels within it.

Remark 8

(Dilation does not select an instrument). Naimark’s theorem realizes every supplied POVM as the compression of a projective measurement, but a POVM fixes outcome statistics rather than a state-update instrument [71], Chapters 3-5. For example, arbitrary output states ${\omega }_a$ give the completely positive maps

$${\mathcal{I}}_a^{\left(\omega \right)}\left(X\right):=Tr\left(E_{a}X\right){\omega }_a$$

(298)

with the same outcome probabilities and trace-preserving sum. Thus the quotient geometry does not select apparatus dynamics or collapse.

Proposition 11

(Finite covariance obstruction and canonical ray POVM). Let $n\ge 2$ and let $(V_{\mathbb{R}},I_V,{\langle ·,·\rangle }_V)$ be the supplied complex Hermitian carrier.

1. 

A finite POVM selected naturally from only the Hermitian carrier and its unitary-invariant entropy geometry cannot be both informative and $\mathrm{U}\left(V\right)$-covariant. More precisely, if the finite outcome labels carry a continuous permutation action and $U{E}_a{U}^{†}=E_{U·a}$, then

$$E_a=c_{a}I,c_a\ge 0,\sum _a{c}_a=1,$$

(299)

so its probability law is state independent.

2. 

Let ${\mu }_{\mathrm{FS}}$ be the normalized $\mathrm{U}\left(V\right)$-invariant measure on $\mathbb{CP}\left(V\right)$ and let P denote the rank-one projector represented by a ray. Then

$$E_{\mathrm{ray}}\left(B\right):=n{\int }_{B}P\mathrm{d}{\mu }_{\mathrm{FS}}\left(P\right)$$

(300)

is the unique normalized $\mathrm{U}\left(V\right)$-covariant POVM whose Radon-Nikodym density at P is rank one and supported on that ray. Its outcome density

$$p_{\rho }\left(P\right):=nTr\left(\rho P\right)$$

(301)

is informationally complete, with reconstruction formula

$$\begin{array}{|c|}\hline \rho =(n+1){\int }_{\mathbb{CP}\left(V\right)}{p}_{\rho }\left(P\right)P\mathrm{d}{\mu }_{\mathrm{FS}}\left(P\right)-I.\\ \hline\end{array}$$

(302)

Its classical divergence and Fisher pullback satisfy

$$\begin{array}{cc}\hfill D_{\mathrm{ray}}(\rho \parallel \sigma )& :=\int p_{\rho }log\frac{p_{\rho }}{p_{\sigma }}\mathrm{d}{\mu }_{\mathrm{FS}}\le D_{\mathrm{U}}(\rho \parallel \sigma ),\hfill \end{array}$$

(303)

$$\begin{array}{cc}\hfill g_{\rho }^{\mathrm{ray}}(V,W)& :=n\int \frac{Tr\left(VP\right)Tr\left(WP\right)}{Tr\left(\rho P\right)}\mathrm{d}{\mu }_{\mathrm{FS}}\left(P\right),g^{\mathrm{ray}}⪯g_{\mathrm{BKM}}.\hfill \end{array}$$

(304)

At the maximally mixed state,

$$\begin{array}{|c|}\hline g_{I/n}^{\mathrm{ray}}=\frac{1}{n+1}{g}_{I/n}^{\mathrm{BKM}}.\\ \hline\end{array}$$

(305)

Thus the supplied carrier admits a distinguished continuous, informationally complete reference readout once the explicit rank-one ray-covariance criterion is imposed. Its projective complex structure is induced functorially from $I_V$, while its $\mathrm{U}\left(V\right)$ covariance uses the full supplied Hermitian pair $(I_V,{\langle ·,·\rangle }_V)$; neither comes from a Born structure. The instrument limitation is Remark 8.

The Haar-moment, measurable-partition, and Fisher-Hessian argument is given in Appendix F.1.

6. Shifted-Cone Geometry and Canonical Compact Holonomy

6.1. The Ambient Log-Determinant Hessian Structure

As an open convex subset of $Herm\left(n\right)$, the full cone ${\mathcal{P}}_n$ carries the canonical flat matrix-coordinate connection ${\nabla }^{\mathrm{aff}}$. Set

$$F\left(P\right)=-\alpha logdetP,\alpha >0.$$

(306)

Its Hessian metric and identity-point cubic are

$$\begin{array}{cc}\hfill g_P(U,V)& =\alpha Tr\left(P^{-1}U{P}^{-1}V\right),\hfill \end{array}$$

(307)

$$\begin{array}{cc}\hfill {\mathcal{C}}_I(U,V,W)& =-\alpha Tr\left(U\left\{V,W\right\}\right).\hfill \end{array}$$

(308)

The cubic $\mathcal{C}=D^{3}F$ is fully symmetric; the directional derivatives are recorded in Appendix D.3. Under the Nesterov-Todd self-scaling axioms, the logarithmic determinant is the unique barrier on this irreducible symmetric cone up to positive scale and an additive constant [72,73]; positivity alone does not imply this uniqueness.

• Ambient versus determinant-one geometry. Because $-logdetP$ is constant on ${\mathcal{P}}_n^1$, its restriction does not make that slice intrinsically Hessian. The dually flat structure belongs to ${\mathcal{P}}_n\cong \mathrm{GL}(n,\mathbb{C})/\mathrm{U}\left(n\right)$; below, its metric and cubic are restricted to slice tangents, and divergences between determinant-one points remain ambient Bregman divergences. Thus no dual flatness of $\mathrm{SL}(n,\mathbb{C})/\mathrm{SU}\left(n\right)$ is claimed from the restricted potential.

The ambient Bregman divergence of Equation (306) is the scaled LogDet (Burg) divergence [74]

$$D_{\mathrm{LD}}^{\left(\alpha \right)}(A\parallel B):=\alpha \left[Tr\left(B^{-1}A\right)-logdet\left(B^{-1}A\right)-n\right]=B_{-\alpha logdet}(A,B).$$

(309)

Frenkel’s identity-ray derivative and limit formulas [75], Lemmas 4-5, integrate to the first resolution below; his Theorem 6 instead gives a negative-spectral-part formula for affine matrix pencils. We retain the shifted LogDet-Bregman proof and metric counterpart to fix conventions for the subsequent determinant-one shape split, cotangent infimal convolution, pairwise curvature resolution, and endpoint constructions, which are not contained in that source.

Theorem 11

(Shifted LogDet resolution of relative entropy). For $A,B\in {\mathcal{P}}_n$, put

$${\mathcal{D}}_{+}(A\parallel B):=Tr\left[A(logA-logB)-A+B\right].$$

(310)

Then the improper integral below converges and

$$\begin{array}{|c|}\hline {\mathcal{D}}_{+}(A\parallel B)=\frac{1}{\alpha }{\int }_0^{\infty }{D}_{\mathrm{LD}}^{\left(\alpha \right)}(A+tI\parallel B+tI)\mathrm{d}t.\\ \hline\end{array}$$

(311)

In particular, for faithful states,

$$\begin{array}{|c|}\hline D_{\mathrm{U}}({\rho }_1\parallel {\rho }_0)=\frac{1}{\alpha }{\int }_0^{\infty }{D}_{\mathrm{LD}}^{\left(\alpha \right)}({\rho }_1+tI\parallel {\rho }_0+tI)\mathrm{d}t.\\ \hline\end{array}$$

(312)

At the infinitesimal level, if $g_X^{\mathrm{KM}}(V,W):=Tr\left[VD{log}_X\left[W\right]\right]$, then

$$\begin{array}{|c|}\hline g_X^{\mathrm{KM}}(V,W)={\int }_0^{\infty }Tr\left[{(X+tI)}^{-1}V{(X+tI)}^{-1}W\right]\mathrm{d}t=\frac{1}{\alpha }{\int }_0^{\infty }{g}_{X+tI}^{\mathrm{AI},\left(\alpha \right)}(V,W)\mathrm{d}t,\\ \hline\end{array}$$

(313)

where $g_P^{\mathrm{AI},\left(\alpha \right)}(U,V)=\alpha Tr\left(P^{-1}U{P}^{-1}V\right)$. On trace-one state tangents $g^{\mathrm{KM}}=g_{\mathrm{BKM}}$. Thus BKM geometry is a positive resolvent average of affine-invariant metrics at the shifted cone points $X+tI$, not the pullback of a single affine-invariant metric at X.

Corollary 19

(Optimal shifted-cone cotangent response). Fix $X\in {\mathcal{P}}_n$ and a finite-dimensional real tangent subspace $\mathcal{V}\subset Herm\left(n\right)$. All forms and inverses below are restricted to $\mathcal{V}$. For $t\ge 0$, define

$$a_{X,t}(U,W):=Tr\left[{(X+tI)}^{-1}U{(X+tI)}^{-1}W\right]={\alpha }^{-1}{g}_{X+tI}^{\mathrm{AI},\left(\alpha \right)}(U,W)$$

(314)

and let $a_{X,t}^{♭}:\mathcal{V}\to {\mathcal{V}}^{\ast }$ be its musical map; write $g_X^{\mathrm{KM},♭}$ similarly. Set ${\ell }_{X,t}\left(U\right):=Tr\left[{(X+tI)}^{-1}U\right]$ and $a_{X,t}^{\mathrm{sh}}:=a_{X,t}-n^{-1}{\ell }_{X,t}\otimes {\ell }_{X,t}$. If ${\widehat{X}}_t:=(X+tI)/{(det(X+tI))}^{1/n}$, then the exact shape-radial splitting is

$$a_{X,t}(U,W)=a_{X,t}^{\mathrm{sh}}(U,W)+\frac{1}{n}{\ell }_{X,t}\left(U\right){\ell }_{X,t}\left(W\right),$$

(315)

where $a_{X,t}^{\mathrm{sh}}$ is ${\alpha }^{-1}$ times the pullback of the determinant-one affine-invariant metric at ${\widehat{X}}_t$. Consequently, Equation (313) integrates both a determinant-one shape response and a radial determinant response.

For $p\in {\mathcal{V}}^{\ast }$, let $\mathfrak{A}\left(p\right)$ be the strongly measurable fields $t\mapsto p_t\in {\mathcal{V}}^{\ast }$ whose Bochner integral is ${\int }_0^{\infty }{p}_t\mathrm{d}t=p$ and whose quadratic energy below is finite. Then

$$\begin{array}{|c|}\hline {\left(g_X^{\mathrm{KM}}\right)}^{-1}(p,p)={\displaystyle \underset{p_{•}\in \mathfrak{A}\left(p\right)}{min}}{\int }_0^{\infty }{a}_{X,t}^{-1}(p_t,p_t)\mathrm{d}t.\\ \hline\end{array}$$

(316)

The minimizing field, unique up to equality almost everywhere, is

$$p_t=a_{X,t}^{♭}v,v={\left(g_X^{\mathrm{KM},♭}\right)}^{-1}p.$$

(317)

Thus the BKM cotangent kinetic energy is an exact continuous infimal convolution of shifted affine-invariant cotangent responses, rather than the generally false integral of their inverse metrics. Related matrix-metric and cometric structures are surveyed in [76].

Proof. 

Differentiating determinant normalization gives Equation (315). For the dual statement, put $v={\left(g_X^{\mathrm{KM},♭}\right)}^{-1}p$. The candidate $a_{X,t}^{♭}v=O\left({(1+t)}^{-2}\right)$ is Bochner integrable. Write any admissible field as $p_t=a_{X,t}^{♭}v+q_t$, so ${\int }_0^{\infty }{q}_t\mathrm{d}t=0$. Expanding the integrand gives

$${\int }_0^{\infty }{a}_{X,t}^{-1}(p_t,p_t)\mathrm{d}t={\left(g_X^{\mathrm{KM}}\right)}^{-1}(p,p)+{\int }_0^{\infty }{a}_{X,t}^{-1}(q_t,q_t)\mathrm{d}t,$$

(318)

because the cross term is $2(\int q_t\mathrm{d}t)\left(v\right)=0$. This proves the minimum and uniqueness almost everywhere. □

Proposition 12

(Pairwise resolvent representation of BKM curvature). Fix $Q\in {\mathcal{P}}_n$ and $U,V,W\in Herm\left(n\right)$, regarded as constant ambient tangent fields in affine matrix coordinates. Put $R_t:={(Q+tI)}^{-1}$, and use the full-cone metric $g_Q^{\mathrm{KM}}$ and shifted forms $a_{Q,t}$ from Equations (313) and (314). Let ${\mathcal{C}}_Q^{\mathrm{KM}}:=D{g}^{\mathrm{KM}}{|}_Q$ and

$$c_{Q,t}(U,V,W):={\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|}_{s=0}{a}_{Q+sU,t}(V,W).$$

(319)

Define the half-difference tensors by

$$\begin{array}{cc}\hfill g_Q^{\mathrm{KM}}(K_U^{\mathrm{KM}}V,W)& =-\frac{1}{2}{\mathcal{C}}_Q^{\mathrm{KM}}(U,V,W),\hfill \\ \hfill a_{Q,t}(J_U^{\left(t\right)}V,W)& =-\frac{1}{2}{c}_{Q,t}(U,V,W).\hfill \end{array}$$

(320)

Then

$$J_U^{\left(t\right)}V=\frac{1}{2}\left(U{R}_{t}V+V{R}_{t}U\right).$$

(321)

With

$${\mathsf{T}}_{Q,t}:={\left(g_Q^{\mathrm{KM},♭}\right)}^{-1}{a}_{Q,t}^{♭},{\mathsf{A}}_{Q,t}\left(U\right):={\mathsf{T}}_{Q,t}\circ J_U^{\left(t\right)},$$

(322)

one has the operator-norm convergent identities

$$\begin{array}{|c|}\hline {\int }_0^{\infty }{\mathsf{T}}_{Q,t}\mathrm{d}t=\mathrm{id},K_U^{\mathrm{KM}}={\int }_0^{\infty }{\mathsf{A}}_{Q,t}\left(U\right)\mathrm{d}t,\\ \hline\end{array}$$

(323)

and

$$\begin{array}{|c|}\hline R_Q^{\mathrm{KM}}(U,V)=-{\int }_0^{\infty }{\int }_0^{\infty }\left[{\mathsf{A}}_{Q,s}\left(U\right),{\mathsf{A}}_{Q,t}\left(V\right)\right]\mathrm{d}s\mathrm{d}t.\\ \hline\end{array}$$

(324)

For each fixed t,

$$R_{Q+tI}^{\mathrm{AI}}(U,V)=-[J_U^{\left(t\right)},J_V^{\left(t\right)}]$$

(325)

under affine translation of tangent spaces. Thus BKM curvature is an exact metric-weighted pairwise resolvent commutator, not an integral of the shifted affine-invariant curvatures: inverse-Hessian weighting and all pairs of scales are essential. Related direct matrix-curvature calculations appear in [27,28].

When $Q\in {\mathcal{S}}_n^{+}$, the same construction gives normalized-state BKM curvature after all forms and musical maps are restricted to ${Herm}_0\left(n\right)$ and $J_U^{\left(t\right)}V$ is replaced by its $a_{Q,t}$-orthogonal tangent projection

$${\mathsf{\Pi }}_{Q,t}Y:=Y-\frac{TrY}{Tr\left[{(Q+tI)}^2\right]}{(Q+tI)}^2.$$

(326)

More explicitly, on $\mathfrak{p}={Herm}_0\left(n\right)$ set

$${\mathsf{T}}_{Q,t}^0:={\left({\left(g_Q^{\mathrm{KM}}{|}_{\mathfrak{p}}\right)}^{♭}\right)}^{-1}{\left(a_{Q,t}{|}_{\mathfrak{p}}\right)}^{♭},{\mathsf{A}}_{Q,t}^0\left(U\right):={\mathsf{T}}_{Q,t}^0\circ {\mathsf{\Pi }}_{Q,t}\circ J_U^{\left(t\right)}.$$

(327)

Equations (323) and (324) then hold on the state slice with $\mathsf{T},\mathsf{A}$ replaced by ${\mathsf{T}}^0,{\mathsf{A}}^0$. The determinant-one component of each $a_{Q,t}$ is the pulled-back $\mathrm{SL}(n,\mathbb{C})/\mathrm{SU}\left(n\right)$ metric in Equation (315); the complementary radial term and the projection cannot generally be discarded. A derivation is given in Appendix D.4.

Corollary 20

(Radial and determinant-one shape resolution). For $A_i\in {\mathcal{P}}_n$, define

$$r_i:={(det{A}_i)}^{1/n},P_i:=r_i^{-1}{A}_i\in {\mathcal{P}}_n^1,q:=\frac{r_1}{r_0},$$

(328)

and

$$Z:=log\left(P_0^{-1/2}{P}_1{P}_0^{-1/2}\right)\in \mathfrak{p}.$$

(329)

Then

$$\begin{array}{|c|}\hline \frac{1}{\alpha }{D}_{\mathrm{LD}}^{\left(\alpha \right)}(A_1\parallel A_0)=n(q-1-logq)+q(Tr{\mathrm{e}}^Z-n).\\ \hline\end{array}$$

(330)

Both terms are nonnegative. For $A_i\left(t\right):={\rho }_i+tI$, define $r_i\left(t\right),P_i\left(t\right),q_t,Z_t$ in the same way. Equation (312) becomes

$$\begin{array}{|c|}\hline D_{\mathrm{U}}({\rho }_1\parallel {\rho }_0)={\int }_0^{\infty }\left[n(q_t-1-log{q}_t)+q_t(Tr{\mathrm{e}}^{Z_t}-n)\right]\mathrm{d}t.\\ \hline\end{array}$$

(331)

Thus the exact bridge resolves Umegaki entropy into a scalar radial part and determinant-one symmetric-space shape contrasts along the identity-shifted family.

Corollary 21

(Jeffreys control of shifted affine-invariant distance). For $A,B\in {\mathcal{P}}_n$, set $A_t:=A+tI$, $B_t:=B+tI$ and

$$d_{\mathrm{AI}}^{\left(\alpha \right)}{(A_t,B_t)}^2:=\alpha Tr\left[log{\left(A_t^{-1/2}{B}_t{A}_t^{-1/2}\right)}^2\right].$$

(332)

Then

$$\begin{array}{|c|}\hline {\mathcal{D}}_{+}(A\parallel B)+{\mathcal{D}}_{+}(B\parallel A)\ge \frac{1}{\alpha }{\int }_0^{\infty }{d}_{\mathrm{AI}}^{\left(\alpha \right)}{(A_t,B_t)}^2\mathrm{d}t.\\ \hline\end{array}$$

(333)

For states, the left-hand side is the Jeffreys-Umegaki divergence. Writing ${\widehat{A}}_t:=A_t/{(det{A}_t)}^{1/n}$ and similarly for $B_t$, the integrand splits exactly as

$$d_{\mathrm{AI}}^{\left(\alpha \right)}{(A_t,B_t)}^2=d_{\mathrm{AI}}^{\left(\alpha \right)}{({\widehat{A}}_t,{\widehat{B}}_t)}^2+\frac{\alpha }{n}{\left[logdet{A}_t-logdet{B}_t\right]}^2.$$

(334)

Thus symmetrized relative entropy globally controls the integrated determinant-one shape displacement and its radial complement. Equality in Equation (333) holds only for $A=B$, and the two sides have the same quadratic expansion at the diagonal.

Proof. 

Let ${\mathrm{e}}^{r_j\left(t\right)}$ be the generalized eigenvalues of $(B_t,A_t)$. Then the sum of the two shifted LogDet divergences divided by $\alpha$ is $2{\sum }_j(cosh{r}_j\left(t\right)-1)\ge {\sum }_j{r}_j{\left(t\right)}^2$. Integrating and using Theorem 11 proves Equation (333). Separating the scalar part of the matrix logarithm from its traceless part proves Equation (334). The equality and common-quadratic-expansion statements follow from $2(coshr-1)=r^2+O\left(r^4\right)$; pointwise, the gap is fourth order in the relative logarithm. □

The metric integral in Equation (313) commutes with the energy of any fixed path. It does not generally commute with minimization over paths: the infimum of an integral is only bounded below by the integral of the separate infima, with equality requiring one path to minimize almost every shifted affine-invariant energy. Therefore Equation (333) is an endpoint comparison, not an identification of BKM and symmetric-space principal actions.

Remark 9

(The cone Laplace origin). For $Y\in {\mathcal{P}}_n$ and $\beta >n-1$, the symmetric-cone gamma integral is

$${\mathcal{Z}}_{\beta }\left(Y\right):={\int }_{Q>0}{\mathrm{e}}^{-Tr\left(YQ\right)}{(detQ)}^{\beta -n}\mathrm{d}Q={\mathsf{\Gamma }}_n^{\mathbb{C}}\left(\beta \right){(detY)}^{-\beta },{\mathsf{\Gamma }}_n^{\mathbb{C}}\left(\beta \right):={\pi }^{n(n-1)/2}{\displaystyle \prod _{j=1}^n}\mathsf{\Gamma }(\beta -j+1).$$

(335)

Here $\mathrm{d}Q$ is the coordinate Lebesgue measure on $Herm\left(n\right)$; a different Euclidean normalization changes only the displayed constant. Hence $-\alpha logdetY=(\alpha /\beta )log{\mathcal{Z}}_{\beta }\left(Y\right)+\mathrm{const}$: the LogDet potential is a positive multiple of a cone log-Laplace function [77] Chapter VII. This proportionality does not assert that ${(detY)}^{-\alpha }$ is itself a positive-measure cone Laplace transform for every geometric $\alpha >0$. Together with Theorems 2 and 11, this yields two exact but differently organized partition structures: the trace-exponential partition is Legendre-dual to Umegaki entropy, whereas the determinant log-partition, after the displayed positive rescaling, generates the affine-invariant LogDet contrasts whose shifted resolvent average reconstructs it.

Corollary 22

(LogDet Type-II determinant generator). On the full positive cone, let $\kappa >0$, $P_0\in {\mathcal{P}}_n$, and let $\mathsf{\Pi }={\mathsf{\Pi }}^{†}$ satisfy

$$I-\frac{1}{\kappa \alpha }{P}_0^{1/2}\mathsf{\Pi }{P}_0^{1/2}>0.$$

(336)

Then the partial Legendre-Fenchel transform is attained uniquely and equals

$$\begin{array}{cc}\hfill G_{\mathrm{LD},\kappa }^{+}(P_0,\mathsf{\Pi })& :=\underset{P_1>0}{sup}\left\{Tr\left(\mathsf{\Pi }{P}_1\right)-\kappa D_{\mathrm{LD}}^{\left(\alpha \right)}(P_1\parallel P_0)\right\}\hfill \\ & =-\kappa \alpha logdet\left(I-\frac{1}{\kappa \alpha }{P}_0^{1/2}\mathsf{\Pi }{P}_0^{1/2}\right),\hfill \end{array}$$

(337)

with

$$P_1={\left(P_0^{-1}-\frac{1}{\kappa \alpha }\mathsf{\Pi }\right)}^{-1}.$$

(338)

Outside the strict positivity domain Equation (336), the supremum is $+\infty$. Near $\mathsf{\Pi }=0$,

$$G_{\mathrm{LD},\kappa }^{+}(P_0,\mathsf{\Pi })=Tr\left(P_0\mathsf{\Pi }\right)+\frac{1}{2\kappa \alpha }Tr\left[{\left(P_0^{1/2}\mathsf{\Pi }{P}_0^{1/2}\right)}^2\right]+O\left({∥\mathsf{\Pi }∥}^3\right).$$

(339)

This is the determinant analogue of the trace-exponential generator in Theorem 2. It generates the affine-invariant cometric on the full cone: more precisely, its Hessian at $\mathsf{\Pi }=0$ is ${\kappa }^{-1}$ times that cometric. Imposing $det{P}_1=1$ instead gives a constrained conjugate and not the closed formula above.

Proof. 

Differentiating the variational objective gives $P_1^{-1}=P_0^{-1}-\mathsf{\Pi }/\left(\kappa \alpha \right)$, whose positivity is exactly Equation (336). Substitution gives Equation (337); the convergent series for $-logdet(I-X)$ gives Equation (339). If the strict domain fails, let v be a unit eigenvector of $C:=P_0^{1/2}\mathsf{\Pi }{P}_0^{1/2}$ with eigenvalue $c\ge \kappa \alpha$ and take $Q_s:=I+(s-1)v{v}^{†}$, $P_1=P_0^{1/2}{Q}_s{P}_0^{1/2}$. The objective equals $TrC+(s-1)(c-\kappa \alpha )+\kappa \alpha logs$, which tends to $+\infty$ as $s\to \infty$. □

• Resolvent averaging does not identify the symmetries. Each $D_{\mathrm{LD}}^{\left(\alpha \right)}$ is congruence invariant, but its fixed central resolvent ray $tI$ is equivariant only under unitary congruence. Symmetrization preserves data processing of the two directed Umegaki terms without enlarging this symmetry.

6.2. Convention map between the log-determinant cubic and K

Choose $\nabla ={\nabla }^{\mathrm{aff}}$ as the primal flat connection and let ${\nabla }^{\ast }$ be its metric dual. In affine coordinates, ${\mathsf{\Gamma }}_{ijk}^{\ast }={\mathcal{C}}_{ijk}$ and ${\mathsf{\Gamma }}_{ijk}^{\mathrm{LC}}={\mathcal{C}}_{ijk}/2$. Since $K=(\nabla -{\nabla }^{\ast })/2$,

$$C_K(X,Y,Z)=g(K_{X}Y,Z)=-\frac{1}{2}\mathcal{C}(X,Y,Z).$$

(340)

Thus the odd Bregman expansion uses $\mathcal{C}=D^{3}F$, whereas $R^{\mathrm{LC}}=-[K,K]$ uses $C_K=-\mathcal{C}/2$ in this convention.

At the identity $K_{X}Y=\left\{X,Y\right\}/2$, so the Jordan-operator calculation in Appendix D.3 reproduces Equation (347). Thus the dual-flat log-determinant and symmetric-space calculations give the same curvature on the full positive cone. This concerns $g_{\mathrm{AI}}$ and its barrier cubic, not the BKM metric or entropy cubic.

6.3. Baker-Campbell-Hausdorff Composition and the Compact Polar Residue

For $x,y\in \mathfrak{p}$ and small $s,t$, apply the BCH series [78], Section 2, and the standard right polar decomposition ${\mathrm{e}}^{sx}{\mathrm{e}}^{ty}=P(s,t)U(s,t)$ [46,79]. Expanding the positive and unitary factors gives

$$\begin{array}{cc}\hfill log\left({\mathrm{e}}^{sx}{\mathrm{e}}^{ty}\right)& =sx+ty+\frac{st}{2}[x,y]+O\left(3\right),\hfill \end{array}$$

(341)

$$\begin{array}{cc}\hfill logP(s,t)& =sx+ty+O\left(3\right),\hfill \end{array}$$

(342)

$$\begin{array}{cc}\hfill logU(s,t)& =\frac{st}{2}[x,y]+O\left(3\right),\hfill \end{array}$$

(343)

where $P>0$, $U\in \mathrm{SU}\left(n\right)$, and $O\left(3\right)$ has total degree at least three in $(s,t)$. Thus noncommuting noncompact directions leave a compact residue. For embedded matrix tangents, the residue is $st[U,V]/8+O\left(3\right)$ because

$$\mathrm{d}{\pi }_e\left(x\right)=2x,U=2x,{\left(\mathrm{d}{s}_0\right)}_I\left(U\right)=E_I\left(U\right)=\frac{1}{2}U.$$

(344)

The normalization and antisymmetrized recovery formula are verified in Appendix D.1.1; the surrounding matrix and invariant-tensor calculations are collected in Appendix D. The factor does not change curvature span or holonomy.

6.4. The Invariant Metric and Nonpositive Curvature

At I, the isotropy action on $T_I{\mathcal{P}}_n^1=\mathfrak{p}$ is the irreducible adjoint representation, so the invariant positive inner product is unique up to scale. With

$$g_I(X,Y)=\alpha Tr\left(XY\right),\alpha >0.$$

(345)

congruence transport gives the standard affine-invariant metric [46]

$$g_P(U,V)=\alpha Tr\left(P^{-1}U{P}^{-1}V\right).$$

(346)

In the embedded normalization of Equation (344), standard symmetric-space curvature gives [26]

$$R_I(U,V)W=-\frac{1}{4}[[U,V],W],$$

(347)

for $U,V,W\in T_I{\mathcal{P}}_n^1$. The coefficient is independent of the overall metric scale $\alpha$, while sectional curvatures scale inversely with $\alpha$. Hence

$$g\left(R\right(U,V)V,U)\le 0,$$

with equality exactly when $[U,V]=0$. After normalization, $n=2$ gives $H^3$ with constant negative curvature. For $n>2$, commuting diagonal directions span flat planes; the space has rank $n-1$ and is not constant-curvature hyperbolic.

For $n=2$, the transvection bracket is the boost-boost source of the Thomas-Wigner compact residue. The explicit mass-shell and principal-log normalization is placed beside the boost-generator-rapidity result in Section 8.5.

Proposition 13

(Separation of the BKM and affine-invariant metrics). For $n\ge 2$, the pullback ${\mathsf{\Phi }}^{\ast }{g}_{\mathrm{AI}}$ under Equation (48) is not a state-independent multiple of the BKM metric. At the maximally mixed state ${\rho }_{\ast }=I/n$, the chosen normalizations satisfy

$${\left.{\mathsf{\Phi }}^{\ast }{g}_{\mathrm{AI}}\right|}_{{\rho }_{\ast }}=\alpha n{\left.g_{\mathrm{BKM}}\right|}_{{\rho }_{\ast }}.$$

(348)

The full-tangent calculation at ${\rho }_{\ast }$ and the diagonal-state counterexamples proving the proposition are given in Appendix D.5; retaining the statement here keeps the distinction between the two metrics explicit without interrupting the holonomy argument.

Theorem 12

(Legendre-Cartan curvature bridge and central decomposition). For $X,H,K,L\in \mathfrak{p}$, put

$${\rho }_X:=\frac{{\mathrm{e}}^{-X}}{Tr{\mathrm{e}}^{-X}},P_X:=\mathsf{\Phi }\left({\rho }_X\right)={\mathrm{e}}^{-X},{\mathsf{S}}_X:=sinhc\left(\frac{{ad}_X}{2}\right),$$

(349)

where $sinhcz:=sinh\left(z\right)/z$, continuously extended by $sinhc0=1$. Define the Legendre Hessian operator

$$\begin{array}{cc}\hfill {\mathsf{G}}_{X}H& :={\mathsf{K}}_{{\rho }_X}\left(H\right)-{\rho }_{X}Tr\left({\rho }_{X}H\right)\hfill \\ & ={\rho }_X^{1/2}{\mathsf{S}}_X\left(H\right){\rho }_X^{1/2}-{\rho }_{X}Tr\left({\rho }_{X}H\right).\hfill \end{array}$$

(350)

It is a positive isomorphism of $\mathfrak{p}$, and

$$D{\rho }_X\left[H\right]=-{\mathsf{G}}_{X}H,D{\mathsf{\Phi }}_{{\rho }_X}[-{\mathsf{G}}_{X}H]=D{P}_X\left[H\right].$$

(351)

Thus the determinant-one map factors as the Gibbs coordinate, which is Legendre-dual to Ψ, followed by the symmetric-space exponential. Its invertible whitened embedded-tangent map is

$${\mathsf{T}}_{X}H:=P_X^{-1/2}D{P}_X\left[H\right]P_X^{-1/2}=-{\mathsf{S}}_X\left(H\right).$$

(352)

In the quotient normalization of Equation (344), the corresponding Cartan solder form is ${\mathsf{E}}_X:={\mathsf{T}}_X/2=-{\mathsf{S}}_X/2$.

Let

$$g_X^{\mathsf{\Psi }}(H,K):=Tr\left(H{\mathsf{G}}_{X}K\right)=Hess{\mathsf{\Psi }}_X(H,K),{\mathsf{Q}}_X\left(H\right):={\mathsf{G}}_X^{-1}\circ D{\mathsf{G}}_X\left[H\right].$$

(353)

The Gibbs coordinate $\mathcal{X}\left(\rho \right)=-log\mathsf{\Phi }\left(\rho \right)$ is an isometry from the state BKM metric to $g^{\mathsf{\Psi }}$. Hence it exactly intertwines the two BKM curvature representations:

$$D{\mathcal{X}}_{\rho }\left(R_{\rho }^{\mathrm{BKM}}(V,W)Z\right)=R_X^{\mathsf{\Psi }}(D{\mathcal{X}}_{\rho }V,D{\mathcal{X}}_{\rho }W)D{\mathcal{X}}_{\rho }Z.$$

(354)

Here $\rho ={\rho }_X$ and $V,W,Z\in T_{\rho }{\mathcal{S}}_n^{+}$. If ${\overline{R}}^{\mathrm{AI}}$ denotes the pullback of the canonical affine-invariant curvature by $X\mapsto P_X$, then the two curvature operators have the exact common-coordinate representations

$$\begin{array}{cc}\hfill R_X^{\mathsf{\Psi }}(H,K)L& =-\frac{1}{4}[{\mathsf{Q}}_X\left(H\right),{\mathsf{Q}}_X\left(K\right)]L,\hfill \end{array}$$

(355)

$$\begin{array}{cc}\hfill {\overline{R}}_X^{\mathrm{AI}}(H,K)L& =-\frac{1}{4}{\mathsf{T}}_X^{-1}\left[[{\mathsf{T}}_{X}H,{\mathsf{T}}_{X}K],{\mathsf{T}}_{X}L\right].\hfill \end{array}$$

(356)

Consequently their global curvature defect is explicit but generally nonzero:

$${\mathsf{\Delta }}_X^{\mathrm{curv}}(H,K)L:=R_X^{\mathsf{\Psi }}(H,K)L-{\overline{R}}_X^{\mathrm{AI}}(H,K)L.$$

(357)

Equivalently, if $H=D{\mathcal{X}}_{\rho }V$, $K=D{\mathcal{X}}_{\rho }W$, $L=D{\mathcal{X}}_{\rho }Z$, and $P_X=\mathsf{\Phi }\left(\rho \right)$, then

$$\begin{array}{cc}\hfill & D{\mathsf{\Phi }}_{\rho }\left(R_{\rho }^{\mathrm{BKM}}(V,W)Z\right)-R_{P_X}^{\mathrm{AI}}\left(D{\mathsf{\Phi }}_{\rho }V,D{\mathsf{\Phi }}_{\rho }W\right)D{\mathsf{\Phi }}_{\rho }Z\hfill \\ \hfill & =D{P}_X\left[{\mathsf{\Delta }}_X^{\mathrm{curv}}(H,K)L\right].\hfill \end{array}$$

(358)

Thus Legendre duality and determinant normalization give an exactdefectivecurvature intertwiner. At the center, the residual after subtracting one quarter of the symmetric-space curvature is precisely the trace-normalization space-form term displayed below; on commuting planes it is the Fisher-simplex contribution.

At the maximally mixed state, $X=0$, write $g_0^{\mathsf{\Psi }}(H,K)=n^{-1}Tr\left(HK\right)$. Then

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill R_0^{\mathsf{\Psi }}(H,K)L& =\frac{1}{4}{\overline{R}}_0^{\mathrm{AI}}(H,K)L+\frac{1}{4}\left[g_0^{\mathsf{\Psi }}(K,L)H-g_0^{\mathsf{\Psi }}(H,L)K\right]\hfill \\ & =-\frac{1}{16}[[H,K],L]+\frac{1}{4n}\left[Tr\left(KL\right)H-Tr\left(HL\right)K\right].\hfill \end{array}\\ \hline\end{array}$$

(359)

For the calibration $\alpha =1/n$, the two metrics agree at the center, and therefore

$${sec}_{\mathrm{BKM},0}=\frac{1}{4}+\frac{1}{4}{sec}_{\mathrm{AI},0}.$$

(360)

In particular, for $n\ge 3$, independent commuting directions have BKM sectional curvature $1/4$ but affine-invariant sectional curvature zero. For $n=2$, the traceless Pauli identity makes the two terms in Equation (359) cancel, so $R_0^{\mathsf{\Psi }}=0$, whereas the symmetric space is negatively curved. Contracting the decomposition gives the central values

$$\begin{array}{cccc}\hfill {Ric}_0^{\mathsf{\Psi }}& =\frac{n^2-4}{8}{g}_0^{\mathsf{\Psi }},\hfill & \hfill {\overline{Ric}}_0^{\mathrm{AI}}& =-\frac{n^2}{2}{g}_0^{\mathsf{\Psi }},\hfill \\ \hfill {Scal}_0^{\mathsf{\Psi }}& =\frac{(n^2-1)(n^2-4)}{8}.\hfill \end{array}$$

(361)

in the normalization of [28,47]. Any real linear curvature conjugacy would preserve the trace defining the Ricci tensor. For $n\ge 3$ the two Ricci forms have opposite definite signs, and for $n=2$ the BKM form vanishes while the affine-invariant form is negative definite. Thus neither the Legendre-exponential factorization noranyreal linear isomorphism at the center conjugates the two curvature tensors: no $L\in \mathrm{GL}\left(\mathfrak{p}\right)$ satisfies

$$L\left(R_0^{\mathsf{\Psi }}(H,K)Z\right)={\overline{R}}_0^{\mathrm{AI}}(LH,LK)LZ\mathit{for}\mathit{all}H,K,Z.$$

(362)

In particular, there is no local connection isomorphism sending the BKM center to the affine-invariant center. The proof and normalization checks are in Appendix D.6.

Corollary 23

(Identity-soldered cubic first jet and raw transport defect). Let $D^{\mathsf{\Psi }}$ be the Levi-Civita connection of $g^{\mathsf{\Psi }}$, let ${\overline{D}}^{\mathrm{AI}}$ be the pullback by $X\mapsto P_X$ of the affine-invariant Levi-Civita connection, and put $\mathcal{A}:=D^{\mathsf{\Psi }}-{\overline{D}}^{\mathrm{AI}}$. The operator ${\mathsf{Q}}_X\left(H\right)$ is exactly the metric-raised entropy cubic:

$$g_X^{\mathsf{\Psi }}\left({\mathsf{Q}}_X\left(H\right)K,L\right)=D^3{\mathsf{\Psi }}_X(H,K,L),{\mathsf{\Delta }}^{\mathrm{curv}}=d^{{\overline{D}}^{\mathrm{AI}}}\mathcal{A}+[\mathcal{A},\mathcal{A}].$$

(363)

For $\alpha =1/n$, the bridge metrics agree at $X=0$, but

$${\left.\mathrm{d}{\overline{g}}^{\mathrm{AI}}\right|}_0=0,{\mathcal{A}}_0\left(H\right)K=\frac{1}{2}{\mathsf{Q}}_0\left(H\right)K=-\frac{1}{4}{\mathsf{\Pi }}_0\{H,K\}.$$

(364)

Hence for $n\ge 3$ the entropy cubic is simultaneously the first metric jet, connection, and Born-shear obstruction to the natural bridge. For $n=2$ this central cubic vanishes by the Pauli anticommutator identity, but the curvature and Ricci obstruction in Theorem 12 remains at the next jet.

The connection defect also gives an exact pathwise comparison. Along $\gamma :[0,T]\to \mathfrak{p}$, let $P_t^{\mathsf{\Psi }}$ and ${\overline{P}}_t^{\mathrm{AI}}$ denote the two parallel transports from $\gamma \left(0\right)$ to $\gamma \left(t\right)$. Then

$${\left({\overline{P}}_T^{\mathrm{AI}}\right)}^{-1}{P}_T^{\mathsf{\Psi }}=\mathcal{P}exp\left[-{\int }_0^T{\left({\overline{P}}_t^{\mathrm{AI}}\right)}^{-1}{\mathcal{A}}_{\gamma \left(t\right)}\left(\dot{\gamma }\left(t\right)\right){\overline{P}}_t^{\mathrm{AI}}\mathrm{d}t\right].$$

(365)

For a sufficiently small coordinate rectangle ${\gamma }_{\epsilon }$ based at X with initial sides $\epsilon H,\epsilon K$,

$$\left(P_{{\gamma }_{\epsilon }}^{\mathsf{\Psi }}-{\overline{P}}_{{\gamma }_{\epsilon }}^{\mathrm{AI}}\right)L=-{\epsilon }^2{\mathsf{\Delta }}_X^{\mathrm{curv}}(H,K)L+O\left({\epsilon }^3\right).$$

(366)

Thus the natural bridge relates the transports by a path-ordered defect, not by endpoint conjugacy. Because this comparison uses the identity soldering while the two connections preserve different metrics, $\mathcal{A}$ need not be $g^{\mathsf{\Psi }}$-skew and the displayed relative map need not be orthogonal. It is therefore a raw connection diagnostic, not yet a compact relative transporter.

The proof is included in Appendix D.6.

6.5. Compact Duality and the Canonical Reductive Connection

Standard symmetric-space duality gives [26]

$${\left(\mathrm{SL}(n,\mathbb{C})/\mathrm{SU}\left(n\right)\right)}^c\simeq \frac{\mathrm{SU}\left(n\right)\times \mathrm{SU}\left(n\right)}{\mathrm{SU}{\left(n\right)}_{diag}}\simeq \mathrm{SU}\left(n\right).$$

(367)

With quotient coordinate $x=\mathrm{i}A\in \mathfrak{p}$, embedded tangent $U=2x$, and $A\in \mathfrak{k}$, distinguish

$$\begin{array}{cccc}\hfill c_q:{\mathfrak{p}}_{\mathrm{quot}}& ⟶{\mathfrak{k}}_{\mathrm{comp}},\hfill & \hfill c_q\left(\mathrm{i}A\right)& =2A,\hfill \\ \hfill c_{\mathrm{emb}}\left(U\right)& :=c_q(U/2),\hfill & \hfill c_{\mathrm{emb}}\left(\mathrm{i}A\right)& =A.\hfill \end{array}$$

(368)

The first line maps homogeneous quotient coordinates to the compact group tangent; the second transports that map to the embedded matrix tangent. The factors are derived in Appendix D.8; for $n=2$ the duality is $H^3\leftrightarrow S^3$.

The reductive quotient gives the principal bundle

$$\mathrm{SU}\left(n\right)\hookrightarrow \mathrm{SL}(n,\mathbb{C})\stackrel{q}{\to }\mathrm{SL}(n,\mathbb{C})/\mathrm{SU}\left(n\right).$$

(369)

For a local section s, split its Maurer-Cartan form by Equation (53) [80], Section 3

$$\mathsf{\Theta }:=s^{-1}\mathrm{d}s=A+E,A\in {\mathsf{\Omega }}^1\left(\mathfrak{su}\left(n\right)\right),E\in {\mathsf{\Omega }}^1\left(\mathfrak{p}\right).$$

(370)

Under $s\mapsto sk$, $k:M\to \mathrm{SU}\left(n\right)$,

$$\begin{array}{cc}\hfill A& ⟼k^{-1}Ak+k^{-1}\mathrm{d}k,\hfill \end{array}$$

(371)

$$\begin{array}{cc}\hfill E& ⟼k^{-1}Ek.\hfill \end{array}$$

(372)

The Maurer-Cartan equation and the symmetric-pair brackets give

$$\begin{array}{cc}\hfill F_A:=\mathrm{d}A+A\wedge A& =-E\wedge E,\hfill \end{array}$$

(373)

$$\begin{array}{cc}\hfill D_{A}E:=\mathrm{d}E+A\wedge E+E\wedge A& =0.\hfill \end{array}$$

(374)

Thus local $\mathrm{SU}\left(n\right)$ redundancy obeys the non-Abelian gauge law, while noncommuting noncompact coframes generate compact curvature:

$$F_A(X,Y)=-[E_X,E_Y]\in \mathfrak{su}\left(n\right).$$

(375)

This is the standard composite connection of a reductive Cartan geometry, not an independent gauge field. For $n=2$, boost-boost brackets generate this compact residue; mixed boost-translation brackets yield the separate boost-generator-rapidity result only after a Poincaré representation is supplied.

Proposition 14

(Levi-Civita connection induced by the reductive split). Under the canonical identification

$$T{\mathcal{P}}_n^1\simeq \mathrm{SL}(n,\mathbb{C}){\times }_{\mathrm{SU}\left(n\right)}\mathfrak{p},$$

(376)

the connection ${\nabla }^A$ induced by A is the Levi-Civita connection of $g_{\mathrm{AI}}$.

Proof from the reductive symmetric-space theorem. 

Nomizu’s theorem identifies the canonical reductive connection of a Riemannian symmetric pair with the Levi-Civita connection of its invariant metric [24,81]. Here $E_I\left(U\right)=U/2$ and $4\alpha Tr\left(XY\right)$ induces exactly $g_{\mathrm{AI},I}(U,V)=\alpha Tr\left(UV\right)$, so no further rescaling occurs. The normalization also gives $F_{A,I}(U,V)=-[U,V]/4$ and hence the curvature convention in (347). □

6.6. The Full Holonomy Group

Theorem 13

(Canonical $\mathrm{SU}\left(n\right)$ holonomy). For $n\ge 2$, the canonical connection $A={pr}_{\mathfrak{k}}\left(g^{-1}\mathrm{d}g\right)$ on Equation (369) has

$${Hol}_e^0\left(A\right)={Hol}_e\left(A\right)=\mathrm{SU}\left(n\right),$$

(377)

at the identity lift e of the identity coset.

Proof from Ambrose-Singer. 

At the identity, $F_{A,I}(U,V)=-[U,V]/4$. Since $\mathfrak{p}=\mathrm{i}\mathfrak{su}\left(n\right)$ and $\mathfrak{su}\left(n\right)$ is perfect, $[\mathfrak{p},\mathfrak{p}]=\mathfrak{su}\left(n\right)$. Ambrose-Singer therefore gives ${\mathfrak{hol}}_e^0\left(A\right)=\mathfrak{su}\left(n\right)$ [45]. Connectedness of $\mathrm{SU}\left(n\right)$ yields restricted holonomy $\mathrm{SU}\left(n\right)$, and ${Hol}_e^0\left(A\right)\subseteq {Hol}_e\left(A\right)\subseteq \mathrm{SU}\left(n\right)$ yields the full equality. □

Corollary 24

(Principal versus tangent holonomy). At the identity coset, the Levi-Civita tangent connection has restricted and full holonomy

$${Hol}_I^0\left({\nabla }^{\mathrm{LC}}\right)={Hol}_I\left({\nabla }^{\mathrm{LC}}\right)=Ad\left(\mathrm{SU}\left(n\right)\right){|}_{\mathfrak{p}}\simeq \mathrm{SU}\left(n\right)/Z_n=\mathrm{PSU}\left(n\right).$$

(378)

For $n=2$ this is $\mathrm{SO}\left(3\right)$, not the defining $\mathrm{SU}\left(2\right)$ doublet.

Proof from associated-bundle holonomy. 

Associated-bundle holonomy is the image of principal holonomy under the fiber representation [81,82]. Here it is Ad on ${Herm}_0\left(n\right)$, whose kernel is exactly $Z_n$; applying Theorem 13 gives (378) and the $n=2$ specialization. □

Proposition 15

(Canonical connection and the local Yang-Mills equation). Let $M={\mathcal{P}}_n^1=\mathrm{SL}(n,\mathbb{C})/\mathrm{SU}\left(n\right)$ carry $g_{\mathrm{AI}}$, and equip $adP$ with any $Ad\left(\mathrm{SU}\right(n\left)\right)$-invariant positive inner product. For the canonical reductive connection A of Equation (370),

$$\begin{array}{|c|}\hline {\nabla }^{A\otimes \mathrm{LC}}{F}_A=0,{\mathrm{d}}_A{F}_A=0,{\mathrm{d}}_A^{{\ast }_{g_{\mathrm{AI}}}}{F}_A=0.\\ \hline\end{array}$$

(379)

Hence A satisfies the local source-free Yang-Mills equation on the statistical symmetric-space base.

These noncompact symmetric spaces have infinite invariant volume [26]. Since the parallel curvature $F_A$ has nonzero constant norm, their unrenormalized total Yang-Mills action is infinite. Nevertheless, Equation (379) is the local Euler-Lagrange equation, equivalently the first variation vanishes for compactly supported connection variations. On any supplied compact locally symmetric quotient to which the bundle descends, the connection is a finite-action Yang-Mills critical point. This is a composite stationary field on parameter/state space, not a Lorentzian spacetime propagation law.

Proof from canonical symmetric-space Yang-Mills theory. 

Canonical invariant connections on Riemannian symmetric spaces have parallel curvature and solve the source-free Yang-Mills equation [24,25]. By Proposition 14, the present A is precisely that connection, proving all three local identities. The statement’s infinite-volume qualification is therefore unaffected. □

6.7. Metric-Normalized Compact Relative Transport

The metric distortion canonically selects the positive isometry J. This symbol denotes a positive g-self-adjoint metric normalizer. On every nonzero fiber, $J^2$ has positive spectrum and therefore cannot equal −id; thus J is not a complex structure. The following results compute its compact relative transport and connection and curvature defects, using the canonical holonomy already established in Sections 6.5 and 6.6.

6.7.1. Positive Metric Normalization and Compact Relative Transport

The positive metric polar factorization is standard [46,79]; the article-specific theorem following the lemma applies it to compare the BKM and affine-invariant connection laws through an exact compact transporter, cocycle, and curvature defect.

Lemma 2

(Positive normalizer and metric polar factor). Let $E\to N$ be an oriented real vector bundle equipped with smooth positive metrics g and $\overline{g}$. There is a unique smooth, positive, g-self-adjoint bundle endomorphism $\mathsf{D}$ such that

$$\overline{g}(u,v)=g(\mathsf{D}u,v).$$

Its positive square root $J:={\mathsf{D}}^{1/2}$ is the unique positive g-self-adjoint bundle isometry from $(E,\overline{g})$ to $(E,g)$:

$$g(Ju,Jv)=\overline{g}(u,v).$$

(380)

It preserves the chosen orientation. More generally, if $L:E\to F$ is a bundle isomorphism into a metric bundle $(F,h)$ and $\overline{g}=L^{\ast }h$, then

$$L=\left(L{J}^{-1}\right)J$$

is its metric polar factorization and $L{J}^{-1}:(E,g)\to (F,h)$ is an isometry.

Proof from positive polar decomposition. 

Smooth positive functional calculus gives the unique smooth root $J={\mathsf{D}}^{1/2}$; positivity makes it orientation preserving, and direct substitution gives both the isometry and polar factorization [46,79]. Fiberwise uniqueness makes the local roots glue globally. □

Theorem 14

(Metric-normalized BKM-affine-invariant transport). Let $n\ge 2$ and let the affine-invariant scale be any $\alpha >0$. Write $g:=g^{\mathsf{\Psi }}$ and $\overline{g}:={\overline{g}}^{\mathrm{AI}}$ on $M=\mathfrak{p}$, and define the smooth positive g-self-adjoint distortion by

$${\overline{g}}_X(H,K)=g_X({\mathsf{D}}_X^{\left(\alpha \right)}H,K),{\mathsf{D}}_X^{\left(\alpha \right)}:={\left(g_X^{♭}\right)}^{-1}\circ {\overline{g}}_X^{♭}={\mathsf{G}}_X^{-1}\circ \left(\alpha {\mathsf{S}}_X^2\right).$$

(381)

Here $\mathsf{G}$ and $\mathsf{S}$ are the operators in Equations (349) and (350). Let $J_X^{\left(\alpha \right)}:={\left({\mathsf{D}}_X^{\left(\alpha \right)}\right)}^{1/2}$. By Lemma 2, it is the unique positive isometry relative to the identity soldering:

$$J^{\left(\alpha \right)}:(TM,\overline{g})⟶(TM,g),g_X(J_X^{\left(\alpha \right)}H,J_X^{\left(\alpha \right)}K)={\overline{g}}_X(H,K).$$

(382)

Equivalently, before pulling the cone geometry back to M, the polar factor of the Legendre-exponential differential is

$${\tilde{J}}_X^{\left(\alpha \right)}:=D{P}_X\circ {\left(J_X^{\left(\alpha \right)}\right)}^{-1}:(T_{X}M,g_X)⟶(T_{P_X}{\mathcal{P}}_n^1,g_{P_X}^{\mathrm{AI}}),g_{P_X}^{\mathrm{AI}}({\tilde{J}}_X^{\left(\alpha \right)}H,{\tilde{J}}_X^{\left(\alpha \right)}K)=g_X(H,K).$$

(383)

It is a canonical bundle isometry covering $X\mapsto P_X$, but it is not generally the differential of a local Riemannian isometry. Transport the affine-invariant connection to the g-metric bundle and define its relative defect by

$$\begin{array}{cccc}\hfill {\widehat{D}}_X^{\mathrm{AI}}Y& :=J^{\left(\alpha \right)}\left({\overline{D}}_X^{\mathrm{AI}}\left({\left(J^{\left(\alpha \right)}\right)}^{-1}Y\right)\right),\hfill & \hfill {\mathcal{B}}_X^{\mathrm{rel}}& :=D_X^{\mathsf{\Psi }}-{\widehat{D}}_X^{\mathrm{AI}}.\hfill \end{array}$$

(384)

Then both connections in the difference are g-metric and

$$\begin{array}{cc}\hfill D_X^{\mathsf{\Psi }}\left(J^{\left(\alpha \right)}Y\right)& =J^{\left(\alpha \right)}{\overline{D}}_X^{\mathrm{AI}}Y+{\mathcal{B}}_X^{\mathrm{rel}}\left(J^{\left(\alpha \right)}Y\right),\hfill \end{array}$$

(385)

$$\begin{array}{cc}\hfill g({\mathcal{B}}_X^{\mathrm{rel}}U,V)& +g(U,{\mathcal{B}}_X^{\mathrm{rel}}V)=0.\hfill \end{array}$$

(386)

Thus ${\mathcal{B}}^{\mathrm{rel}}\in {\mathsf{\Omega }}^1\left(\mathfrak{so}(TM,g)\right)$ is an adjoint-valued connection defect, or relative gauge potential; it is not a connection by itself. If $\mathcal{A}:=D^{\mathsf{\Psi }}-{\overline{D}}^{\mathrm{AI}}$ is the raw identity-soldered defect, the normalized and raw defects are related exactly by

$$\begin{array}{|c|}\hline {\mathcal{B}}^{\mathrm{rel}}=\mathcal{A}+\left({\overline{D}}^{\mathrm{AI}}{J}^{\left(\alpha \right)}\right){\left(J^{\left(\alpha \right)}\right)}^{-1}.\\ \hline\end{array}$$

(387)

Indeed, ${\widehat{D}}^{\mathrm{AI}}={\overline{D}}^{\mathrm{AI}}-\left({\overline{D}}^{\mathrm{AI}}{J}^{\left(\alpha \right)}\right){\left(J^{\left(\alpha \right)}\right)}^{-1}$.

The normalized connection ${\widehat{D}}^{\mathrm{AI}}$, connection defect ${\mathcal{B}}^{\mathrm{rel}}$, relative transporter, and relative curvature below are independent of the constant affine-invariant scale. The choice $\alpha =1/n$ is only the identity calibration at the center. For readability, the superscript $\left(\alpha \right)$ on $\mathsf{D}$ and J is suppressed below whenever the scale is fixed.

For a piecewise $C^1$ path $\gamma :[0,T]\to M$, let $P_t^{\mathsf{\Psi }}$ and ${\overline{P}}_t^{\mathrm{AI}}$ be parallel transport from $x=\gamma \left(0\right)$ to $\gamma \left(t\right)$ and put

$${\widehat{P}}_t^{\mathrm{AI}}:=J_{\gamma \left(t\right)}^{\left(\alpha \right)}{\overline{P}}_t^{\mathrm{AI}}{\left(J_x^{\left(\alpha \right)}\right)}^{-1},U_{\gamma }^{\mathrm{rel}}:={\left({\widehat{P}}_T^{\mathrm{AI}}\right)}^{-1}{P}_T^{\mathsf{\Psi }}.$$

(388)

Then

$$\begin{array}{|c|}\hline U_{\gamma }^{\mathrm{rel}}=\mathcal{P}exp\left[-{\int }_0^T{\left({\widehat{P}}_t^{\mathrm{AI}}\right)}^{-1}{\mathcal{B}}_{\gamma \left(t\right)}^{\mathrm{rel}}\left(\dot{\gamma }\left(t\right)\right){\widehat{P}}_t^{\mathrm{AI}}\mathrm{d}t\right]\in \mathrm{SO}(T_{x}M,g_x).\\ \hline\end{array}$$

(389)

For composable paths ${\gamma }_1:x\to y$ and ${\gamma }_2:y\to z$ it obeys the exact twisted cocycle

$$U_{{\gamma }_2\circ {\gamma }_1}^{\mathrm{rel}}={\left({\widehat{P}}_{{\gamma }_1}^{\mathrm{AI}}\right)}^{-1}{U}_{{\gamma }_2}^{\mathrm{rel}}{\widehat{P}}_{{\gamma }_1}^{\mathrm{AI}}{U}_{{\gamma }_1}^{\mathrm{rel}}.$$

(390)

Thus its natural composition law is twisted; in general it is not an ordinary path-groupoid or holonomy representation. Its relative curvature is the exact transgression

$$\begin{array}{|c|}\hline {\mathcal{F}}^{\mathrm{rel}}:=R^{\mathsf{\Psi }}-J{\overline{R}}^{\mathrm{AI}}{J}^{-1}={\mathrm{d}}^{{\widehat{D}}^{\mathrm{AI}}}{\mathcal{B}}^{\mathrm{rel}}+{\mathcal{B}}^{\mathrm{rel}}\wedge {\mathcal{B}}^{\mathrm{rel}}\in {\mathsf{\Omega }}^2\left(\mathfrak{so}(TM,g)\right).\\ \hline\end{array}$$

(391)

Here ${\mathrm{d}}^{{\widehat{D}}^{\mathrm{AI}}}$ is the induced covariant exterior derivative on endomorphism-valued forms, and the wedge convention is

$$({\mathcal{B}}^{\mathrm{rel}}\wedge {\mathcal{B}}^{\mathrm{rel}})(X,Y):=[{\mathcal{B}}^{\mathrm{rel}}\left(X\right),{\mathcal{B}}^{\mathrm{rel}}\left(Y\right)].$$

(392)

Writing ${\widehat{R}}^{\mathrm{AI}}:=J{\overline{R}}^{\mathrm{AI}}{J}^{-1}$, the two ordinary Bianchi identities impose the exact relative identity

$$\begin{array}{|c|}\hline {\mathrm{d}}^{{\widehat{D}}^{\mathrm{AI}}}{\mathcal{F}}^{\mathrm{rel}}+[{\mathcal{B}}^{\mathrm{rel}}\wedge ({\widehat{R}}^{\mathrm{AI}}+{\mathcal{F}}^{\mathrm{rel}})]=0.\\ \hline\end{array}$$

(393)

Here the bracket of an endomorphism-valued one-form B and two-form Q is cyclic alternation:

$$\begin{array}{cc}\hfill (X,Y,Z):=& \left[B\right(X),Q(Y,Z\left)\right]+\left[B\right(Y),Q(Z,X\left)\right]\hfill \\ & +\left[B\right(Z),Q(X,Y\left)\right].\hfill \end{array}$$

Thus ${\mathcal{F}}^{\mathrm{rel}}$ is a covariant curvature difference, not the curvature of ${\mathcal{B}}^{\mathrm{rel}}$ viewed as an independent connection. For a sufficiently small positively oriented coordinate rectangle used in Equation (366), the principal logarithm is defined and

$$log{U}_{{\gamma }_{\epsilon }}^{\mathrm{rel}}=-{\epsilon }^2{\mathcal{F}}_X^{\mathrm{rel}}(H,K)+O\left({\epsilon }^3\right).$$

(394)

Under a simultaneous change of oriented g-orthonormal frame for the two connections, $U_{\gamma }^{\mathrm{rel}}$ changes only by conjugation at x; hence its conjugacy class, characteristic polynomial, unordered eigenangles, and representation characters are frame independent, even when the base path is open.

The proof is in Appendix D.6. Before specializing at the tracial center, we record the construction’s natural pullback to the faithful-state manifold.

6.7.2. State-Space Naturality

Proposition 16

(State-space pullback of compact relative transport). Let $E_{-}:\mathfrak{p}\to {\mathcal{P}}_n^1$ be $E_{-}\left(X\right)={\mathrm{e}}^{-X}$, so that $\mathsf{\Phi }=E_{-}\circ \mathcal{X}$. Fix any affine-invariant scale $\alpha >0$ (with $\alpha =1/n$ the optional identity calibration) and, on ${\mathcal{S}}_n^{+}$, put

$$\begin{array}{c}{g}_S:=g_{\mathrm{BKM}}={\mathcal{X}}^{\ast }{g}^{\mathsf{\Psi }},{\overline{g}}_S:={\mathsf{\Phi }}^{\ast }{g}_{\mathrm{AI}},\\ D_S:={\nabla }_{g_S}^{\mathrm{LC}}={\mathcal{X}}^{\ast }{D}^{\mathsf{\Psi }},{\overline{D}}_S:={\mathsf{\Phi }}^{\ast }{\nabla }_{g_{\mathrm{AI}}}^{\mathrm{LC}}.\end{array}$$

(395)

Define the unique positive $g_S$-self-adjoint distortion and its positive square root by

$${\overline{g}}_S(U,V)=g_S({\mathcal{D}}_{S}U,V),J_S:={\mathcal{D}}_S^{1/2},$$

(396)

and set

$${\widehat{D}}_U^{S}V:=J_S\left({\overline{D}}_{S,U}\left(J_S^{-1}V\right)\right),{\mathcal{B}}_S^{\mathrm{rel}}:=D_S-{\widehat{D}}^S.$$

(397)

Then $\mathcal{X}$ intertwines $({\mathcal{D}}_S,J_S,{\widehat{D}}^S,{\mathcal{B}}_S^{\mathrm{rel}})$ with $(\mathsf{D},J,{\widehat{D}}^{\mathrm{AI}},{\mathcal{B}}^{\mathrm{rel}})$ of Theorem 14. In particular, ${\widehat{D}}^S$ is $g_S$-metric and

$${\mathcal{B}}_S^{\mathrm{rel}}\in {\mathsf{\Omega }}^1\left(\mathfrak{so}(T{\mathcal{S}}_n^{+},g_S)\right).$$

(398)

For a piecewise-$C^1$ path $\gamma :{\rho }_0\to {\rho }_1$, with $\zeta :=\mathcal{X}\circ \gamma$, its relative transporter is

$$\begin{array}{|c|}\hline U_{\gamma }^{\mathrm{rel},S}:={\left(P_{\gamma }^{{\widehat{D}}^S}\right)}^{-1}{P}_{\gamma }^{D_S}={\left(D{\mathcal{X}}_{{\rho }_0}\right)}^{-1}{U}_{\zeta }^{\mathrm{rel}}D{\mathcal{X}}_{{\rho }_0}\in \mathrm{SO}(T_{{\rho }_0}{\mathcal{S}}_n^{+},g_{S,{\rho }_0}).\\ \hline\end{array}$$

(399)

It obeys the pullback of the twisted cocycle Equation (390), and its curvature defect is

$${\mathcal{F}}_S^{\mathrm{rel}}:=R^{D_S}-R^{{\widehat{D}}^S}={\mathrm{d}}^{{\widehat{D}}^S}{\mathcal{B}}_S^{\mathrm{rel}}+{\mathcal{B}}_S^{\mathrm{rel}}\wedge {\mathcal{B}}_S^{\mathrm{rel}}.$$

(400)

These statements are natural under the diffeomorphism $\mathcal{X}$. Combining them with the canonical principal and tangent holonomy already proved in Theorem 13 and Corollary 24 gives the corollary immediately below.

Proof from naturality. 

The factorization $\mathsf{\Phi }=E_{-}\circ \mathcal{X}$ identifies the two pulled-back metrics and connections. Naturality of positive functional calculus gives $D\mathcal{X}\circ J_S=J\circ D\mathcal{X}$, while standard naturality of pullback connections, curvature, and parallel transport [81,82] then gives the normalized connection, transporter, skew-defect, and transgression formulas. Thus no new state-space connection is assumed: every displayed object is the pullback of the already constructed Gibbs-chart object. □

Corollary 25

(Normalized affine-invariant and state-space holonomy). For every $\alpha >0$, the normalized affine-invariant connection of Theorem 14 has

$${Hol}_x\left({\widehat{D}}^{\mathrm{AI}}\right)=J_x{Hol}_x\left({\overline{D}}^{\mathrm{AI}}\right)J_x^{-1}\simeq \mathrm{PSU}\left(n\right).$$

(401)

On state space, the pullback principal bundle

$${\mathcal{P}}_{\mathsf{\Phi }}:=\{(\rho ,G)\in {\mathcal{S}}_n^{+}\times \mathrm{SL}(n,\mathbb{C}):G{G}^{†}=\mathsf{\Phi }\left(\rho \right)\}⟶{\mathcal{S}}_n^{+}$$

(402)

with the pulled-back canonical connection has principal holonomy $\mathrm{SU}\left(n\right)$, its associated tangent connection has holonomy $\mathrm{PSU}\left(n\right)$, and the normalized state-space connection satisfies

$${Hol}_{\rho }\left({\widehat{D}}^S\right)=J_{S,\rho }{Hol}_{\rho }\left({\overline{D}}_S\right)J_{S,\rho }^{-1}\simeq \mathrm{PSU}\left(n\right).$$

(403)

These are connection holonomies. By contrast, $U_{\gamma }^{\mathrm{rel},S}$ is a compact comparison transporter with twisted composition, and ${\mathcal{B}}_S^{\mathrm{rel}}$ is not a connection.

Proof from holonomy naturality. 

Holonomy is preserved by diffeomorphic pullback and conjugated by a bundle gauge [81,82]. Apply these two standard facts to $E_{-}$, ${\mathsf{\Phi }}^{-1}$, and the positive gauges $J,J_S$, then use Theorem 13 and Corollary 24. This proves both principal and tangent assertions while leaving the comparison transporter logically distinct from a connection holonomy. □

We now return through the Gibbs chart to the tracial center, where the BKM and normalized affine-invariant curvature maps and their relative defect admit an explicit Casimir decomposition.

6.7.3. Central Spectra and Qutrit Resolution

Lemma 3

(Central bracket-Casimir decomposition). Let $V_n={Herm}_0\left(n\right)$ carry $g_0(H,K)=Tr\left(HK\right)/n$, identify ${\mathsf{\Lambda }}^2{V}_n\simeq \mathfrak{so}(V_n,g_0)$ by $H\wedge K\mapsto W_{H,K}$, and equip $\mathfrak{su}\left(n\right)$ with $\langle A,B\rangle =-Tr\left(AB\right)/n$. For

$$b:{\mathsf{\Lambda }}^2{V}_n⟶\mathfrak{su}\left(n\right),b(H\wedge K)=[H,K],$$

one has

$$W\circ b^{\ast }\left(A\right)={ad}_A,b{b}^{\ast }=n^2{\mathrm{id}}_{\mathfrak{su}\left(n\right)}.$$

(404)

Consequently

$${\mathsf{\Pi }}_{\mathrm{ad}}:=\frac{1}{n^2}{b}^{\ast }b,{\mathsf{\Pi }}_{\perp }:=\mathrm{id}-{\mathsf{\Pi }}_{\mathrm{ad}}$$

(405)

are the orthogonal projectors onto $ad\mathfrak{su}\left(n\right)$ and its orthogonal complement.

Proof from the adjoint Casimir. 

Trace invariance gives $W\circ b^{\ast }\left(A\right)={ad}_A$. The standard $\mathrm{SU}\left(n\right)$ adjoint-Casimir contraction [83,84], converted to $g_0=Tr(··)/n$ and $\langle A,B\rangle =-Tr\left(AB\right)/n$, is $b{b}^{\ast }=n^2\mathrm{id}$. Hence $b^{\ast }b$ is $n^2$ on $im{b}^{\ast }=ad\mathfrak{su}\left(n\right)$ and zero on $kerb$, which proves the two normalization-dependent projector formulas. □

Proposition 17

(Central cancellation, spectra, and generated mismatch). At the maximally mixed point, the normalizer at arbitrary $\alpha >0$ satisfies

$${\mathsf{D}}_0^{\left(\alpha \right)}=\alpha n\mathrm{id},J_0^{\left(\alpha \right)}=\sqrt{\alpha n}\mathrm{id},{\left(\mathrm{D}{\mathsf{D}}^{\left(\alpha \right)}\right)}_0\left[H\right]=-\alpha n{\mathsf{Q}}_0\left(H\right),{\left(\mathrm{D}{J}^{\left(\alpha \right)}\right)}_0\left[H\right]=-\frac{\sqrt{\alpha n}}{2}{\mathsf{Q}}_0\left(H\right).$$

(406)

The scalar factors cancel from the transported connection, so for every $\alpha >0$,

$$\begin{array}{|c|}\hline {\mathcal{B}}_0^{\mathrm{rel}}=0,{\mathcal{F}}_0^{\mathrm{rel}}={\mathsf{\Delta }}_0^{\mathrm{curv}}.\\ \hline\end{array}$$

(407)

The matched calibration $\alpha =1/n$ additionally gives $J_0=\mathrm{id}$ and ${\left(\mathrm{D}J\right)}_0\left[H\right]=-{\mathsf{Q}}_0\left(H\right)/2$. More explicitly, with $W_{H,K}L:=g_0(K,L)H-g_0(H,L)K$,

$${\mathcal{F}}_0^{\mathrm{rel}}(H,K)L=\frac{1}{4}{W}_{H,K}L+\frac{3}{16}[[H,K],L].$$

(408)

With the projectors of Lemma 3, the three central curvature maps are

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill {\widehat{R}}_0^{\mathrm{AI}}& =-\frac{n^2}{4}{\mathsf{\Pi }}_{\mathrm{ad}},\hfill \\ \hfill {\widehat{R}}_0^{\mathsf{\Psi }}& =\frac{4-n^2}{16}{\mathsf{\Pi }}_{\mathrm{ad}}+\frac{1}{4}{\mathsf{\Pi }}_{\perp },\hfill \\ \hfill {\widehat{\mathcal{F}}}_0^{\mathrm{rel}}& =\frac{4+3n^2}{16}{\mathsf{\Pi }}_{\mathrm{ad}}+\frac{1}{4}{\mathsf{\Pi }}_{\perp }.\hfill \end{array}\\ \hline\end{array}$$

(409)

The relative curvature map is therefore an isomorphism, and the linear span of the area-order relative defects at the center is all of $\mathfrak{so}(n^2-1)$. Define the closed generated mismatch group by

$$G_x^{\mathrm{mismatch}}:=\overline{〈U_{\gamma }^{\mathrm{rel}}:\gamma \mathit{is}\mathit{a}\mathit{based}\mathit{loop}\mathit{at}x〉}.$$

(410)

Small rectangles and their reversals then give

$$G_0^{\mathrm{mismatch}}=\mathrm{SO}(n^2-1).$$

(411)

This generated mismatch group is not the holonomy group of ${\mathcal{B}}^{\mathrm{rel}}$, which is not a connection.

Proposition 18

(Plane-wise complementarity and sharp central curvature bounds). Retain the norms of Lemma 3; in particular, $H\wedge K\mapsto W_{H,K}$ is an isometry when $\mathfrak{so}(V_n,g_0)$ carries ${\langle A,B\rangle }_{\mathfrak{so}}=-\frac{1}{2}{Tr}_{V_n}\left(AB\right)$. For independent $X,Y\in V_n$, put

$$\xi :=X\wedge Y,\mathsf{\Delta }(X,Y):={∥X∥}^2{∥Y∥}^2-g_0{(X,Y)}^2={∥\xi ∥}^2.$$

(412)

Then the adjoint and transverse responses obey the exact Pythagorean law

$$\begin{array}{|c|}\hline \mathsf{\Delta }(X,Y)=\frac{1}{n^2}{∥[X,Y]∥}_{\mathfrak{su}\left(n\right)}^2+{∥{\mathsf{\Pi }}_{\perp }\xi ∥}^2.\\ \hline\end{array}$$

(413)

The sharp Böttcher-Wenzel inequality [85] becomes

$${∥[X,Y]∥}_{\mathfrak{su}\left(n\right)}^2\le 2n\mathsf{\Delta }(X,Y),$$

(414)

and hence

$$\begin{array}{|c|}\hline \frac{{∥{\mathsf{\Pi }}_{\mathrm{ad}}\xi ∥}^2}{{∥\xi ∥}^2}\le \frac{2}{n},\frac{{∥{\mathsf{\Pi }}_{\perp }\xi ∥}^2}{{∥\xi ∥}^2}\ge \frac{n-2}{n}.\\ \hline\end{array}$$

(415)

Equivalently, every simple tangent bivector makes an angle

$$\theta \ge arccos\sqrt{\frac{2}{n}}$$

(416)

with $ad\mathfrak{su}\left(n\right)$.

The central BKM sectional curvature is

$$\begin{array}{|c|}\hline {sec}_0^{\mathsf{\Psi }}(X,Y)=\frac{1}{4}-\frac{1}{16}\frac{{∥[X,Y]∥}_{\mathfrak{su}\left(n\right)}^2}{\mathsf{\Delta }(X,Y)}.\\ \hline\end{array}$$

(417)

For $n\ge 3$ this gives the sharp range

$$\begin{array}{|c|}\hline \frac{2-n}{8}\le {sec}_0^{\mathsf{\Psi }}(X,Y)\le \frac{1}{4}.\\ \hline\end{array}$$

(418)

The upper endpoint is attained by independent commuting directions. The lower endpoint is attained, up to unitary conjugacy, scaling, and replacing Y by $Y+cX$, by an embedded two-level Pauli pair. For $n=2$ the transverse space vanishes and every two-plane has curvature zero; the formal upper value $1/4$ is not attained.

Equivalently, the normalized incompatibility

$$\chi (X,Y):=\frac{{∥[X,Y]∥}_{\mathfrak{su}\left(n\right)}^2}{2n\mathsf{\Delta }(X,Y)}\in [0,1]$$

(419)

is invariant under simultaneous unitary conjugation and any invertible real change of the two plane generators. Thus, unlike its unnormalized numerator, it is a scale-independent invariant of the unoriented tangent two-plane relative to the supplied matrix algebra. It simultaneously determines

$$\begin{array}{|c|}\hline {sec}_0^{\mathsf{\Psi }}=\frac{2-n\chi }{8},r_{\mathrm{ad}}:=\frac{{∥{\mathsf{\Pi }}_{\mathrm{ad}}\xi ∥}^2}{{∥\xi ∥}^2}=\frac{2\chi }{n},r_{\perp }=1-\frac{2\chi }{n}.\\ \hline\end{array}$$

(420)

Finally,

$$dim{(ad\mathfrak{su}\left(n\right))}^{\perp }=\frac{(n^2-1)(n^2-4)}{2},$$

(421)

so qutrits are the first matrix-state dimension with a nonzero transverse curvature channel.

Proof from the sharp commutator inequality. 

By Lemma 3, ${∥{\mathsf{\Pi }}_{\mathrm{ad}}\xi ∥}^2=n^{-2}{∥b\xi ∥}^2$; orthogonal projection gives Equation (413). The sharp Böttcher-Wenzel theorem, including the normal/Hermitian equality case used here [85], becomes (414) after replacing Y by its $g_0$-orthogonal component to X and converting the Frobenius norm to $g_0$. Its Hermitian equality case is the embedded two-level Pauli pair, up to the transformations stated in the proposition. Pairing the BKM line of Equation (409) with $\xi /∥\xi ∥$ gives Equation (417); the projector fractions, curvature range, incompatibility formulas, and final dimension then follow algebraically. For $(X^{\prime },Y^{\prime })=(aX+bY,cX+dY)$, both $X^{\prime }\wedge Y^{\prime }$ and $[X^{\prime },Y^{\prime }]$ acquire the factor $ad-bc$; their squared norms acquire its square, proving plane-basis invariance of $\chi$. Simultaneous unitary conjugation preserves the traces defining both norms. Thus only the matrix inequality is imported; all geometric normalizations and curvature consequences are retained here. □

Corollary 26

(Isotropic plane averages and transverse concentration). Let a central tangent two-plane be Haar-uniform on ${Gr}_2\left(V_n\right)$. Then

$$\begin{array}{|c|}\hline \mathbb{E}{r}_{\mathrm{ad}}=\frac{2}{n^2-2},\mathbb{E}{r}_{\perp }=\frac{n^2-4}{n^2-2},\mathbb{E}{sec}_0^{\mathsf{\Psi }}=\frac{n^2-4}{8(n^2-2)}.\\ \hline\end{array}$$

(422)

Moreover, for every $\epsilon >0$,

$$Pr(r_{\mathrm{ad}}>\epsilon )\le \frac{2}{(n^2-2)\epsilon }.$$

(423)

Thus transverse norm dominance is typical at large n, but it must not be confused with curvature dominance: the adjoint eigenvalue $(4-n^2)/16$ grows in magnitude.

Proof. 

Choose an orientation of the Haar-uniform plane and let $\xi$ be its unit simple bivector. In an orthonormal wedge basis, invariance under coordinate sign changes kills the off-diagonal second moments and invariance under permutations makes all diagonal moments equal. Thus $\mathrm{O}\left(V_n\right)$-invariant second moment

$$\mathbb{E}(\xi \otimes \xi )=\frac{1}{dim{\mathsf{\Lambda }}^2{V}_n}{\mathrm{id}}_{{\mathsf{\Lambda }}^2{V}_n};$$

the scalar follows by taking the trace. Therefore the expected squared norm of the projection of $\xi$ onto any fixed subspace is the ratio of dimensions. Here $dimad\mathfrak{su}\left(n\right)=n^2-1$ and $dim{\mathsf{\Lambda }}^2{V}_n=(n^2-1)(n^2-2)/2$. The curvature average follows from Equation (409), and the probability bound is Markov’s inequality. □

Remark 10

(Tracial squared commutator and coherence diagnostic). At ${\rho }_{\ast }=I_n/n$,

$${∥[X,Y]∥}_{\mathfrak{su}\left(n\right)}^2=Tr\left({\rho }_{\ast }{[X,Y]}^{†}[X,Y]\right),Tr\left({\rho }_{\ast }[X,Y]\right)=0.$$

(424)

Thus curvature detects a second-order incompatibility invisible to the tracial first moment. If $X=diag(x_1,\dots ,x_n)$, then

$${∥[X,Y]∥}_{\mathfrak{su}\left(n\right)}^2=\frac{2}{n}\sum _{i<j}{(x_i-x_j)}^2|Y_{ij}{|}^2,{sec}_0^{\mathsf{\Psi }}(X,Y)=\frac{1}{4}-\frac{1}{8n\mathsf{\Delta }(X,Y)}\sum _{i<j}{(x_i-x_j)}^2{\left|Y_{ij}\right|}^2.$$

(425)

The $W/4$ term is therefore a Fisher-type space-form background, agreeing with the radius-two square-root realization on commuting probability simplices, while off-diagonal coherence supplies the commutator correction. For $Y\left(t\right)=U_t^{†}Y{U}_t$, the same formula is OTOC-like, but raw growth of $Tr\left({\rho }_{\ast }{[X,Y\left(t\right)]}^{†}[X,Y\left(t\right)]\right)$ lowers curvature only when $\mathsf{\Delta }(X,Y(t\left)\right)$ is fixed; in general it is growth of the normalized ratio that matters. Likewise, for thermodynamic controls the formula applies to centered dimensionless score directions at the tracial reference, not to a generic Gibbs state without recomputing its curvature.

Corollary 27

(Full BKM tangent holonomy). For every $n\ge 3$, both eigenvalues in the BKM row of Equation (409) are nonzero, so the BKM curvature values at the center span $\mathfrak{so}(n^2-1)$. Ambrose-Singer therefore gives the full BKM Levi-Civita tangent holonomy

$${Hol}_0\left(D^{\mathsf{\Psi }}\right)=\mathrm{SO}(n^2-1),n\ge 3.$$

(426)

This is a holonomy statement about the genuine BKM connection, unlike the earlier generated-group statement for a relative comparison.

The proof, including the passage from restricted to full holonomy, is in Appendix D.6.

Thus $D^{\mathsf{\Psi }}={\widehat{D}}^{\mathrm{AI}}+{\mathcal{B}}^{\mathrm{rel}}$ is an exact connection bridge between full BKM frame holonomy for $n\ge 3$, by Corollary 27, and canonical $\mathrm{PSU}\left(n\right)$ tangent holonomy by Corollary 25. The derived $\mathfrak{so}(n^2-1)$-valued compensator is not itself an $\mathfrak{su}\left(n\right)$ connection. Although ${\widehat{D}}^{\mathrm{AI}}$ is g-metric, it need not be torsion free because J is a bundle isometry between two metric bundles, not the differential of a base diffeomorphism. Positive normalization cancels the self-adjoint cubic first connection jet at the center, but not the curvature mismatch: by Proposition 17, its leading relative defect is an area-order compact rotation. For $n\ge 3$, the entropy cubic enters $R^{\mathsf{\Psi }}=-[\mathsf{Q}/2,\mathsf{Q}/2]$; for $n=2$ the central cubic vanishes and the residual comes from the affine-invariant curvature.

Monotone quantum metrics are classified in [6], while relevant curvature calculations appear in [27,28]. The article-specific result below is the operator-level qutrit $8\oplus 20$ resolution of the central BKM curvature, including its plane-wise fractions and bounds.

Theorem 15

(Qutrit $\mathfrak{so}\left(8\right)$$8+20$ curvature-channel and d-symbol resolution). Specialize Lemma 3 to $n=3$, retaining its maps $W,b,b^{\ast }$ and inner products. At the tracial state ${\rho }_{\ast }=I_3/3$, use the Gibbs tangent isometry with $V:={Herm}_0\left(3\right)$ and $g_0(H,K)=Tr\left(HK\right)/3$. Then

$$b{b}^{\ast }=9\mathrm{id},\mathfrak{so}\left(8\right)={\mathfrak{h}}_8\oplus {\mathfrak{m}}_{20},$$

(427)

where ${\mathfrak{h}}_8:=im{b}^{\ast }=ad\mathfrak{su}\left(3\right)$ and ${\mathfrak{m}}_{20}:=kerb$. The general projectors specialize to ${\mathsf{\Pi }}_8={\mathsf{\Pi }}_{\mathrm{ad}}$ and ${\mathsf{\Pi }}_{20}={\mathsf{\Pi }}_{\perp }$, acting explicitly as

$$\begin{array}{|c|}\hline {\mathsf{\Pi }}_8\left(W_{H,K}\right)=\frac{1}{9}{ad}_{[H,K]},{\mathsf{\Pi }}_{20}\left(W_{H,K}\right)=W_{H,K}-\frac{1}{9}{ad}_{[H,K]}.\\ \hline\end{array}$$

(428)

Here and below a hat denotes the curvature operator ${\mathsf{\Lambda }}^{2}V\to \mathfrak{so}(V,g_0)$. In the matched calibration $\alpha =1/3$, the three tracial curvature maps split as

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill {\widehat{R}}_0^{\mathrm{AI}}& =-\frac{9}{4}{\mathsf{\Pi }}_8,\hfill \\ \hfill {\widehat{R}}_0^{\mathsf{\Psi }}& =\frac{1}{4}{\widehat{R}}_0^{\mathrm{AI}}+\frac{1}{4}\mathrm{id}=-\frac{5}{16}{\mathsf{\Pi }}_8+\frac{1}{4}{\mathsf{\Pi }}_{20},\hfill \\ \hfill {\widehat{\mathcal{F}}}_0^{\mathrm{rel}}& =\frac{31}{16}{\mathsf{\Pi }}_8+\frac{1}{4}{\mathsf{\Pi }}_{20}.\hfill \end{array}\\ \hline\end{array}$$

(429)

The cubic origin of this split is coefficient-explicit. After complexification, the Weyl-character decomposition is

$${Sym}^3\left(\mathbf{8}\right)=\mathbf{1}\oplus \mathbf{8}\oplus \mathbf{10}\oplus \overline{\mathbf{10}}\oplus \mathbf{27}\oplus \mathbf{64}.$$

(430)

The standard $\mathrm{SU}\left(3\right)$ tensor-product data used in this character calculation are tabulated in [33], Table 24; the symmetric-cube projection itself is derived below. Here $\mathbf{10}=V_{(3,0)}$ and $\overline{\mathbf{10}}=V_{(0,3)}$ in Dynkin-label notation. The invariant $d_{abc}$ spans the unique singlet, so the central cubic has no direct $\mathbf{10}\oplus \overline{\mathbf{10}}$ component. Instead, let $T_a$ be Hermitian generators normalized by

$$Tr\left(T_a{T}_b\right)=\frac{1}{2}{\delta }_{ab},\{T_a,T_b\}=\frac{1}{3}{\delta }_{ab}I+d_{abc}{T}_c,$$

put $e_a:=\sqrt{6}{T}_a$, and define

$${\left({\mathsf{d}}_a\right)}_{bc}:=d_{abc},J_{ab}:=W_{e_a,e_b}.$$

(431)

Then $\left\{e_a\right\}$ is $g_0$-orthonormal. Write $K^{\mathsf{\Psi }}:=-\mathsf{Q}/2$ for the dual-connection half difference. The following identities hold. Notice that d itself is the fixed singlet, whereas $e_a\mapsto {\mathsf{d}}_a$ is an equivariant copy of the adjoint $\mathbf{8}$; antisymmetrizing two such contracted shears is what produces the map from ${\mathsf{\Lambda }}^2\mathbf{8}$.

$$\begin{array}{|c|}\hline \begin{array}{cccc}\hfill D^3{\mathsf{\Psi }}_0(e_a,e_b,e_c)& =-\sqrt{\frac{3}{2}}{d}_{abc},\hfill & \hfill {\mathsf{Q}}_0\left(e_a\right)& =-\sqrt{\frac{3}{2}}{\mathsf{d}}_a,\hfill \\ \hfill K_{0,e_a}^{\mathsf{\Psi }}& =-\frac{1}{2}{\mathsf{Q}}_0\left(e_a\right)=\sqrt{\frac{3}{8}}{\mathsf{d}}_a,\hfill & \hfill [{\mathsf{d}}_a,{\mathsf{d}}_b]& =\frac{5}{6}{\mathsf{\Pi }}_8{J}_{ab}-\frac{2}{3}{\mathsf{\Pi }}_{20}{J}_{ab},\hfill \\ \hfill R_0^{\mathsf{\Psi }}(e_a,e_b)& =-\frac{3}{8}[{\mathsf{d}}_a,{\mathsf{d}}_b]=-\frac{5}{16}{\mathsf{\Pi }}_8{J}_{ab}+\frac{1}{4}{\mathsf{\Pi }}_{20}{J}_{ab}.\hfill \end{array}\\ \hline\end{array}$$

(432)

Thus, under the metric identification $W:{\mathsf{\Lambda }}^{2}V\stackrel{\sim }{\to }\mathfrak{so}(V,g_0)$, the equivariant endomorphism

$${\mathcal{S}}_d:\mathfrak{so}(V,g_0)⟶\mathfrak{so}(V,g_0),{\mathcal{S}}_d\left(J_{ab}\right):=[{\mathsf{d}}_a,{\mathsf{d}}_b](1\le a<b\le 8),$$

(433)

acts by $5/6$ on ${\mathfrak{h}}_8$ and by $-2/3$ on ${\mathfrak{m}}_{20}$. Both eigenvalues are nonzero, so

$${\mathcal{S}}_d^{-1}=\frac{6}{5}{\mathsf{\Pi }}_8-\frac{3}{2}{\mathsf{\Pi }}_{20}.$$

(434)

In particular, the 28 pairwise d-symbol shear commutators form a basis of $\mathfrak{so}\left(8\right)$; the 20 arises at the commutator/curvature level, not by linearly projecting the symmetric cubic. Thus the transverse term ${\mathsf{\Pi }}_{20}/4$ comes entirely from trace normalization. The full space-form contribution is $\mathrm{id}/4=({\mathsf{\Pi }}_8+{\mathsf{\Pi }}_{20})/4$: it also shifts the ${\mathfrak{h}}_8$ coefficient, which includes one quarter of the affine-invariant curvature. Equations (427)- (429) are orthogonal decompositions of curvature channels, not a direct-product decomposition of $\mathrm{SO}\left(8\right)$.

Corollary 28

(Qutrit plane fractions and a sharp interpolation). For a nonzero simple qutrit bivector $\xi =X\wedge Y$, use $(r_8,r_{20})=(r_{\mathrm{ad}},r_{\perp })$ from Equation (420). Then

$$\begin{array}{|c|}\hline r_8=\frac{2}{3}\chi ,r_{20}=1-\frac{2}{3}\chi \ge \frac{1}{3},{sec}_0^{\mathsf{\Psi }}=-\frac{5}{16}+\frac{9}{16}{r}_{20}.\\ \hline\end{array}$$

(435)

In particular,

$$r_{20}\gtrless \frac{5}{9}⟺{sec}_0^{\mathsf{\Psi }}\gtrless 0,$$

(436)

and a qutrit plane is purely transverse, ${\mathsf{\Pi }}_{20}\xi =\xi$, if and only if $[X,Y]=0$. Hence the real 20 is a normalization-transverse sector, not simply a synonym for noncommutativity. The fractions are squared projections of the input bivector; their signed sectional contributions are $-5r_8/16$ and $r_{20}/4$, not probabilities for curvature magnitudes.

The bounds are realized continuously. Let

$$X=\sqrt{\frac{3}{2}}\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right),Y_0=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -2\end{array}\right),Y_1=\sqrt{\frac{3}{2}}\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),$$

(437)

and $Y\left(\vartheta \right):=cos\vartheta Y_0+sin\vartheta Y_1$. These are unit directions with $g_0(X,Y\left(\vartheta \right))=0$, and

$$\begin{array}{|c|}\hline \chi \left(\vartheta \right)={sin}^2\vartheta ,{sec}_0^{\mathsf{\Psi }}\left(\vartheta \right)=\frac{1}{4}-\frac{3}{8}{sin}^2\vartheta ,r_8\left(\vartheta \right)=\frac{2}{3}{sin}^2\vartheta ,r_{20}\left(\vartheta \right)=1-\frac{2}{3}{sin}^2\vartheta .\\ \hline\end{array}$$

(438)

Thus the endpoint $\vartheta =0$ is the commuting Fisher plane, whereas $\vartheta =\pi /2$ is an embedded two-level plane with ${∥[X,Y_1]∥}^2=6$ and ${sec}_0^{\mathsf{\Psi }}=-1/8$. This interpolation varies the plane under one fixed tracial curvature operator; it is not a dynamical transfer between curvature channels.

Proof. 

Specialize Equation (420) to $n=3$ and use Equation (429). Direct multiplication of the three displayed matrices gives the interpolation formulas. □

6.7.4. Jordan Normalization and the Algebraic Origin of the Tracial Split

The scalar normalization calculation below is a specialization of a general compression identity. Recording it first separates the exact algebra of a reduction from later geometric or dynamical interpretations.

Proposition 19

(Compression commutator and ordered-excursion defect). Let E be a finite-dimensional real or complex vector space, let $P\in End\left(E\right)$ satisfy $P^2=P$, and put

$$V:=imP,W:=kerP,Q:={\mathrm{id}}_E-P.$$

(439)

If $\mathfrak{l}$ is a real vector space and $L:\mathfrak{l}\to End\left(E\right)$ is linear, define

$$M_X:=P{L}_{X}P{{|}_V,{\mathsf{\Omega }}_P(X,Y):=P(L_{X}Q{L}_Y-L_{Y}Q{L}_X)P|}_V.$$

(440)

Then

$$\begin{array}{|c|}\hline [M_X,M_Y]=P[L_X,L_Y]{P|}_V-{\mathsf{\Omega }}_P(X,Y).\\ \hline\end{array}$$

(441)

Relative to $E=V\oplus W$, write

$$L_X=\left(\begin{array}{cc}{A}_X& B_X\\ C_X& D_X\end{array}\right).$$

(442)

Then

$$M_X=A_X,\begin{array}{|c|}\hline {\mathsf{\Omega }}_P(X,Y)=B_X{C}_Y-B_Y{C}_X.\\ \hline\end{array}$$

(443)

Thus ${\mathsf{\Omega }}_P$ is the difference between the two ordered excursions $V\to W\to V$. If L is a representation of a Lie algebra, then

$$[M_X,M_Y]-M_{[X,Y]}=-{\mathsf{\Omega }}_P(X,Y),$$

(444)

and the Jacobi identities imply the algebraic constraint

$$\begin{array}{|c|}\hline {\displaystyle \sum _{cyc(X,Y,Z)}}\left([M_X,{\mathsf{\Omega }}_P(Y,Z)]+{\mathsf{\Omega }}_P(X,[Y,Z])\right)=0.\\ \hline\end{array}$$

(445)

Either $Q{L}_{X}P=0$ for every X, or $P{L}_{X}Q=0$ for every X, is sufficient for ${\mathsf{\Omega }}_P=0$; neither condition is necessary in general.

Proof. 

Insert $P={\mathrm{id}}_E-Q$ between the two factors in $M_X{M}_Y-M_Y{M}_X$. This gives Equation (441); block multiplication gives Equation (443). The representation statement follows by substitution. Finally, insert Equation (444) into the Jacobi identity for the endomorphisms $M_X$ and use the Jacobi identity in $\mathfrak{l}$. □

Proposition 20

(Positive Gram response of an orthogonal reduction). Suppose in addition that E is a finite-dimensional Hilbert space, $P=P^{†}$, and $L_X=L_X^{†}$. Define the unresolved coupling and its operator-valued Gram kernel by

$${\mathsf{U}}_X:=Q{L}_X{P|}_V:V⟶W,{\mathfrak{G}}_P(X,Y):={\mathsf{U}}_X^{†}{\mathsf{U}}_Y\in End\left(V\right).$$

(446)

Then

$${\mathfrak{G}}_P(Y,X)={\mathfrak{G}}_P{(X,Y)}^{†},\sum _{i,j}\langle v_i,{\mathfrak{G}}_P(X_i,X_j)v_j\rangle ={∥{\sum }_i{\mathsf{U}}_{X_i}{v}_i∥}^2\ge 0,$$

(447)

for all $v_i\in V$. Its Hermitian and skew-Hermitian parts are

$${\mathfrak{S}}_P(X,Y):=\frac{1}{2}\left({\mathfrak{G}}_P(X,Y)+{\mathfrak{G}}_P(Y,X)\right),{\mathfrak{A}}_P(X,Y):=\frac{1}{2}\left({\mathfrak{G}}_P(X,Y)-{\mathfrak{G}}_P(Y,X)\right),$$

(448)

and satisfy

$${\mathfrak{S}}_P(X,X)={\mathsf{U}}_X^{†}{\mathsf{U}}_X⪰0,\begin{array}{|c|}\hline {\mathsf{\Omega }}_P(X,Y)=2{\mathfrak{A}}_P(X,Y).\\ \hline\end{array}$$

(449)

The scalar unresolved-coupling form

$$g_P^{\mathrm{unr}}(X,Y):={ReTr}_V{\mathfrak{G}}_P(X,Y)=Re{\langle {\mathsf{U}}_X,{\mathsf{U}}_Y\rangle }_{\mathrm{HS}}$$

(450)

is positive semidefinite, with

$$g_P^{\mathrm{unr}}(X,X)={∥Q{L}_{X}P∥}_{\mathrm{HS}}^2,rad{g}_P^{\mathrm{unr}}=ker(X\mapsto Q{L}_{X}P).$$

(451)

Moreover,

$$\begin{array}{|c|}\hline {∥{\mathsf{\Omega }}_P(X,Y)∥}_{\mathrm{HS}}\le 2\sqrt{g_P^{\mathrm{unr}}(X,X)g_P^{\mathrm{unr}}(Y,Y)}.\\ \hline\end{array}$$

(452)

The positivity assertion for ${\mathfrak{S}}_P$ is its diagonal quadratic positivity; ${\mathfrak{S}}_P(X,Y)$ need not be positive when $X\ne Y$.

Proof. 

Orthogonality and self-adjointness give ${\mathsf{U}}_X^{†}=P{L}_X{Q|}_W$, so all identities follow from the Gram factorization. For the estimate, use the triangle inequality and ${∥A^{†}B∥}_{\mathrm{HS}}\le {∥A∥}_{\mathrm{HS}}{∥B∥}_{\mathrm{HS}}$. □

Proposition 21

(Composition of nested orthogonal reductions). Let $P_2\le P_1$ be orthogonal projectors, and put

$$Q_1:=I-P_1,R:=P_1-P_2,I-P_2=Q_1+R.$$

(453)

Let ${\mathsf{\Omega }}_1$ be the defect of the reduction $E\to im{P}_1$ and let ${\mathsf{\Omega }}_2$ be the defect of the subsequent reduction $im{P}_1\to im{P}_2$, computed from $M_X^{\left(1\right)}:=P_1{L}_X{P}_1{|}_{im{P}_1}$ and unresolved projector R. Then the direct defect satisfies the exact additive law

$$\begin{array}{|c|}\hline {\mathsf{\Omega }}_{E\to im{P}_2}(X,Y)=P_2{\mathsf{\Omega }}_1(X,Y)P_2+{\mathsf{\Omega }}_2(X,Y).\\ \hline\end{array}$$

(454)

If the $L_X$ are self-adjoint, the total coupling decomposes into orthogonal channels,

$$(I-P_2)L_X{P}_2=Q_1{L}_X{P}_2\oplus R{L}_X{P}_2,$$

(455)

so the operator-valued Gram response and $g^{\mathrm{unr}}$ add without a mixed term. Mixed terms can occur when the reductions are not nested or not orthogonal.

Proof. 

Expand the direct defect using $I-P_2=Q_1+R$. The $Q_1$ terms are the $P_2$-compression of ${\mathsf{\Omega }}_1$, and insertion of $P_1$ around R turns the remaining terms into ${\mathsf{\Omega }}_2$. Orthogonality $Q_{1}R=0$ proves the Gram decomposition. □

Remark 11

(Moving projectors and quantum-geometric response). For a smooth constant-rank orthogonal projector $P\left(\lambda \right)$ on a fixed Hilbert space, differentiation of $P^2=P$ gives $P\left(\mathrm{d}P\right)P=Q\left(\mathrm{d}P\right)Q=0$. With

$${\mathsf{U}}_X^{\mathrm{proj}}:=Q{\left(\mathrm{d}P\right)}_{X}P,{\mathfrak{G}}^{\mathrm{proj}}(X,Y):=P{\left(\mathrm{d}P\right)}_{X}Q{\left(\mathrm{d}P\right)}_{Y}P,$$

(456)

the traced metric is the higher-rank extension of the Provost-Vallée quantum metric [86], while ${\nabla }^P=P\mathrm{d}$ is the projector form of the non-Abelian geometric connection [87]. Direct differentiation gives the metric/curvature pair

$$ReTr{\mathfrak{G}}^{\mathrm{proj}}(X,Y)=\frac{1}{2}Tr\left[{\left(\mathrm{d}P\right)}_X{\left(\mathrm{d}P\right)}_Y\right],F^P(X,Y)=P[{\left(\mathrm{d}P\right)}_X,{\left(\mathrm{d}P\right)}_Y]P=2Alt{\mathfrak{G}}^{\mathrm{proj}}(X,Y),$$

(457)

for the projected connection ${\nabla }^P=P\mathrm{d}$ and the curvature convention used here. The trace-normalization projector $P_0$ below is fixed, so $\mathrm{d}{P}_0=0$; its analogous response comes from the direction-dependent multiplications $L_X$, not from motion of the projector itself.

Shared Jordan reconstruction.

For every $n\ge 2$, let

$$J_n:=Herm\left(n\right)=\mathbb{R}I\oplus V_n,V_n:={Herm}_0\left(n\right),g_0(X,Y):=\frac{1}{n}Tr\left(XY\right),$$

(458)

and let $P_0:J_n\to V_n$ and $P_{\mathbf{1}}:=\mathrm{id}-P_0$ be the trace-free and scalar orthogonal projectors. For $X\in V_n$, write

$$L_{X}Z:=X\circ Z:=\frac{1}{2}(XZ+ZX),M_X:=P_0{L}_X{|}_{V_n}.$$

(459)

Trace associativity and the central cumulant Equation (A113) give

$$g_0(M_{X}Y,Z)=\frac{1}{2n}Tr\left(X\{Y,Z\}\right)=-{\mathcal{C}}_0^{\mathsf{\Psi }}(X,Y,Z).$$

(460)

Thus the metric and cubic recover the compressed product. Adjoining the distinguished unit recovers the full product by the algebraic identity

$$(aI+X)\circ (bI+Y)=(ab+g_0(X,Y))I+aY+bX+M_{X}Y,a,b\in \mathbb{R}.$$

(461)

Both identities hold for $n=2$ as well; the stabilizer classification below is stated for $n\ge 3$.

Proposition 22

(General central metric-cubic stabilizer). For every $n\ge 3$, the central metric and cubic reconstruct $H_n\left(\mathbb{C}\right)$ by Equations (460) and (461). Their exact orthogonal stabilizer is

$$\begin{array}{|c|}\hline {Stab}_{\mathrm{O}(V_n,g_0)}\left({\mathcal{C}}_0^{\mathsf{\Psi }}\right)=\mathrm{PSU}\left(n\right)⋊\langle {\tau }_{\mathrm{J}}\rangle ,{\tau }_{\mathrm{J}}\left(X\right)=X^{\mathsf{T}}.\\ \hline\end{array}$$

(462)

Since ${det}_{V_n}{\tau }_{\mathrm{J}}={(-1)}^{n(n-1)/2}$, the oriented stabilizer is

$$\begin{array}{|c|}\hline {Stab}_{\mathrm{SO}(V_n,g_0)}\left({\mathcal{C}}_0^{\mathsf{\Psi }}\right)=\left\{\begin{array}{cc}\mathrm{PSU}\left(n\right)⋊{\mathbb{Z}}_2,\hfill & n\equiv 0,1(mod4),\hfill \\ \mathrm{PSU}\left(n\right),\hfill & n\equiv 2,3(mod4).\hfill \end{array}\right.\\ \hline\end{array}$$

(463)

Higher primitive trace tensors may organize higher responses, but are not needed to recover the connected compact group. No collection of the single-matrix trace invariants $Tr\left(X^k\right)$ removes transpose, since each is transpose-invariant.

Proof. 

If $R\in \mathrm{O}(V_n,g_0)$ preserves ${\mathcal{C}}_0^{\mathsf{\Psi }}$, Equation (460) and nondegeneracy give $R\left(M_{X}Y\right)=M_{RX}\left(RY\right)$. By Equation (461), the extension $\tilde{R}(aI+X):=aI+RX$ is a unital Jordan automorphism. The classification for $H_n\left(\mathbb{C}\right)$ [77] gives precisely unitary conjugations and their compositions with transpose; these directly preserve $g_0$ and ${\mathcal{C}}_0^{\mathsf{\Psi }}$. Precisely the scalar unitaries act trivially, proving Equation (462). Transpose negates the $n(n-1)/2$ imaginary off-diagonal directions, proving Equation (463). □

Theorem 16

(Jordan normalization completion and the intrinsic qutrit cubic). With the preceding notation, for every $n\ge 2$, $a\in \mathbb{R}$, and $Z\in V_n$,

$$L_X(aI+Z)=g_0(X,Z)I+aX+M_{X}Z,$$

(464)

and the full and compressed multiplication commutators obey

$$\begin{array}{|c|}\hline [L_X,L_Y]{|}_{V_n}=\frac{1}{4}{ad}_{[X,Y]},[M_X,M_Y]=\frac{1}{4}{ad}_{[X,Y]}-W_{X,Y},\\ \hline\end{array}$$

(465)

where $W_{X,Y}Z:=g_0(Y,Z)X-g_0(X,Z)Y$. The correction is exactly the omitted scalar intermediate channel:

$$P_0{L}_X{P}_{\mathbf{1}}{L}_Y{P}_0-P_0{L}_Y{P}_{\mathbf{1}}{L}_X{P}_0=W_{X,Y}.$$

(466)

Thus commutators of the full Jordan multiplication operators take values in $Der\left(J_n\right)$, whereas trace normalization omits the scalar excursion $W_{X,Y}$, producing the $-W_{X,Y}$ commutator defect and hence the $+W_{X,Y}/4$ BKM-curvature correction below.At the maximally mixed BKM point, with the conventions of Proposition 17,

$$\begin{array}{|c|}\hline {\mathcal{C}}_0^{\mathsf{\Psi }}(X,Y,Z)=-g_0(M_{X}Y,Z),{\mathsf{Q}}_0\left(X\right)=-M_X,K_{0,X}^{\mathsf{\Psi }}=\frac{1}{2}{M}_X.\\ \hline\end{array}$$

(467)

Consequently

$$\begin{array}{|c|}\hline R_0^{\mathsf{\Psi }}(X,Y)=-\frac{1}{16}{ad}_{[X,Y]}+\frac{1}{4}{W}_{X,Y}=:{\tilde{R}}_0^J(X,Y)+\frac{1}{4}{W}_{X,Y},\\ \hline\end{array}$$

(468)

where ${\tilde{R}}_0^J:=-\frac{1}{4}[L_X,L_Y]{|}_{V_n}$ is an algebraic full-Jordan comparison, not the curvature of a nondegenerate BKM geometry on $J_n$. Indeed, extending the log-partition function to $J_n$ gives $\mathsf{\Psi }(X+cI)=\mathsf{\Psi }\left(X\right)-c$ and $D^2{\mathsf{\Psi }}_X[I,·]=0$. The restored scalar is therefore Gibbs normalization gauge, not an additional physical state tangent.

The BKM row of Equation (409) therefore identifies the full $+\frac{1}{4}{\mathrm{id}}_{{\mathsf{\Lambda }}^2{V}_n}$ term as the trace-normalization correction; it also shifts the adjoint coefficient.

For $n=3$, the tracial information cubic is the negative polarization of the determinant:

$${\mathcal{C}}_0^{\mathsf{\Psi }}(X,X,X)=-\frac{1}{3}Tr\left(X^3\right)=-detX,F\left(X\right):=-\sqrt{2}{\mathcal{C}}_0^{\mathsf{\Psi }}(X,X,X)=\sqrt{2}detX.$$

(469)

In the generator normalization $Tr\left(T_a{T}_b\right)={\delta }_{ab}/2$, this reads explicitly

$$X=x^a{T}_a⟹detX=\frac{1}{12}{d}_{abc}{x}^a{x}^b{x}^c.$$

(470)

With respect to $g_0$, F is Cartan’s normalized eight-dimensional isoparametric cubic: ${∥gradF∥}_{g_0}^2=9{∥X∥}_{g_0}^4$ and ${\mathsf{\Delta }}_{g_0}F=0$. Specializing Proposition 22 to $n=3$ gives

$$\begin{array}{|c|}\hline {Stab}_{\mathrm{SO}(V_3,g_0)}\left({\mathcal{C}}_0^{\mathsf{\Psi }}\right)=\mathrm{PSU}\left(3\right),{Stab}_{\mathrm{O}(V_3,g_0)}\left({\mathcal{C}}_0^{\mathsf{\Psi }}\right)=\mathrm{PSU}\left(3\right)⋊\langle {\tau }_{\mathrm{J}}\rangle ,\\ \hline\end{array}{\tau }_{\mathrm{J}}\left(X\right)=X^{\mathsf{T}}.$$

(471)

The outer Jordan involution ${\tau }_{\mathrm{J}}$ reverses orientation. Thus the oriented metric and central BKM cubic determine the embedded subgroup

$$H_0:={Stab}_{\mathrm{SO}(V_3,g_0)}\left({\mathcal{C}}_0^{\mathsf{\Psi }}\right)=Ad\left(\mathrm{SU}\left(3\right)\right)\simeq \mathrm{PSU}\left(3\right)$$

(472)

and the corresponding $8\oplus 20$ split in the tracial tangent fiber. This is a one-fiber statement and does not by itself define a reduction away from 0. The global reduction used below is independently defined by the ${\widehat{D}}^{\mathrm{AI}}$-parallel Cartan form in Theorem 17. Only after aligning a fixed central adjoint frame with the actual embedded holonomy subgroup ${Hol}_0\left({\widehat{D}}^{\mathrm{AI}}\right)=H_0$ does the central cubic seed the separate parallel comparison cubic ${\mathcal{C}}^J$ constructed there. The state-dependent BKM cubic ${\mathcal{C}}_X^{\mathsf{\Psi }}$ is not used for this globalization, and neither cubic supplies the principal $\mathrm{SU}\left(3\right)$ lift.

The same completion realizes the symmetric pair itself:

$$\begin{array}{|c|}\hline {\mathfrak{str}}_0\left(J_3\right)=Der\left(J_3\right)\oplus L_{V_3}\simeq \mathfrak{sl}{(3,\mathbb{C})}_{\mathbb{R}},\\ \hline\end{array}\rho \left(A\right)Z:=AZ+Z{A}^{†}.$$

(473)

Anti-Hermitian A acts by derivations and Hermitian trace-free A acts by $2L_A$. Thus the Jordan completion is the algebraic form of the same $\mathfrak{sl}(3,\mathbb{C})=\mathfrak{su}\left(3\right)\oplus V_3$ Cartan decomposition used by the $\mathrm{SL}(3,\mathbb{C})/\mathrm{SU}\left(3\right)$ mechanics.

Finally, in the qutrit split $\mathfrak{so}\left(8\right)={\mathfrak{h}}_8\oplus {\mathfrak{m}}_{20}$,

$$\begin{array}{|c|}\hline {\mathsf{\Pi }}_{20}[M_X,M_Y]=-{\mathsf{\Pi }}_{20}{W}_{X,Y},{\pi }_{20}{R}_0^{\mathsf{\Psi }}(X,Y)=\frac{1}{4}{\mathsf{\Pi }}_{20}{W}_{X,Y}.\\ \hline\end{array}$$

(474)

After complexification, ${\left({\mathfrak{m}}_{20}\right)}_{\mathbb{C}}\simeq \mathbf{10}\oplus \overline{\mathbf{10}}$. Thus the entire complementary tracial curvature channel is the scalar-compression defect, although compression also changes the adjoint coefficient. This statement is pointwise at the tracial state: the global one-form ${\mathsf{\Phi }}_{20}$ is not a curvature two-form and is not identified with the right-hand side of Equation (474).

Proposition 23

(Scalar-channel Gram response and the qutrit response modules). Equip $J_n$ with $g_0(A,B)=Tr\left(AB\right)/n$, so that $∥I∥=1$. The unresolved coupling of the fixed scalar compression is

$${\mathsf{U}}_X\left(Z\right)=P_{\mathbf{1}}{L}_X{P}_0\left(Z\right)=g_0(X,Z)I,{\mathsf{U}}_X^{†}\left(aI\right)=aX.$$

(475)

Hence its Gram response is the rank-one endomorphism

$$\begin{array}{|c|}\hline {\mathfrak{G}}_0^J(X,Y)Z=g_0(Y,Z)X,{\mathfrak{G}}_0^J(X,Y)=X\otimes Y^{♭}.\\ \hline\end{array}$$

(476)

It packages both the scalar metric contraction and the normalization commutator defect:

$${Tr}_{End\left(V_n\right)}{\mathfrak{G}}_0^J(X,Y)=g_0(X,Y),2Alt{\mathfrak{G}}_0^J(X,Y)=W_{X,Y}.$$

(477)

Moreover,

$${span}_{X,Y}\left\{{\mathfrak{G}}_0^J(X,Y)\right\}=End\left(V_n\right).$$

(478)

Thus a one-dimensional eliminated scalar channel can have full endomorphism span through its direction-dependent entrance and exit couplings.

For the ordinary endomorphism Hilbert-Schmidt norm, write ${\mathfrak{S}}_0^J=Sym{\mathfrak{G}}_0^J$ and ${\mathfrak{A}}_0^J=Alt{\mathfrak{G}}_0^J$. Then

$$\begin{array}{cc}\hfill {∥{\mathfrak{G}}_0^J(X,Y)∥}_{End,\mathrm{HS}}^2& ={∥X∥}^2{∥Y∥}^2,\hfill \end{array}$$

(479)

$$\begin{array}{cc}\hfill {∥{\mathfrak{S}}_0^J(X,Y)∥}_{End,\mathrm{HS}}^2& =\frac{1}{2}\left({∥X∥}^2{∥Y∥}^2+g_0{(X,Y)}^2\right),\hfill \end{array}$$

(480)

$$\begin{array}{cc}\hfill {∥{\mathfrak{A}}_0^J(X,Y)∥}_{End,\mathrm{HS}}^2& =\frac{1}{2}\mathsf{\Delta }(X,Y).\hfill \end{array}$$

(481)

This endomorphism norm differs from the curvature convention ${∥W_{X,Y}∥}_{\mathfrak{so}}^2=\mathsf{\Delta }$ by the factor ${∥W_{X,Y}∥}_{End,\mathrm{HS}}^2=2\mathsf{\Delta }$.

For qutrits, the real response decomposition is

$$\begin{array}{|c|}\hline {End}_{\mathbb{R}}\left(V_3\right)=\underset{{Sym}^2{V}_3}{\underbrace{\mathbf{1}\oplus {\mathbf{8}}_s\oplus \mathbf{27}}}\oplus \underset{{\mathsf{\Lambda }}^2{V}_3}{\underbrace{{\mathbf{8}}_a\oplus {\mathbf{20}}_{\mathbb{R}}}},{\left({\mathbf{20}}_{\mathbb{R}}\right)}_{\mathbb{C}}=\mathbf{10}\oplus \overline{\mathbf{10}}.\\ \hline\end{array}$$

(482)

In the standard adjoint coordinates with $d_{amn}{d}_{bmn}=5{\delta }_{ab}/3$, the symmetric projectors are

$${\left({\mathsf{\Pi }}_{1}s\right)}_{ab}=\frac{1}{8}{\delta }_{ab}{s}_{cc},{\left({\mathsf{\Pi }}_{8_s}s\right)}_{ab}=\frac{3}{5}{d}_{abe}{d}_{cde}{s}_{cd},{\mathsf{\Pi }}_{27}=I_{\mathrm{sym}}-{\mathsf{\Pi }}_1-{\mathsf{\Pi }}_{8_s}.$$

(483)

Equation (477) should be read as a reconstruction within the response package, not as a metric generated from nothing: $g_0$ already enters $Y^{♭}$ and the orthogonal scalar projection.

Proof. 

Equation (464) gives the coupling and its adjoint. Rank-one operators span $End\left(V_n\right)$, their traces and Hilbert-Schmidt products give the displayed norm identities, and the qutrit decomposition is the standard real form of $\mathbf{8}\otimes \mathbf{8}$. □

Corollary 29

(Associators and fixed-element Jordan compatibility). The full Hermitian Jordan associator and its trace-free compression are

$$\begin{array}{cc}\hfill {Assoc}_{\circ }(X,Y,Z)& :=(X\circ Y)\circ Z-X\circ (Y\circ Z)=\frac{1}{4}[Y,[X,Z]],\hfill \\ \hfill {Assoc}_{\ast }(X,Y,Z)& :=(X\ast Y)\ast Z-X\ast (Y\ast Z)\hfill \end{array}$$

(484)

$$\begin{array}{c}={Assoc}_{\circ }(X,Y,Z)-g_0(X,Y)Z+g_0(Y,Z)X.\hfill \end{array}$$

(485)

Consequently,

$$\begin{array}{|c|}\hline R_0^{\mathsf{\Psi }}(X,Y)Z=\frac{1}{4}\left({Assoc}_{\ast }(X,Y,Z)-{Assoc}_{\ast }(Y,X,Z)\right).\\ \hline\end{array}$$

(486)

The factor $1/4$ is essential because $K_{0,X}^{\mathsf{\Psi }}=M_X/2$.

Let $X^{\ast 2}:=X\ast X$. The compressed Jordan-identity defect is

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill {\mathfrak{J}}_{\ast }(X;Y)& :=(X\ast X)\ast (X\ast Y)-X\ast \left(\right(X\ast X)\ast Y)\hfill \\ & =g_0(X\ast X,Y)X-g_0(X,Y)(X\ast X).\hfill \end{array}\\ \hline\end{array}$$

(487)

For $n=2$, $X\ast Y=0$ for every $X,Y\in V_2$, so the compressed algebra is the associative zero algebra. For every $n\ge 3$, $(V_n,\ast )$ is commutative but not Jordan. More precisely, for $X\ne 0$,

$$\begin{array}{|c|}\hline {\mathfrak{J}}_{\ast }(X;·)=0⟺X\ast X=\lambda X⟺X\mathit{has}\mathit{exactly}\mathit{two}\mathit{distinct}\mathit{eigenvalues}.\\ \hline\end{array}$$

(488)

This is a fixed-element compatibility locus; it does not restore the Jordan identity for the whole compressed algebra.

Proof. 

Direct expansion of the anticommutators gives Equation (484); substituting $X\ast Y=X\circ Y-g_0(X,Y)I$ gives Equation (485). Antisymmetrization and Equation (467) prove Equation (486). Since $[X\ast X,X]=0$ as matrices, Equation (465) gives Equation (487). For $n=2$, the anticommutator of two traceless Hermitian matrices is scalar. For $n\ge 3$, $X=diag(1,0,-1,0,\dots ,0)$ has a nonzero defect. Finally, ${\mathfrak{J}}_{\ast }(X;·)=0$ is equivalent to linear dependence of X and $X\ast X$, hence to $X^2-\lambda X-g_0(X,X)I=0$. Hermiticity makes this equivalent to a quadratic spectrum. Conversely, if the eigenvalues are $a,b$ with multiplicities $r,n-r$, tracelessness gives $ab=-g_0(X,X)$ and $X\ast X=(a+b)X$. □

6.7.5. Low-Rank Relative-Transport Diagnostics

Corollary 30

(Diagonal-qutrit $\mathrm{U}\left(1\right)$ rotation, qubit spin lift, and qutrit obstruction). The compact relative bridge has the following sharply different low-rank consequences.

Diagonal qutrits. Apply the construction intrinsically to

$${\mathcal{C}}_3:=\{diag(p_1,p_2,p_3):p_i>0,{\sum }_i{p}_i=1\}.$$

(489)

By Corollary 18, its BKM metric $g_{\mathrm{F}}$ is the Fisher metric with the radius-2 spherical realization. The pulled-back affine-invariant metric is flat in diagonal logarithmic coordinates. Fixing the orientation by an ordering of the spectral labels, let $\mathsf{j}$ be the positive quarter-turn. If a positively oriented piecewise smooth simple loop $\gamma =\partial \mathsf{\Sigma }$ bounds a region in ${\mathcal{C}}_3$, then

$$\begin{array}{|c|}\hline U_{\gamma }^{\mathrm{rel}}=exp\left[\frac{{Area}_{g_{\mathrm{F}}}\left(\mathsf{\Sigma }\right)}{4}\mathsf{j}\right]\in \mathrm{SO}\left(2\right)\simeq \mathrm{U}\left(1\right).\\ \hline\end{array}$$

(490)

For a nonsimple loop, the area is signed and counted with winding multiplicity; reversing the loop inverts the transporter. This is an exact relative tangent-frame $\mathrm{U}\left(1\right)$ transporter comparing the two loop holonomies, not by itself an electromagnetic or Berry phase.

Qubits. On the faithful qubit manifold, $U_{\gamma }^{\mathrm{rel}}\in \mathrm{SO}\left(3\right)=\mathrm{PSU}\left(2\right)$. The Gibbs-coordinate manifold is contractible and oriented, so its oriented frame bundle has a spin lift, unique up to equivalence, and the relative transporter lifts to $Spin\left(3\right)\simeq \mathrm{SU}\left(2\right)$. At the maximally mixed state,

$$R_0^{\mathsf{\Psi }}=0,{\mathcal{F}}_0^{\mathrm{rel}}=-{\overline{R}}_0^{\mathrm{AI}},log{U}_{{\gamma }_{\epsilon }}^{\mathrm{rel}}={\epsilon }^2{\overline{R}}_0^{\mathrm{AI}}(H,K)+O\left({\epsilon }^3\right).$$

(491)

Thus the leading relative rotation is the inverse of the affine-invariant, equivalently Thomas-Wigner, tangent rotation in the normalization established above. It becomes a unitary two-component mixing only after a spinor carrier and its physical coupling are supplied.

Full qutrits. Here $U_{\gamma }^{\mathrm{rel}}$ is naturally $\mathrm{SO}\left(8\right)$-valued, but it is not confined to the canonical adjoint subgroup $Ad\left(\mathrm{SU}\right(3\left)\right)\simeq \mathrm{PSU}\left(3\right)\subset \mathrm{SO}\left(8\right)$. Indeed, for independent commuting diagonal $H,K$ at the center,

$$\begin{array}{|c|}\hline {\mathcal{F}}_0^{\mathrm{rel}}(H,K)=\frac{1}{4}\left(H{\wedge }_{g}K\right),\left(H{\wedge }_{g}K\right)L:=g_0(K,L)H-g_0(H,L)K,\\ \hline\end{array}$$

(492)

Taking the alternating part of Equation (482) gives $\mathfrak{so}(8,\mathbb{C})\simeq {\mathsf{\Lambda }}^2\mathbf{8}=\mathbf{8}\oplus \mathbf{10}\oplus \overline{\mathbf{10}}$, with ${\left({\mathfrak{m}}_{20}\right)}_{\mathbb{C}}=\mathbf{10}\oplus \overline{\mathbf{10}}$ [33], Table 24. The real complement ${\mathfrak{m}}_{20}$ is irreducible [31]. This is a single real curvature module: the two complex summands are conjugate, not independently supplied fields. The automorphism in Equation (432) realizes this branching by cubic-shear commutators; its real 8- and 20-dimensional eigenspaces complexify as $\mathbf{8}$ and $\mathbf{10}\oplus \overline{\mathbf{10}}$, respectively. This is a ${\mathbb{Z}}_3$-graded reductive decomposition, not a symmetric-pair or direct-product decomposition: after complexification,

$$[\mathbf{10},\mathbf{10}]=\overline{\mathbf{10}},[\overline{\mathbf{10}},\overline{\mathbf{10}}]=\mathbf{10},[\mathbf{10},\overline{\mathbf{10}}]=\mathbf{8},$$

(493)

so $[{\mathfrak{m}}_{20},{\mathfrak{m}}_{20}]=\mathfrak{so}\left(8\right)$ [32], Remark 3.1. The quotient $\mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right)$ is locally three-symmetric; its globally three-symmetric universal cover is $Spin\left(8\right)/\mathrm{PSU}\left(3\right)$ [88].

The obstruction is stronger than the displayed commuting example. For independent $H,K$, $W_{H,K}$ has rank two. Every nonzero ${ad}_A$ on $\mathfrak{su}\left(3\right)$ instead has rank four or six, according as A has a repeated eigenvalue or is regular. Hence no nonzero decomposable bivector belongs to ${\mathfrak{h}}_8$, and

$${\mathsf{\Pi }}_{20}\left(W_{H,K}\right)\ne 0\mathit{for}\mathit{every}\mathit{independent}H,K.$$

(494)

When $[H,K]=0$, Equation (428) further gives ${\mathsf{\Pi }}_8\left(W_{H,K}\right)=0$: its tracial infinitesimal diagonal generator is purely an ${\mathfrak{m}}_{20}$ direction, not a color Cartan or hypercharge direction. This does not assert that the finite Fisher-area transporter away from the center remains in a fixed complement, since ${\mathfrak{m}}_{20}$ is not a Lie algebra. Thus the normalized relative defect does not itself define a color-$\mathrm{SU}\left(3\right)$ connection. Its closed generated mismatch group is $\mathrm{SO}\left(8\right)$ by Equation (411), but this does not turn the relative comparisons into the holonomy of a single $\mathrm{SO}\left(8\right)$ connection. The genuine BKM tangent holonomy is independently $\mathrm{SO}\left(8\right)$, whereas the canonical symmetric-space principal connection has $\mathrm{SU}\left(3\right)$ holonomy and its tangent representation sees only $\mathrm{PSU}\left(3\right)$.

Corollary 31

(Classical generators of the tracial qutrit transverse channel). At ${\rho }_{\ast }=I_3/3$, use the Gibbs tangent realization $V={Herm}_0\left(3\right)$ with $g_0(X,Y)=Tr\left(XY\right)/3$, and identify ${\mathsf{\Lambda }}^{2}V$ with $\mathfrak{so}(V,g_0)$ by $X\wedge Y\mapsto W_{X,Y}$. A nonzero simple bivector $X\wedge Y$ lies in ${\mathfrak{m}}_{20}$ if and only if its two-plane is tangent at ${\rho }_{\ast }$ to some fixed commuting simplex (295). Moreover,

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill {\mathfrak{m}}_{20}& ={span}_{\mathbb{R}}\{X\wedge Y:X,Y\in V,[X,Y]=0\}\hfill \\ & ={span}_{\mathbb{R}}\{R_0^{\mathsf{\Psi }}(X,Y):X,Y\in V,[X,Y]=0\}.\hfill \end{array}\\ \hline\end{array}$$

(495)

These curvature values, taken over all such simplices, already Lie-generate $\mathfrak{so}\left(8\right)$. The span statement does not make ${\mathfrak{m}}_{20}$ a classical state space: it also contains nondecomposable bivectors.

Proof. 

Since ${\mathfrak{m}}_{20}=kerb$ and $b(X\wedge Y)=[X,Y]$, membership for a simple bivector is exactly commutation. Simultaneous unitary diagonalization identifies each independent commuting pair with a diagonal Cartan plane. The Gibbs parametrization $X\mapsto {\rho }_X$ has differential $D{\rho }_0\left[X\right]=-X/3$, so the same plane is tangent to the corresponding ${\mathcal{C}}_{\mathcal{A}}$. Choose a nonzero diagonal Cartan bivector ${\xi }_0$. The real span of its $\mathrm{PSU}\left(3\right)$ conjugation orbit is a nonzero invariant subspace of ${\mathfrak{m}}_{20}$, hence is all of it by the real irreducibility established in Corollary 30 [31], Section 2. Every orbit element remains commuting, proving the first equality. The central formula $R_0^{\mathsf{\Psi }}(X,Y)=W_{X,Y}/4$ for commuting $X,Y$ proves the second; on the Cartan plane this is precisely the Fisher curvature from Corollary 18. Finally, $[{\mathfrak{m}}_{20},{\mathfrak{m}}_{20}]=\mathfrak{so}\left(8\right)$ by Equation (493), proving Lie generation.

For an explicit nonsimple element, take the independent diagonal $X,Y_0$ in Equation (437), let $S=E_{12}+E_{23}+E_{31}$, and put $A=S+S^{†}$, $B=\mathrm{i}(S-S^{†})$. Then $[A,B]=0$ and the four Hermitian matrices $X,Y_0,A,B$ are real-linearly independent. Thus $\zeta =X\wedge Y_0+A\wedge B\in {\mathfrak{m}}_{20}$ but $\zeta \wedge \zeta =2X\wedge Y_0\wedge A\wedge B\ne 0$, which rules out decomposability. □

Together with Ambrose-Singer, this gives another proof of the full BKM tangent holonomy in Corollary 27 for $n=3$. It combines curvature from differently oriented classical sectors in the same quantum tangent space; one fixed classical qutrit simplex instead has only its intrinsic $\mathrm{SO}\left(2\right)$ Fisher holonomy. Neither the span nor the Lie-closure statement identifies the 20 with an integrable distribution or a Lie subalgebra.

Remark 12

(Complex type, oriented polarization, and nonintegrability of the real 20). The branching ${\left({\mathfrak{m}}_{20}\right)}_{\mathbb{C}}=\mathbf{10}\oplus \overline{\mathbf{10}}$ makes ${\mathfrak{m}}_{20}$ a real irreducible module of complex type. On its complexification define

$$J_{\perp }{{|}_{\mathbf{10}}=\mathrm{i}\mathrm{id},J_{\perp }|}_{\overline{\mathbf{10}}}=-\mathrm{i}\mathrm{id}.$$

(496)

This commutes with real conjugation and descends to a $\mathrm{PSU}\left(3\right)$-equivariant orthogonal complex structure on ${\mathfrak{m}}_{20}$, unique up to sign by real Schur theory. The normalizing outer involution exchanges the two complex summands and sends $J_{\perp }$ to $-J_{\perp }$.

The supplied orientation of the eight-dimensional tangent fiber does allow an intrinsic choice. Let $(E,g,\phi )$ be a compatible metric-Cartan pair: the bracket defined by $g\left(\beta \right(X,Y),Z)=\phi (X,Y,Z)$ makes E a compact Lie algebra isomorphic to $\mathfrak{su}\left(3\right)$. Let $d_{\beta }$ be its algebraic Chevalley-Eilenberg differential, not a derivative on the state manifold, and let * be the oriented Hodge star. Identify ${\mathsf{\Lambda }}^2{E}^{\ast }$ with $\mathfrak{so}(E,g)$ and set

$$\begin{array}{cccc}\hfill {\mathsf{A}}_{\phi }& :=d_{\beta }^{\ast }\circ \ast \circ (\phi \wedge ·),\hfill & \hfill a_{\phi }& :={\left[-\frac{1}{20}{tr}_{{\mathsf{\Lambda }}^2{E}^{\ast }}\left({\mathsf{A}}_{\phi }^2\right)\right]}^{1/2},\hfill \\ \hfill J_{\perp }^{\mathrm{or}}& :={\left.a_{\phi }^{-1}{\mathsf{A}}_{\phi }\right|}_{{\mathfrak{m}}_{20}},\hfill & \hfill {\left(J_{\perp }^{\mathrm{or}}\right)}^2& =-\mathrm{id}.\hfill \end{array}$$

(497)

Here ${\mathsf{A}}_{\phi }$ is skew-adjoint, annihilates ${\mathfrak{h}}_8$, and satisfies ${\mathsf{A}}_{\phi }^2=-a_{\phi }^2{\mathsf{\Pi }}_{20}$ with $a_{\phi }>0$; the algebraic proof is in Appendix D.6. Thus $J_{\perp }^{\mathrm{or}}$ is an orthogonal member of $\{\pm J_{\perp }\}$. Reversing the orientation of E reverses it, whereas replacing φ by any nonzero real multiple leaves it unchanged. The equal complex orientations induced by $\pm J_{\perp }$ on the real 20 do not obstruct this construction: that orientation is not the supplied orientation of E, which the Cartan outer automorphism reverses. The oriented central metric-cubic data also determine the invariant Cartan line of Equation (472), hence this polarization in the tracial fiber, but not its global parallel extension.

On the homogeneous space $\mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right)$ this defines an invariant almost complex structure, but it is not integrable. Indeed, choosing ${\mathfrak{m}}^{1,0}=\mathbf{10}$, the bracket $[\mathbf{10},\mathbf{10}]=\overline{\mathbf{10}}\ne 0$ in Equation (493) violates the homogeneous integrability criterion. On the eight-dimensional qutrit state manifold, $J_{\perp }$ instead acts on the value index of ${\mathsf{\Phi }}_{20}\in {\mathsf{\Omega }}^1(M,{\mathfrak{m}}_{20})$; it is not an almost complex structure on the state tangent bundle, and its homogeneous integrability is not an intrinsic-torsion condition there. The scalar $\mathrm{i}$ in Equation (496) belongs to the algebraic complexification ${\left({\mathfrak{m}}_{20}\right)}_{\mathbb{C}}$, not to the amplitude structure $I_V$. No principal $\mathrm{SU}\left(3\right)$ lift is needed to construct $J_{\perp }^{\mathrm{or}}$ or its complex rank-ten value bundle. With a specified defining triplet ${\mathcal{H}}_3$, identifying its $+\mathrm{i}$-eigenspace with ${Sym}^3{\mathcal{H}}_3$, rather than the conjugate representation, must respect the orientation and sign convention in Equation (497). This is a representation-labeling question, not missing geometric polarization. The transverse structure remains distinct from both $I_V$ and $I_B^{\nabla }$.

Proposition 24

(Compact-subgroup criterion and relative Wilson characters). Let $G\subseteq \mathrm{SO}\left(d\right)$ be closed and suppose ${\widehat{D}}^{\mathrm{AI}}$ preserves a G-reduction $P_G$ of the oriented g-orthonormal frame bundle. The following are equivalent:

1. 

$D^{\mathsf{\Psi }}={\widehat{D}}^{\mathrm{AI}}+{\mathcal{B}}^{\mathrm{rel}}$ preserves the same reduction;

2. 

${\mathcal{B}}^{\mathrm{rel}}\in {\mathsf{\Omega }}^1(ad{P}_G)$;

3. 

in ${\widehat{D}}^{\mathrm{AI}}$-parallel G-frames, $U_{\gamma }^{\mathrm{rel}}\in G$ for every sufficiently short path from every base point.

Therefore ${\mathcal{F}}^{\mathrm{rel}}\in {\mathsf{\Omega }}^2(ad{P}_G)$ is necessary, and any component outside $\mathfrak{g}$ is a local obstruction; curvature membership alone is not generally sufficient for the global reduction.

For every finite-dimensional unitary representation ϱ of $\mathrm{SO}\left(d\right)$, the relative Wilson character

$$W_{\varrho }\left(\gamma \right):=\frac{1}{dim{V}_{\varrho }}Tr\varrho \left(U_{\gamma }^{\mathrm{rel}}\right)$$

(498)

is invariant under endpoint-frame changes. This remains true for an open base path because the two transports close each other in the initial fiber; it is not the gauge invariance of a single open Wilson line. In the defining representation, the small-loop character begins only at squared area:

$$\frac{1}{d}Tr{U}_{{\gamma }_{\epsilon }}^{\mathrm{rel}}=1-\frac{{\epsilon }^4}{2d}{∥{\mathcal{F}}_X^{\mathrm{rel}}(H,K)∥}_{\mathrm{HS}}^2+O\left({\epsilon }^5\right),$$

(499)

whereas eigenangles and represented off-diagonal mixing amplitudes are generically $O\left({\epsilon }^2\right)$.

Proposition 25

(Invariant Cartan three-form). For $n\ge 2$ and $\alpha >0$, equip $T_I{\mathcal{P}}_n^1=\mathfrak{p}$ with $g_I(U,V)=\alpha Tr\left(UV\right)$ and set

$$H_I(U,V,W):=\frac{\mathrm{i}\alpha }{2}Tr\left([U,V]W\right).$$

(500)

This tensor is real, totally antisymmetric, and invariant under the isotropy representation of $K=\mathrm{SU}\left(n\right)$. It therefore has a unique $G=\mathrm{SL}(n,\mathbb{C})$-invariant extension to $G/K={\mathcal{P}}_n^1$. Equivalently, it is the unique Levi-Civita-parallel extension with the prescribed value at I; invariance under the global K-action alone would not imply uniqueness. Since $\mathfrak{su}\left(n\right)$ is simple, ${\left({\mathsf{\Lambda }}^3{\mathfrak{p}}^{\ast }\right)}^K$ is one-dimensional, so every nonzero G-invariant three-form is a real multiple of this one. The metric scale α fixes its norm, while $H\mapsto -H$ is an independent discrete choice. For $n=2$, H is a top-degree volume form and its sign selects an orientation; for $n\ge 3$, $3<dim\left({\mathcal{P}}_n^1\right)=n^2-1$, so its sign is not merely an orientation of the base.

Trace cyclicity gives alternation, and the invariant-tensor correspondence on a simply connected symmetric space gives the unique full-G-invariant, equivalently Levi-Civita-parallel, extension [24,26,81]. The standard Cartan three-form classification gives the one-dimensional invariant space [29,30]. Only the compact-dual normalization is retained in Appendix D.8.

6.7.6. Adapted Qutrit Color Connection and Exact Reconstruction

Theorem 17

(Projected qutrit $\mathrm{PSU}\left(3\right)$ tangent connection and exact $\mathfrak{so}\left(8\right)$$8+20$ reconstruction). Set $n=3$ and let H be the nonzero affine-invariant, equivalently Levi-Civita-parallel, Cartan three-form of 25. Pull it back to the Gibbs-coordinate manifold and denote it by $\overline{\phi }$. Transport it through the positive isometry J by

$${\phi }_X(U,V,W):={\overline{\phi }}_X(J_X^{-1}U,J_X^{-1}V,J_X^{-1}W).$$

(501)

Then ${\widehat{D}}^{\mathrm{AI}}\phi =0$. At the tracial model retain ${\tau }_{\mathrm{J}}\left(X\right)=X^{\mathsf{T}}$ from Equation (471) and define the Hermitian-model outer Lie automorphism

$${\tau }_{\mathfrak{g}}\left(X\right):=-X^{\mathsf{T}}=-{\tau }_{\mathrm{J}}\left(X\right).$$

(502)

For ${\phi }_0{:=\phi |}_{X=0}$, the exact sign and stabilizer relations are

$$\begin{array}{cccc}\hfill {\tau }_{\mathrm{J}}^{\ast }{\mathcal{C}}_0^{\mathsf{\Psi }}& ={\mathcal{C}}_0^{\mathsf{\Psi }},\hfill & \hfill {\tau }_{\mathrm{J}}^{\ast }{\phi }_0& =-{\phi }_0,\hfill \\ \hfill {\tau }_{\mathfrak{g}}^{\ast }{\mathcal{C}}_0^{\mathsf{\Psi }}& =-{\mathcal{C}}_0^{\mathsf{\Psi }},\hfill & \hfill {\tau }_{\mathfrak{g}}^{\ast }{\phi }_0& ={\phi }_0.\hfill \end{array}$$

(503)

$$\begin{array}{cc}\hfill {Stab}_{\mathrm{GL}\left(V_3\right)}\left({\phi }_0\right)& ={Stab}_{\mathrm{O}(V_3,g_0)}\left({\phi }_0\right)\hfill \\ & =Aut\left(\mathfrak{su}\left(3\right)\right)=\mathrm{PSU}\left(3\right)⋊\langle {\tau }_{\mathfrak{g}}\rangle ,\hfill \\ \hfill {Stab}_{\mathrm{SO}(V_3,g_0)}\left({\phi }_0\right)& =\mathrm{PSU}\left(3\right).\hfill \end{array}$$

(504)

Moreover,

$${Stab}_{\mathrm{O}(V_3,g_0)}({\mathcal{C}}_0^{\mathsf{\Psi }},{\phi }_0)=\mathrm{PSU}\left(3\right).$$

(505)

Thus φ defines a ${\widehat{D}}^{\mathrm{AI}}$-parallel $\mathrm{PSU}\left(3\right)$-reduction $P_{\phi }$ of the oriented g-orthonormal frame bundle [29,30,31]. Recall $H_0=Ad\left(\mathrm{SU}\left(3\right)\right)$ from Equation (472) and fix the standard oriented central adjoint identification $T_{0}M=V_3$ used above. Since $J_0^{\left(\alpha \right)}=\sqrt{3\alpha }\mathrm{id}$, the normalized tangent holonomy and the two central stabilizer calculations give the exact embedded equality

$$\begin{array}{|c|}\hline {Hol}_0^0\left({\widehat{D}}^{\mathrm{AI}}\right)={Hol}_0\left({\widehat{D}}^{\mathrm{AI}}\right)=H_0={Stab}_{\mathrm{SO}(V_3,g_0)}\left({\mathcal{C}}_0^{\mathsf{\Psi }}\right)={Stab}_{\mathrm{SO}(V_3,g_0)}\left({\phi }_0\right).\\ \hline\end{array}$$

(506)

This is equality of the actual subgroups in the fixed central frame, not only an abstract isomorphism with $\mathrm{PSU}\left(3\right)$.

For $X\in M$ and a piecewise smooth path $\gamma :0\to X$, write $P_{\gamma }:=P_{\gamma }^{{\widehat{D}}^{\mathrm{AI}}}$ and define

$$\begin{array}{|c|}\hline {\mathcal{C}}_X^J(U,V,W):={\mathcal{C}}_0^{\mathsf{\Psi }}(P_{\gamma }^{-1}U,P_{\gamma }^{-1}V,P_{\gamma }^{-1}W).\\ \hline\end{array}$$

(507)

The full-holonomy equality in Equation (506) makes this definition independent of γ. It is the unique ${\widehat{D}}^{\mathrm{AI}}$-parallel symmetric cubic with ${\mathcal{C}}_0^J={\mathcal{C}}_0^{\mathsf{\Psi }}$. If $u_0:V_3\to T_{0}M$ is the chosen oriented $g_0$-orthonormal adjoint frame and $u_{\gamma }:=P_{\gamma }{u}_0$, then the stabilizer-frame fibers are

$${\left(P_{{\mathcal{C}}^J}\right)}_X=u_{\gamma }{H}_0={\left(P_{\phi }\right)}_X,\begin{array}{|c|}\hline P_{{\mathcal{C}}^J}=P_{\phi }.\\ \hline\end{array}$$

(508)

Thus ${\mathcal{C}}^J$ is a connection-dependent holonomy extension of central data, not a globalization supplied by the central BKM cubic alone. The comparison cubic and the alternating Cartan form encode the same oriented reduction but are inequivalent tensors, with different disconnected orthogonal stabilizers as in Equation (503). The actual construction imposes only ${\mathcal{C}}_0^J={\mathcal{C}}_0^{\mathsf{\Psi }}$; it asserts no equality with the state-dependent BKM cubic away from 0, and on an open dense set that cubic does not have this stabilizer, as quantified in Remark 14. Let

$$\mathfrak{so}(TM,g)={\mathfrak{h}}_{\phi }\oplus {\mathfrak{m}}_{\phi },{\pi }_8:\mathfrak{so}(TM,g)\to {\mathfrak{h}}_{\phi },{\pi }_{20}:=\mathrm{id}-{\pi }_8$$

(509)

be its parallel reductive splitting; at the tracial state it is exactly Equation (427).

Split the normalized connection defect and define

$$\begin{array}{|c|}\hline \begin{array}{cccccc}\hfill {\mathcal{B}}^{\mathrm{rel}}& ={\mathcal{B}}_8+{\mathsf{\Phi }}_{20},\hfill & \hfill {\mathcal{B}}_8& :={\pi }_8{\mathcal{B}}^{\mathrm{rel}},\hfill & \hfill {\mathsf{\Phi }}_{20}& :={\pi }_{20}{\mathcal{B}}^{\mathrm{rel}},\hfill \\ \hfill D^{\mathrm{col}}& :={\widehat{D}}^{\mathrm{AI}}+{\mathcal{B}}_8,\hfill & \hfill D^{\mathsf{\Psi }}& =D^{\mathrm{col}}+{\mathsf{\Phi }}_{20}.\hfill \end{array}\\ \hline\end{array}$$

(510)

For the BKM dual-flat pair, let

$$K^{\mathsf{\Psi }}:=\frac{1}{2}(\nabla -{\nabla }^{\ast })=-\frac{1}{2}\mathsf{Q}.$$

(511)

Combining its Levi-Civita midpoint with Equation (510) gives the exact connection decomposition

$$\begin{array}{|c|}\hline \nabla =D^{\mathrm{col}}+{\mathsf{\Phi }}_{20}+K^{\mathsf{\Psi }},{\nabla }^{\ast }=D^{\mathrm{col}}+{\mathsf{\Phi }}_{20}-K^{\mathsf{\Psi }}.\\ \hline\end{array}$$

(512)

The connection $D^{\mathrm{col}}$ is g-metric and preserves $P_{\phi }$. It is the orthogonally projected, or adapted, $\mathrm{PSU}\left(3\right)$ connection of this reduction; in particular,

$$D^{\mathrm{col}}\phi =0,D^{\mathrm{col}}{\mathcal{C}}^J=0.$$

(513)

By Remark 12, the induced connections on ${\mathfrak{m}}_{\phi }$ also satisfy ${\widehat{D}}^{\mathrm{AI}}{J}_{\perp }^{\mathrm{or}}=D^{\mathrm{col}}{J}_{\perp }^{\mathrm{or}}=0$. No $D^{\mathsf{\Psi }}$-parallelness is asserted. The complementary field ${\mathsf{\Phi }}_{20}\in {\mathsf{\Omega }}^1\left({\mathfrak{m}}_{\phi }\right)$ is its intrinsic-torsion component, not a second connection. Since $D^{\mathsf{\Psi }}$ is torsion free,

$$T^{\mathrm{col}}(X,Y)=-\left({\mathsf{\Phi }}_{20}\left(X\right)Y-{\mathsf{\Phi }}_{20}\left(Y\right)X\right).$$

(514)

With the convention

$$({\mathsf{\Phi }}_{20}\wedge {\mathsf{\Phi }}_{20})(X,Y):=[{\mathsf{\Phi }}_{20}\left(X\right),{\mathsf{\Phi }}_{20}\left(Y\right)],[K^{\mathsf{\Psi }},K^{\mathsf{\Psi }}](X,Y):=[K_X^{\mathsf{\Psi }},K_Y^{\mathsf{\Psi }}],$$

(515)

the reconstruction has the exact curvature form

$$\begin{array}{|c|}\hline R^{\mathsf{\Psi }}=R^{\mathrm{col}}+{\mathrm{d}}^{D^{\mathrm{col}}}{\mathsf{\Phi }}_{20}+{\mathsf{\Phi }}_{20}\wedge {\mathsf{\Phi }}_{20}.\\ \hline\end{array}$$

(516)

Because $D^{\mathrm{col}}$ preserves the reductive splitting, ${\mathrm{d}}^{D^{\mathrm{col}}}{\mathsf{\Phi }}_{20}$ is ${\mathfrak{m}}_{\phi }$-valued, and hence

$$\begin{array}{cc}\hfill {\pi }_8{R}^{\mathsf{\Psi }}& =R^{\mathrm{col}}+{\pi }_8({\mathsf{\Phi }}_{20}\wedge {\mathsf{\Phi }}_{20})=-{\pi }_8[K^{\mathsf{\Psi }},K^{\mathsf{\Psi }}],\hfill \end{array}$$

(517)

$$\begin{array}{cc}\hfill {\pi }_{20}{R}^{\mathsf{\Psi }}& ={\mathrm{d}}^{D^{\mathrm{col}}}{\mathsf{\Phi }}_{20}+{\pi }_{20}({\mathsf{\Phi }}_{20}\wedge {\mathsf{\Phi }}_{20})=-{\pi }_{20}[K^{\mathsf{\Psi }},K^{\mathsf{\Psi }}].\hfill \end{array}$$

(518)

Both quadratic projections can be nonzero by the nonsymmetric bracket relations in Equation (493). The second flatness identity is the exact mixed Codazzi equation

$$\begin{array}{|c|}\hline {\mathrm{d}}^{D^{\mathrm{col}}}{K}^{\mathsf{\Psi }}+[{\mathsf{\Phi }}_{20}\wedge K^{\mathsf{\Psi }}]=0,\\ \hline\end{array}[{\mathsf{\Phi }}_{20}\wedge K^{\mathsf{\Psi }}](X,Y):=[{\mathsf{\Phi }}_{20}\left(X\right),K_Y^{\mathsf{\Psi }}]-[{\mathsf{\Phi }}_{20}\left(Y\right),K_X^{\mathsf{\Psi }}].$$

(519)

Since ${\widehat{R}}^{\mathrm{AI}}$ is ${\mathfrak{h}}_{\phi }$-valued, the normalized relative curvature also obeys

$$\begin{array}{|c|}\hline {\pi }_{20}{\mathcal{F}}^{\mathrm{rel}}={\pi }_{20}{R}^{\mathsf{\Psi }},\\ \hline\end{array}$$

(520)

whose cubic and connection-difference expressions are Equation (518).

At $X=0$, corresponding to the tracial state ${\rho }_{\ast }=I_3/3$ in the Gibbs-isometric coordinates used in Equation (429),

$${\mathcal{B}}_0^{\mathrm{rel}}={\mathcal{B}}_{8,0}={\mathsf{\Phi }}_{20,0}=0,{\widehat{R}}_0^{\mathrm{col}}=-\frac{5}{16}{\mathsf{\Pi }}_8,{\widehat{\left({\mathrm{d}}^{D^{\mathrm{col}}}{\mathsf{\Phi }}_{20}\right)}}_0=\frac{1}{4}{\mathsf{\Pi }}_{20}.$$

(521)

Thus ${\mathsf{\Phi }}_{20}$ vanishes pointwise at the tracial state, but its first covariant exterior derivative supplies the nonzero complement curvature, equivalently the ${\mathfrak{m}}_{20}$ projection of the cubic-shear commutator. The one-form ${\mathsf{\Phi }}_{20}$ itself is not that two-form. Equivalently,

$$R_0^{\mathrm{col}}(H,K)=-\frac{5}{144}{ad}_{[H,K]}.$$

(522)

Let $\tilde{P}\to M$ be the pullback by $X\mapsto P_X$ of the canonical Cartan bundle $\mathrm{SL}(3,\mathbb{C})\to {\mathcal{P}}_3^1$ from Theorem 13 and Corollary 24. Transport its adjoint-frame embedding through J; the resulting quotient $\tilde{P}/Z_3$ is $P_{\phi }$. Thus this already constructed canonical principal $\mathrm{SU}\left(3\right)$ bundle is a fixed lift of the tangent reduction. Since $ad:\mathfrak{su}\left(3\right)\to {\mathfrak{h}}_{\phi }$ is an isomorphism, ${\mathcal{B}}_8$ has a unique infinitesimal lift and defines a principal connection $A^{\mathrm{col}}$ whose central curvature is

$$F_0^{\mathrm{col}}(H,K)=-\frac{5}{144}[H,K].$$

(523)

These values span $\mathfrak{su}\left(3\right)$. Ambrose-Singer therefore gives

$$\begin{array}{|c|}\hline {Hol}_0^0\left(D^{\mathrm{col}}\right)={Hol}_0\left(D^{\mathrm{col}}\right)=\mathrm{PSU}\left(3\right),{Hol}_{e_0}^0\left(A^{\mathrm{col}}\right)={Hol}_{e_0}\left(A^{\mathrm{col}}\right)=\mathrm{SU}\left(3\right),\\ \hline\end{array}$$

(524)

where $e_0\in {\tilde{P}}_0$ is a lift of the tracial base point. Thus the tangent construction sees the adjoint quotient, whereas the retained canonical principal lift admits associated fundamental triplet carriers.

Corollary 32

(Sharp $\mathrm{PSU}\left(3\right)$-$\mathrm{SO}\left(8\right)$ holonomy dichotomy). For $s\in \mathbb{R}$, define the metric connection

$$D^{\left(s\right)}:=D^{\mathrm{col}}+s{\mathsf{\Phi }}_{20}.$$

(525)

Then

$$\begin{array}{cc}\hfill R^{\left(s\right)}& =R^{\mathrm{col}}+s{\mathrm{d}}^{D^{\mathrm{col}}}{\mathsf{\Phi }}_{20}+s^2{\mathsf{\Phi }}_{20}\wedge {\mathsf{\Phi }}_{20},\hfill \end{array}$$

(526)

$$\begin{array}{cc}\hfill {\widehat{R}}_0^{\left(s\right)}& =-\frac{5}{16}{\mathsf{\Pi }}_8+\frac{s}{4}{\mathsf{\Pi }}_{20},\hfill \end{array}$$

(527)

$$\begin{array}{cc}\hfill T^{\left(s\right)}(X,Y)& =(s-1)\left({\mathsf{\Phi }}_{20}\left(X\right)Y-{\mathsf{\Phi }}_{20}\left(Y\right)X\right).\hfill \end{array}$$

(528)

Consequently,

$$\begin{array}{|c|}\hline {Hol}_0\left(D^{\left(0\right)}\right)=\mathrm{PSU}\left(3\right),{Hol}_0\left(D^{\left(s\right)}\right)=\mathrm{SO}\left(8\right)(s\ne 0).\\ \hline\end{array}$$

(529)

The principal lift at $s=0$ has holonomy $\mathrm{SU}\left(3\right)$. This sharp connection-theoretic jump is not a thermodynamic or topological phase transition: it says that exact color reduction requires the ${\mathfrak{m}}_{20}$ coefficient to vanish within the family Equation (525). In particular, no orthogonal gauge conjugation can convert the BKM Levi-Civita connection into a $\mathrm{PSU}\left(3\right)$ connection, because such a conjugation preserves its $\mathrm{SO}\left(8\right)$ holonomy up to conjugacy.

6.7.7. Bridge Rigidity, Response, and Smoothing

Theorem 18

(Canonical qutrit bridge rigidity and curvature stiffness). Assume the qutrit construction and normalizations of Theorem 17 and Corollary 32. On $\mathfrak{so}(T_{X}M,g_X)$ use

$${\langle A,B\rangle }_{\mathfrak{so},X}:=-\frac{1}{2}{Tr}_{T_{X}M}(A\circ B),$$

(530)

and contract the one-form index of ${\mathsf{\Theta }}_X\in T_X^{\ast }M\otimes \mathfrak{so}(T_{X}M,g_X)$ with $g_X^{-1}$:

$${∥{\mathsf{\Theta }}_X∥}_X^2:={\displaystyle \sum _{a=1}^8}{\langle {\mathsf{\Theta }}_X\left(E_a\right),{\mathsf{\Theta }}_X\left(E_a\right)\rangle }_{\mathfrak{so},X},$$

(531)

where $\left\{E_a\right\}$ is any $g_X$-orthonormal basis. For constant Gibbs-coordinate vectors $H,K\in T_{0}M={Herm}_0\left(3\right)$, the first covariant jet of the complementary bridge is

$$\begin{array}{|c|}\hline {\left(D_H^{\mathrm{col}}{\mathsf{\Phi }}_{20}\right)}_0\left(K\right)=-\frac{1}{2}{\pi }_{20}[K_{0,H}^{\mathsf{\Psi }},K_{0,K}^{\mathsf{\Psi }}]=\frac{1}{8}{\mathsf{\Pi }}_{20}\left(W_{H,K}\right).\\ \hline\end{array}$$

(532)

By Equation (474), this alternating first jet is the type-correct local carrier of the scalar-normalization closure defect: ${\mathsf{\Phi }}_{20,0}=0$, while ${\left({\mathrm{d}}^{D^{\mathrm{col}}}{\mathsf{\Phi }}_{20}\right)}_0=\frac{1}{4}{\mathsf{\Pi }}_{20}W$. Consequently, the canonical bridge scalar

$${\mathcal{V}}_{\mathrm{br}}\left(X\right):=\frac{1}{2}{∥{\mathsf{\Phi }}_{20,X}∥}_X^2$$

(533)

has the Gibbs-coordinate expansion

$$\begin{array}{|c|}\hline {\mathcal{V}}_{\mathrm{br}}\left(X\right)=\frac{5}{128}{∥X∥}_{g_0}^2+O\left({∥X∥}_{g_0}^3\right),{Hess}_0{\mathcal{V}}_{\mathrm{br}}=\frac{5}{64}{g}_0.\\ \hline\end{array}$$

(534)

Thus the tracial state is a nondegenerate strict local minimum of the squared intrinsic torsion.

For $A\in \mathfrak{so}(TM,g)$ and any covariant three-tensor α, define

$$(A·\alpha )(U,V,W):=-\alpha (AU,V,W)-\alpha (U,AV,W)-\alpha (U,V,AW).$$

(535)

The adapted connection preserves the transported Cartan form, and the full BKM connection therefore satisfies the exact identity

$$\begin{array}{|c|}\hline D^{\mathsf{\Psi }}\phi ={\mathsf{\Phi }}_{20}·\phi ,\\ \hline\end{array}{∥D^{\mathsf{\Psi }}\phi ∥}_X^2=c_{\phi }{∥{\mathsf{\Phi }}_{20,X}∥}_X^2=2c_{\phi }{\mathcal{V}}_{\mathrm{br}}\left(X\right)$$

(536)

Here three-form norms make wedges of orthonormal covectors orthonormal (equivalently, they use $1/3!$ times the ordered-index contraction), with the product metric on covariant derivatives. The constant $c_{\phi }={∥A·\phi ∥}_{{\mathsf{\Lambda }}^3}^2/{∥A∥}_{\mathfrak{so}}^2>0$, for $0\ne A\in {\mathfrak{m}}_{\phi ,X}$, is independent of A by irreducibility and of X by $D^{\mathrm{col}}$-parallel transport. Hence ${\mathcal{V}}_{\mathrm{br}}$ is equivalently the squared failure of the BKM connection to preserve the color-defining Cartan form.

The symmetric cubic gives a coefficient-complete version of the same statement. With the ordinary induced tensor-product norm on ${Sym}^3{T}_X^{\ast }M$ (the sum over ordered indices, with no $1/3!$ factor), define

$${\iota }_{J,X}:\mathfrak{so}(T_{X}M,g_X)⟶{Sym}^3{T}_X^{\ast }M,{\iota }_{J,X}\left(A\right):=A·{\mathcal{C}}_X^J.$$

(537)

Then

$$\begin{array}{|c|}\hline ker{\iota }_{J,X}={\mathfrak{h}}_{\phi ,X},{\pi }_{20}=\frac{1}{27}{\iota }_{J,X}^{\ast }{\iota }_{J,X}.\\ \hline\end{array}$$

(538)

Consequently,

$$\begin{array}{|c|}\hline D^{\mathsf{\Psi }}{\mathcal{C}}^J={\mathsf{\Phi }}_{20}·{\mathcal{C}}^J,{∥D^{\mathsf{\Psi }}{\mathcal{C}}^J∥}_X^2=27{∥{\mathsf{\Phi }}_{20,X}∥}_X^2=54{\mathcal{V}}_{\mathrm{br}}\left(X\right).\\ \hline\end{array}$$

(539)

Moreover, ${\iota }_{J,X}\left({\mathfrak{m}}_{\phi ,X}\right)$ is the tangent space at ${\mathcal{C}}_X^J$ to the orbit $\mathrm{SO}\left(8\right)·{\mathcal{C}}_X^J\simeq \mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right)$; its complexification is $\mathbf{10}\oplus \overline{\mathbf{10}}$. This identifies the same real 20 as the complement in $\mathfrak{so}\left(8\right)$, the infinitesimal deformation orbit of the symmetric cubic, and the target of the intrinsic torsion.

Finally, equip $Hom({\mathsf{\Lambda }}^2{T}_{0}M,\mathfrak{so}(T_{0}M,g_0))$ with the induced Hilbert-Schmidt norm. The central curvature maps of the interpolation Equation (525) obey

$$\begin{array}{|c|}\hline {∥{\widehat{R}}_0^{\left(s\right)}∥}_{\mathrm{HS}}^2=\frac{25}{32}+\frac{5}{4}{s}^2.\\ \hline\end{array}$$

(540)

Thus $s=0$ uniquely minimizes the tracial curvature-map norm within this scalar family.

Corollary 33

(Local bridge Legendre response and Bregman smoothing). The positive Hessian in Equation (534) makes $\mathrm{d}{\mathcal{V}}_{\mathrm{br}}$ a local diffeomorphism at the tracial state. Its local Legendre transform satisfies

$${Hess}_0{\mathcal{V}}_{\mathrm{br}}^{\ast ,\mathrm{loc}}=\frac{64}{5}{g}_0^{-1}.$$

(541)

For every fixed $r>0$, after restricting to sufficiently small Gibbs-coordinate neighborhoods, the local Bregman-Moreau envelope

$${\mathcal{V}}_{\mathrm{br}}^{\left(r\right)}\left(X\right):=\underset{Y}{min}\left\{{\mathcal{V}}_{\mathrm{br}}\left(Y\right)+\frac{1}{r}{B}_{\mathsf{\Psi }}(Y,X)\right\}$$

(542)

has a unique local minimizer and

$$\begin{array}{|c|}\hline {Hess}_0{\mathcal{V}}_{\mathrm{br}}^{\left(r\right)}=\frac{5}{64+5r}{g}_0.\\ \hline\end{array}$$

(543)

These are local statements: global convexity and coercivity of ${\mathcal{V}}_{\mathrm{br}}$ have not been proved. In particular, Equation (542) is distinct from the global entropic and moment-space envelopes in Theorems 6 and 7.

Proof. 

The inverse-function theorem and Equation (534) give the local Legendre statement. The stationarity equation for Equation (542) is

$$\mathrm{d}{\mathcal{V}}_{\mathrm{br}}{|}_Y+\frac{1}{r}(\mathrm{d}{\mathsf{\Psi }}_Y-\mathrm{d}{\mathsf{\Psi }}_X)=0.$$

(544)

Its derivative in Y at $(X,Y)=(0,0)$ is $(5/64+1/r)g_0$, so the implicit-function theorem gives the unique local minimizer and $D{Y}_0={(1+5r/64)}^{-1}\mathrm{id}$. Differentiating the envelope in X then gives Equation (543). □

Remark 13

(Singlet and adjoint quadratic bridge channels). After complexification, ${\left({\mathfrak{m}}_{20}\right)}_{\mathbb{C}}=\mathbf{10}\oplus \overline{\mathbf{10}}$ and

$$\mathbf{10}\otimes \overline{\mathbf{10}}=\mathbf{1}\oplus \mathbf{8}\oplus \mathbf{27}\oplus \mathbf{64}.$$

(545)

This standard tensor product is tabulated in [33], Table 24. The invariant fiber norm in Equation (533) selects the quadratic singlet, whereas ${\pi }_8({\mathsf{\Phi }}_{20}\wedge {\mathsf{\Phi }}_{20})$ in Equation (517) selects the adjoint feedback into projected curvature. These are two inequivalent contractions of the same real bridge data; the complex summands are conjugate components, not two independently supplied fields.

Proposition 26

(Higher qutrit response and the leading color obstruction). Let $C^{\left(r\right)}:=D^r\mathsf{\Psi }$, retain the metric-raised cubic operator $\mathsf{Q}$ of Equation (353), and define the quartic operator by

$$g_X\left({\mathsf{T}}_X(M,H)K,L\right)=C_X^{\left(4\right)}(M,H,K,L).$$

(546)

For constant affine vector fields,

$${\partial }_M{\mathsf{Q}}_X\left(H\right)={\mathsf{T}}_X(M,H)-{\mathsf{Q}}_X\left(M\right){\mathsf{Q}}_X\left(H\right).$$

(547)

The resulting curvature is Equation (355), with coefficient-level tracial qutrit realization Equation (432). Thus $C^{\left(4\right)}$ cancels from pointwise Hessian curvature, but may enter $D^{\mathsf{\Psi }}{R}^{\mathsf{\Psi }}$; more generally ${\left(D^{\mathsf{\Psi }}\right)}^r{R}^{\mathsf{\Psi }}$ is a universal expression in $g^{-1},C^{\left(3\right)},\dots ,C^{(r+3)}$. Higher response tensors therefore control variation and finite-loop memory, not the nonzero leading ${\mathfrak{m}}_{20}$ curvature in Equation (429).

At the qutrit tracial state, Cayley-Hamilton gives, for $q:=Tr{X}^2$ and $r:=Tr{X}^3$,

$$X^3-\frac{q}{2}X-\frac{r}{3}I=0.$$

(548)

Consequently $\mathbb{R}{\left[V\right]}^{\mathrm{SU}\left(3\right)}=\mathbb{R}[q,r]$: every degree-m central symmetric Taylor tensor is a linear combination of polarizations of the monomials $q^a{r}^b$ with $2a+3b=m$. Thus higher cumulants need not vanish, but they introduce no new primitive symmetric qutrit invariant beyond the quadratic metric and cubic d-tensor. The exact spectral log-Laplace recurrence and its first higher cumulants are Equations (A144)–(A146). Away from the center the tensors are state dependent; for $n>3$, primitive invariant degrees through n occur.

For homothetically shrinking loops based at 0, the area term is $O\left({\epsilon }^2\right)$, $C^{\left(4\right)}$ may first enter at $O\left({\epsilon }^3\right)$ through $\nabla {\mathcal{F}}^{\mathrm{rel}}$, and curvature commutators first enter at $O\left({\epsilon }^4\right)$. Higher terms may cancel for specially composed or isolated finite loops, but cannot uniformly cancel the nonzero $O\left({\epsilon }^2\right){\mathfrak{m}}_{20}$ component of a generic shrinking qutrit plane. The full Magnus-order argument is in the proof.

Remark 14

(Generic obstruction to a state-dependent cubic $\mathrm{PSU}\left(3\right)$ stabilizer). The actual BKM cubic away from the tracial state should not be assumed to have the same stabilizer as ${\mathcal{C}}_0^{\mathsf{\Psi }}$. In a fixed Gibbs-affine frame, take its metric trace

$${\tau }_i\left(X\right):=g^{jk}\left(X\right){\mathcal{C}}_{ijk}^{\mathsf{\Psi }}\left(X\right)={\partial }_{i}logdet\left[g_{jk}\left(X\right)\right].$$

(549)

Every orthogonal stabilizer of ${\mathcal{C}}_X^{\mathsf{\Psi }}$ fixes the metric dual of ${\tau }_X$. The latter is generically nonzero, as the following explicit qutrit ray shows. Put

$$X\left(t\right)=diag\left(-\frac{2t}{3},\frac{t}{3},\frac{t}{3}\right),s={\mathrm{e}}^t,{\rho }_t=diag(p,q,q),p=\frac{s}{s+2},q=\frac{1}{s+2}.$$

(550)

Relative to a fixed $g_0$-orthonormal Gell-Mann frame, the BKM Hessian eigenvalues along this ray are

$$3L(p,q)\mathit{with}\mathit{multiplicity}4,3q\mathit{with}\mathit{multiplicity}3,9pq\mathit{with}\mathit{multiplicity}1,$$

(551)

where

$$L(p,q):=\frac{p-q}{logp-logq}=\frac{s-1}{(s+2)t}.$$

(552)

Consequently

$$\begin{array}{|c|}\hline \underset{g_0}{det}{g}_{X\left(t\right)}={\left[\frac{3(s-1)}{(s+2)t}\right]}^4{\left[\frac{3}{s+2}\right]}^3\left[\frac{9s}{{(s+2)}^2}\right].\\ \hline\end{array}$$

(553)

Its limit at $t=0$ is 1, whereas ${det}_{g_0}{g}_{X\left(t\right)}\sim 19683{\mathrm{e}}^{-4t}{t}^{-4}\to 0$ as $t\to +\infty$. Hence $detg$ is nonconstant. Analyticity implies $\tau \ne 0$ on an open dense subset. Since the real adjoint $\mathrm{PSU}\left(3\right)$ representation is irreducible and fixes no nonzero vector, the actual state-dependent metric-cubic pair cannot have that stabilizer on this open dense subset. This obstruction explains why Theorem 17 transports a separate comparison cubic and identifies it with the entropy cubic only at the tracial point.

Remark 15

(Geometric and physical status of the color projection). The intrinsic torsion is a section of $T^{\ast }M\otimes {\mathfrak{m}}_{20}$, a rank-160 module. Its six irreducible $\mathrm{PSU}\left(3\right)$ classes have dimensions $8,20,27,8,27,70$ [31]; the present theorem does not prove vanishing of the 70-dimensional ${\mathcal{W}}_6$ component required for Puhle’s characteristic connection, so $D^{\mathrm{col}}$ is called adapted rather than characteristic. It is necessarily torsionful somewhere: if it were globally torsion free, metric uniqueness would give $D^{\mathrm{col}}=D^{\mathsf{\Psi }}$, contradicting their different holonomy, although Equation (521) shows that its torsion vanishes at the tracial state.

The fixed $d_{abc}$ cubic is an invariant singlet. The real 20 is not a linear component of that singlet: it is its infinitesimal transverse rotational orbit, the complement in $\mathfrak{so}\left(8\right)$, and the target of intrinsic torsion (Equations (430) and (538)). At the tracial state, scalar-channel restoration removes this complementary curvature but also changes the adjoint coefficient. This algebraic operation adds no physical state direction and does not reduce actual BKM holonomy (Theorem 16).

The transported comparison cubic ${\mathcal{C}}^J$ and Cartan form φ encode the same oriented $\mathrm{PSU}\left(3\right)$ reduction but have different disconnected stabilizers (Equations (503) and (508)). The reconstruction in Equations (510) and (520) distinguishes the complementary connection-defect one-form from its curvature contribution: ${\mathsf{\Phi }}_{20,0}=0$ while ${\left({\mathrm{d}}^{D^{\mathrm{col}}}{\mathsf{\Phi }}_{20}\right)}_0\ne 0$. Reduction requires the complementary one-form to vanish throughout the relevant region, not merely at one point.

6.7.8. Conditional Spacetime Completion and Physical Status

The conditional construction of Proposition A3 has three distinct input levels. The reductive curvature split requires a supplied spacetime principal $\mathrm{SO}\left(8\right)$ bundle with $\mathrm{PSU}\left(3\right)$ reduction and connection. Its Euclidean Yang-Mills Hessian additionally uses the compact oriented Riemannian four-metric, coupling $g_8$, and supplied action; a positive transverse zeroth-order coefficient further requires a reduction field and one supplied coefficient f or $f_J$. None of these data follows from the state-space reduction, which by itself supplies no spacetime gauge dynamics, particle spectrum, confinement, or mass-gap claim. The construction also remains distinct from the independently supplied $n=5$ AIII grading of Theorem 20; its proof and physical qualifications are in Appendix D.9.

6.8. Symmetric-Space Endpoint Action and the Spectral Log-Laplace Bridge

For $P_0,P_1\in {\mathcal{P}}_n^1$, set

$$Z_{01}:=log\left(P_0^{-1/2}{P}_1{P}_0^{-1/2}\right)\in \mathfrak{p},d_{\mathrm{AI}}^{\left(\alpha \right)}{(P_0,P_1)}^2:=\alpha Tr{Z}_{01}^2.$$

(554)

Proposition 27

(Free symmetric-space action and LogDet endpoint proxies). Supply an inertial coefficient ${\mu }_{\mathrm{AI}}>0$, independent of the BKM coefficient ${\mu }_{\ast }$ below, and a duration $h>0$. For dimensionless cone coordinates, $\left[{\mu }_{\mathrm{AI}}\right]$ is action times time. For the free Lagrangian

$$L_{\mathrm{AI}}(P,\dot{P}):=\frac{{\mu }_{\mathrm{AI}}}{2}{g}_P^{\mathrm{AI},\left(\alpha \right)}(\dot{P},\dot{P}),$$

(555)

the unique endpoint geodesic and its exact principal action are

$$\begin{array}{cc}\hfill P\left(t\right)& =P_0^{1/2}exp\left(\frac{t}{h}{Z}_{01}\right)P_0^{1/2},0\le t\le h,\hfill \end{array}$$

(556)

$$\begin{array}{cc}\hfill S_{\mathrm{AI}}^E(P_0,P_1;h)& =\frac{{\mu }_{\mathrm{AI}}}{2h}{d}_{\mathrm{AI}}^{\left(\alpha \right)}{(P_0,P_1)}^2.\hfill \end{array}$$

(557)

The ambient LogDet contrast between determinant-one endpoints satisfies

$$\begin{array}{cc}\hfill D_{\mathrm{LD}}^{\left(\alpha \right)}(P_1\parallel P_0)& =\alpha Tr({\mathrm{e}}^{Z_{01}}-I)\hfill \end{array}$$

$$\begin{array}{cc}& =\frac{1}{2}{d}_{\mathrm{AI}}^{\left(\alpha \right)}{(P_0,P_1)}^2+\frac{\alpha }{6}Tr{Z}_{01}^3+O\left({∥Z_{01}∥}^4\right),\hfill \\ \hfill \frac{1}{2}\left[D_{\mathrm{LD}}^{\left(\alpha \right)}(P_1\parallel P_0)+D_{\mathrm{LD}}^{\left(\alpha \right)}(P_0\parallel P_1)\right]& =\alpha Tr(cosh{Z}_{01}-I)\hfill \end{array}$$

(558)

$$\begin{array}{cc}& =\frac{1}{2}{d}_{\mathrm{AI}}^{\left(\alpha \right)}{(P_0,P_1)}^2+\frac{\alpha }{24}Tr{Z}_{01}^4+O\left({∥Z_{01}∥}^6\right).\hfill \end{array}$$

(559)

Since $coshx-1\ge x^2/2$,

$$\frac{1}{2}\left[D_{\mathrm{LD}}^{\left(\alpha \right)}(P_1\parallel P_0)+D_{\mathrm{LD}}^{\left(\alpha \right)}(P_0\parallel P_1)\right]\ge \frac{1}{2}{d}_{\mathrm{AI}}^{\left(\alpha \right)}{(P_0,P_1)}^2.$$

(560)

Hence $({\mu }_{\mathrm{AI}}/h)D_{\mathrm{LD}}^{\left(\alpha \right)}$ has an $O\left(h^2\right)$ one-step action defect when $Z_{01}=O\left(h\right)$, whereas its adjoint average has defect $O\left(h^3\right)$ and globally majorizes the exact free action Equation (557). These are local discrete proxies for the affine-invariant action; they have neither the Umegaki data-processing interpretation nor the BKM continuum metric. The global majorization is strict unless $P_0=P_1$. The exact action and both endpoint contrasts are invariant under the diagonal $\mathrm{SL}(n,\mathbb{C})$ congruence $(P_0,P_1)\mapsto (G{P}_0{G}^{†},G{P}_1{G}^{†})$.

Proof. 

The distance and geodesic formulas are standard for the affine-invariant metric on the positive cone [46]; the determinant-one slice is totally geodesic. Constant speed gives Equation (557). Since $P_0^{-1}{P}_1$ is similar to ${\mathrm{e}}^{Z_{01}}$, while $P_1^{-1}{P}_0$ has the reciprocal spectrum and therefore trace $Tr{\mathrm{e}}^{-Z_{01}}$, and since $Tr{Z}_{01}=0$, Equation (309) gives the first equality in Equation (558). Expanding ${\mathrm{e}}^{\pm Z_{01}}$ proves both series, and the scalar hyperbolic-cosine inequality gives Equation (560). Substitution $Z_{01}=O\left(h\right)$ gives the stated action defects. Phase-map orders additionally require the regular no-caustic and derivative-control hypotheses used in Proposition A8. □

Theorem 19

(Cartan spectral log-Laplace bridge at the symmetric center). Let ${\rho }_{\ast }:=I/n$, choose nonzero $Z\in \mathfrak{p}$, and define

$$P_t:={\mathrm{e}}^{tZ},{\rho }_t:=\frac{{\mathrm{e}}^{tZ}}{Tr{\mathrm{e}}^{tZ}},{\mathcal{Z}}_Z\left(t\right):=\frac{1}{n}Tr{\mathrm{e}}^{tZ},{\ell }_Z\left(t\right):=log{\mathcal{Z}}_Z\left(t\right).$$

(561)

If $z_1,\dots ,z_n$ are the eigenvalues of Z, then

$${\ell }_Z\left(t\right)=log\left(\frac{1}{n}{\displaystyle \sum _{j=1}^n}{\mathrm{e}}^{t{z}_j}\right)$$

(562)

is the literal log-Laplace transform of the uniform Cartan spectral measure. Moreover,

$$\begin{array}{c}\mathsf{\Phi }\left({\rho }_t\right)=P_t,{\ell }_Z^{\prime }\left(t\right)=Tr\left({\rho }_{t}Z\right),{\ell }_Z^{\prime \prime }\left(t\right)=Tr\left[{\rho }_t{\left(Z-Tr\left({\rho }_{t}Z\right)I\right)}^2\right]>0,\end{array}$$

(563)

$$\begin{array}{c}{D}_{\mathrm{U}}({\rho }_t\parallel {\rho }_{\ast })=t{\ell }_Z^{\prime }\left(t\right)-{\ell }_Z\left(t\right),D_{\mathrm{U}}({\rho }_{\ast }\parallel {\rho }_t)={\ell }_Z\left(t\right),\end{array}$$

(564)

$$\begin{array}{c}{D}_{\mathrm{U}}({\rho }_t\parallel {\rho }_{\ast })+D_{\mathrm{U}}({\rho }_{\ast }\parallel {\rho }_t)=t{\ell }_Z^{\prime }\left(t\right),\end{array}$$

(565)

$$\begin{array}{c}{d}_{\mathrm{AI}}^{\left(\alpha \right)}{(I,P_t)}^2=\alpha t^{2}Tr{Z}^2=\alpha n{t}^2{\ell }_Z^{\prime \prime }\left(0\right),\end{array}$$

(566)

$$\begin{array}{c}{D}_{\mathrm{LD}}^{\left(\alpha \right)}(P_t\parallel I)=\alpha n\left({\mathcal{Z}}_Z\left(t\right)-1\right)=\alpha n\left({\mathrm{e}}^{{\ell }_Z\left(t\right)}-1\right).\end{array}$$

(567)

Writing $m_Z\left(t\right):={\ell }_Z^{\prime }\left(t\right)$ and ${\ell }_Z^{\ast }\left(m\right):={sup}_{s\in \mathbb{R}}\{sm-{\ell }_Z\left(s\right)\}$, the map $m_Z:\mathbb{R}\to (z_{min},z_{max})$ is a diffeomorphism and

$${\ell }_Z^{\ast }\left(m_Z\left(t\right)\right)=t{m}_Z\left(t\right)-{\ell }_Z\left(t\right)=D_{\mathrm{U}}({\rho }_t\parallel {\rho }_{\ast }).$$

(568)

Finally,

$$D_{\mathrm{U}}({\rho }_t\parallel {\rho }_{\ast })=\frac{t^2}{2n}Tr{Z}^2+O\left(t^3\right)=\frac{d_{\mathrm{AI}}^{\left(\alpha \right)}{(I,P_t)}^2}{2\alpha n}+O\left(t^3\right).$$

(569)

Thus affine-invariant free mechanics uses the second spectral cumulant at the origin, while the two orientations of Umegaki entropy use the value and slope of the same log-Laplace function at the endpoint. They share an exact center-based carrier and quadratic contact, not a global equality of actions.

Proof. 

The spectral theorem and $TrZ=0$ give Equation (562) and $det{P}_t=1$, hence $\mathsf{\Phi }\left({\rho }_t\right)=P_t$. Differentiation gives the mean and variance in Equation (563); the variance is positive because nonzero traceless Z has at least two distinct eigenvalues. Substitution of

$$log{\rho }_t=tZ-logTr{\mathrm{e}}^{tZ}I,log{\rho }_{\ast }=-(logn)I,$$

(570)

proves the entropy identities. Strict convexity and the limits of the tilted mean at $t\to \pm \infty$ prove the scalar Legendre statement. Equations (554) and (309) give the remaining exact functionals, and Taylor expansion at zero proves Equation (569). □

Corollary 34

(Rank-one rapidity/entropy Legendre pair). For $n=2$, take $Z=\mathit{n}·\sigma$ with $∥\mathit{n}∥=1$ and write $t=\eta$. Then

$$\begin{array}{c}{\rho }_{\eta }=\frac{1}{2}\left(I+tanh\eta \mathit{n}·\sigma \right),{\ell }_Z\left(\eta \right)=logcosh\eta ,{\ell }_Z^{\prime }\left(\eta \right)=tanh\eta ,\end{array}$$

(571)

$$\begin{array}{c}{D}_{\mathrm{U}}({\rho }_{\eta }\parallel I/2)=\eta tanh\eta -logcosh\eta ,D_{\mathrm{U}}(I/2\parallel {\rho }_{\eta })=logcosh\eta ,\end{array}$$

(572)

$$\begin{array}{c}{d}_{\mathrm{AI}}^{\left(\alpha \right)}{(I,{\mathrm{e}}^{\eta Z})}^2=2\alpha {\eta }^2.\end{array}$$

(573)

With $r=tanh\eta \in (-1,1)$,

$${\ell }_Z^{\ast }\left(r\right)=\frac{1}{2}\left[(1+r)log(1+r)+(1-r)log(1-r)\right].$$

(574)

Thus rapidity is the source, the Bloch radius is its expectation coordinate, and relative entropy is the corresponding Legendre rate function. Nevertheless, $D_{\mathrm{U}}({\rho }_{\eta }\parallel I/2)\to log2$ while $d_{\mathrm{AI}}^{\left(\alpha \right)}{(I,{\mathrm{e}}^{\eta Z})}^2\to \infty$ as $\eta \to \infty$.

Corollary 35

(Multivariate Cartan partition dictionary). Let $Z_1,\dots ,Z_d\in \mathfrak{p}$ be linearly independent and pairwise commuting. Put

$$Z\left(\theta \right):={\displaystyle \sum _{a=1}^d}{\theta }^a{Z}_a,\ell \left(\theta \right):=log\left(\frac{1}{n}Tr{\mathrm{e}}^{Z\left(\theta \right)}\right),{\rho }_{\theta }:=\frac{{\mathrm{e}}^{Z\left(\theta \right)}}{Tr{\mathrm{e}}^{Z\left(\theta \right)}},P_{\theta }:={\mathrm{e}}^{Z\left(\theta \right)}.$$

(575)

In a joint eigenbasis, write $z_j=({\left(Z_1\right)}_{jj},\dots ,{\left(Z_d\right)}_{jj})\in {\mathbb{R}}^d$, $\mathcal{C}:=conv\{z_1,\dots ,z_n\}$, and set

$${\mathsf{\Delta }}_n:=\left\{p\in {[0,1]}^n:{\displaystyle \sum _{j=1}^n}{p}_j=1\right\},D(q\parallel p):=\sum _{j:q_j>0}{q}_{j}log\frac{q_j}{p_j},$$

(576)

with $0log0:=0$ and the extended-real convention $D(q\parallel p)=+\infty$ unless $suppq\subseteq suppp$. Define

$${\ell }^{\ast }\left(m\right):=\underset{\vartheta \in {\mathbb{R}}^d}{sup}\{\vartheta ·m-\ell \left(\vartheta \right)\}.$$

(577)

Then $\mathcal{C}$ has nonempty ordinary interior and

$$\ell \left(\theta \right)=log\left(\frac{1}{n}{\displaystyle \sum _{j=1}^n}{\mathrm{e}}^{\theta ·z_j}\right),$$

(578)

is a literal multivariate log-Laplace transform, $\nabla \ell :{\mathbb{R}}^d\to int\mathcal{C}$ is a diffeomorphism, and

$${\ell }^{\ast }\left(m\right)=\underset{\begin{array}{c}p\in {\mathsf{\Delta }}_n\\ {\sum }_j{p}_j{z}_j=m\end{array}}{min}{\displaystyle \sum _{j=1}^n}{p}_{j}log\left(n{p}_j\right),m\in \mathcal{C},$$

(579)

with ${\ell }^{\ast }\left(m\right)=+\infty$ outside $\mathcal{C}$.

For $\delta :=\theta -\eta$, the exact information- and cone-geometric relations are

$$\begin{array}{cccc}\hfill D_{\mathrm{U}}({\rho }_{\theta }\parallel {\rho }_{\eta })& =B_{\ell }(\eta ,\theta ),\hfill & \hfill g_{ab}^{\mathrm{BKM}}\left(\theta \right)& ={\partial }_a{\partial }_b\ell \left(\theta \right),\hfill \\ \hfill d_{\mathrm{AI}}^{\left(\alpha \right)}{(P_{\eta },P_{\theta })}^2& =\alpha TrZ{\left(\delta \right)}^2=\alpha n{\delta }^{\mathsf{T}}{\nabla }^2\ell \left(0\right)\delta ,\hfill & \hfill D_{\mathrm{LD}}^{\left(\alpha \right)}(P_{\theta }\parallel P_{\eta })& =\alpha n\left({\mathrm{e}}^{\ell \left(\delta \right)}-1\right).\hfill \end{array}$$

(580)

Moreover, both entropy orientations satisfy the sharp dimension-independent comparison

$$\begin{array}{|c|}\hline max\left\{D_{\mathrm{U}}({\rho }_{\theta }\parallel {\rho }_{\eta }),D_{\mathrm{U}}({\rho }_{\eta }\parallel {\rho }_{\theta })\right\}\le \frac{d_{\mathrm{AI}}^{\left(\alpha \right)}{(P_{\eta },P_{\theta })}^2}{4\alpha }\le \frac{D_{\mathrm{LD}}^{\left(\alpha \right)}(P_{\theta }\parallel P_{\eta })+D_{\mathrm{LD}}^{\left(\alpha \right)}(P_{\eta }\parallel P_{\theta })}{4\alpha }.\\ \hline\end{array}$$

(581)

The first constant is approached by infinitesimal qubit center rays and cannot be reduced uniformly in dimension. In either entropy orientation, there is no $c>0$ for which $D_{\mathrm{U}}\ge c{d}_{\mathrm{AI}}^2$ holds uniformly, as Corollary 34 shows.

Proof from finite I-projection theory. 

Joint diagonalization reduces the convex core to finite log-sum-exp duality and minimum-relative-entropy projection. The exact classical I-projection result is [89]; the parallel commuting Gibbs variational formula is [8]. The ordinary interior used here is full-dimensional: if $v·z_j=c$ for every j, then ${\sum }_a{v}_a{Z}_a=cI$; tracelessness gives $c=0$, and linear independence gives $v=0$. The cited duality therefore yields the gradient diffeomorphism and Equation (579), with the displayed support convention.

Commutativity gives $log\left(P_{\eta }^{-1/2}{P}_{\theta }{P}_{\eta }^{-1/2}\right)=Z(\theta -\eta )$, proving the matrix-specific BKM, affine-invariant, and LogDet dictionary. Along the source segment, Equation (96) integrates the variance of $(\theta -\eta )·z$. The sharp range-variance bound $Var\left(Z\right)\le range{\left(Z\right)}^2/4$, equivalently the quadratic consequence of Hoeffding’s lemma [90], and $range{\left(Z\right)}^2\le 2Tr{Z}^2$ give the first comparison in Equation (581); $2(coshx-1)\ge x^2$ gives the second. The infinitesimal central qubit family proves sharpness, while Corollary 34 rules out a uniform reverse bound. □

Corollary 36

(Cartan large deviations and the Chernoff boundary). Under the hypotheses and notation of Corollary 35, set

$$p_{\eta ,j}:=\frac{{\mathrm{e}}^{\eta ·z_j}}{{\sum }_k{\mathrm{e}}^{\eta ·z_k}}.$$

(582)

On each of N independent copies of ${\rho }_{\eta }$, perform a common rank-one spectral refinement of the commuting $Z_a$, let $Y_k\in \{z_1,\dots ,z_n\}$ be its record, and put $M_N=N^{-1}{\sum }_{k=1}^N{Y}_k$. Then $M_N$ satisfies the large-deviation principle with good rate

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill I_{\eta }\left(m\right)& ={\displaystyle \underset{u\in {\mathbb{R}}^d}{sup}}\{u·m-[\ell (\eta +u)-\ell \left(\eta \right)]\}\hfill \\ & ={\ell }^{\ast }\left(m\right)-\eta ·m+\ell \left(\eta \right)\hfill \\ & ={\displaystyle \underset{\begin{array}{c}q\in {\mathsf{\Delta }}_n\\ {\sum }_j{q}_j{z}_j=m\end{array}}{min}}D(q\parallel p_{\eta }),m\in \mathcal{C},\hfill \end{array}\\ \hline\end{array}$$

(583)

and $I_{\eta }=+\infty$ outside $\mathcal{C}$ [91]. $I_{\eta }$ is exactly the moment effective potential with reference ${\rho }_{\eta }$ and observable family $Z=(Z_1,\dots ,Z_d)$:

$$I_{\eta }\left(m\right)={\mathsf{\Gamma }}_{{\rho }_{\eta },Z}\left(m\right).$$

(584)

In particular, for $m_{\theta }:=\nabla \ell \left(\theta \right)$,

$$\begin{array}{|c|}\hline I_{\eta }\left(m_{\theta }\right)=D(p_{\theta }\parallel p_{\eta })=D_{\mathrm{U}}({\rho }_{\theta }\parallel {\rho }_{\eta }),\nabla I_{\eta }\left(m_{\theta }\right)=\theta -\eta ,{\nabla }^2{I}_{\eta }\left(m_{\theta }\right)={[{\nabla }^2\ell \left(\theta \right)]}^{-1}.\\ \hline\end{array}$$

(585)

Thus the BKM susceptibility is exactly the inverse macroscopic fluctuation stiffness, and

$$I_{\eta }(m_{\eta }+\delta m)=\frac{1}{2}\delta m^{\mathsf{T}}{[{\nabla }^2\ell \left(\eta \right)]}^{-1}\delta m+O\left({∥\delta m∥}^3\right).$$

(586)

The finite-sample method-of-types bound

$${Pr}_{\eta }\{M_N\in F\}\le {(N+1)}^{n}exp\left[-N\underset{m\in F}{inf}{I}_{\eta }\left(m\right)\right],$$

(587)

holds for every Borel set $F\subseteq \mathcal{C}$.

For $\theta \ne \eta$, write $\delta :=\theta -\eta$ and $\mathsf{\Delta }\ell :=\ell \left(\theta \right)-\ell \left(\eta \right)$. There is a unique $s_{\ast }\in (0,1)$ such that

$$\delta ·\nabla \ell (\eta +s_{\ast }\delta )=\mathsf{\Delta }\ell .$$

(588)

With ${\xi }_{\ast }:=\eta +s_{\ast }\delta$ and $m_{\ast }:=\nabla \ell \left({\xi }_{\ast }\right)$, the commuting-state quantum Chernoff exponent for asymptotic symmetric iid discrimination with fixed nonzero priors is

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill C_{\mathrm{Q}}({\rho }_{\theta },{\rho }_{\eta })& =-\underset{0\le s\le 1}{min}\{\ell (\eta +s\delta )-(1-s)\ell \left(\eta \right)-s\ell \left(\theta \right)\}\hfill \\ & =I_{\eta }\left(m_{\ast }\right)=I_{\theta }\left(m_{\ast }\right)\hfill \\ & =D_{\mathrm{U}}({\rho }_{{\xi }_{\ast }}\parallel {\rho }_{\eta })=D_{\mathrm{U}}({\rho }_{{\xi }_{\ast }}\parallel {\rho }_{\theta }).\hfill \end{array}\\ \hline\end{array}$$

(589)

Moreover, $m_{\ast }$ is the unique minimizer of either rate on the equal-prior normalized log-likelihood decision hyperplane $\delta ·m=\mathsf{\Delta }\ell$. Fixed unequal nonzero priors shift the finite-N normalized threshold only by $O\left(N^{-1}\right)$ and hence leave the exponent unchanged. Thus the Chernoff exponent is exactly the relative-entropy cost of the least unlikely decision-boundary fluctuation [55].

Proof from Cramér, I-projection, and Chernoff theory. 

The scaled cumulant generator is $\ell (\eta +u)-\ell \left(\eta \right)$. Cramér duality [91] gives the first line of Equation (583); $v=\eta +u$ gives the second, and finite I-projection theory [89] gives the constrained-relative-entropy form. At $m_{\theta }$, Legendre differentiation and the commuting dictionary give Equation (585); hence the rate is exactly ${\mathsf{\Gamma }}_{{\rho }_{\eta },Z}$ and its Hessian is the inverse BKM susceptibility. The usual type count (at most ${(N+1)}^n$ types), together with ${Pr}_{\eta }\{\widehat{p}=q\}\le {\mathrm{e}}^{-ND(q\parallel p_{\eta })}$, proves Equation (587).

For commuting states the quantum Chernoff theorem [55] reduces to the displayed scalar convex objective. Its derivative is $\delta ·\nabla \ell (\eta +s\delta )-\mathsf{\Delta }\ell$; strict convexity and the endpoint tangent inequalities give the unique zero $s_{\ast }$. Substitution there gives all equal-cost identities in Equation (589). Moreover, $\nabla I_{\eta }\left(m_{\ast }\right)=s_{\ast }\delta$ and $\nabla I_{\theta }\left(m_{\ast }\right)=-(1-s_{\ast })\delta$. On the decision hyperplane the corresponding supporting terms vanish, so the global strong-convexity bound Equation (121) makes $m_{\ast }$ the unique constrained minimizer of both rates, including against boundary competitors. The cited probability theorems supply the asymptotic cores; the effective-potential, BKM-stiffness, and equal-boundary identifications are the retained bridge. □

For arbitrary, possibly noncommuting, $H,K\in \mathfrak{p}$, Equation (99) still controls both entropy orientations by $\frac{1}{4}Tr{(H-K)}^2$. Affine-invariant cone geometry, however, uses the distinct displacement $log\left({\mathrm{e}}^{-K/2}{\mathrm{e}}^H{\mathrm{e}}^{-K/2}\right)$. The proof therefore supplies no bound in terms of the noncommuting affine-invariant distance, and no such extension of Equation (581) is asserted. Likewise, without a specified measurement protocol, ${\mathsf{\Gamma }}_{\sigma ,O}$ for noncommuting observables is a static quantum effective potential, not automatically a classical joint-record large-deviation rate.

6.9. A Conditional Standard-Model-Compatible AIII Sector

The full $\mathrm{SU}\left(5\right)$ holonomy above admits no invariant nontrivial rank-three projector, so it cannot be reduced on the whole cone by declaring a fixed $3+2$ grading parallel. A block-diagonal restriction does not solve the problem either: for

$${\mathfrak{p}}_{\mathrm{bd}}:=\left\{diag(H_3,H_2):H_j=H_j^{†},Tr{H}_3+Tr{H}_2=0\right\},span[{\mathfrak{p}}_{\mathrm{bd}},{\mathfrak{p}}_{\mathrm{bd}}]=\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right),$$

(590)

so its relative scalar direction is flat. Within the canonical curvature mechanism $F=-E\wedge E$, the missing Abelian direction must instead be generated by noncommuting directions.

The AIII symmetric pair and the Georgi-Glashow representation data are standard [26,34]. The claim below is not a new identification of that group, but its realization as the holonomy of the explicitly supplied off-diagonal statistical state sector.

Theorem 20

(AIII Standard-Model-compatible principal holonomy). Set $n=5$ on the supplied complex carrier and independently supply an orthogonal grading ${\mathbb{C}}^5=V_c\oplus V_w$, with $dim{V}_c=3$, $dim{V}_w=2$, projectors $P_c,P_w$, and ${\mathsf{\Gamma }}_{3,2}:=P_c-P_w$. Here ${\mathsf{\Gamma }}_{3,2}^2=\mathrm{id}$ and ${\mathsf{\Gamma }}_{3,2}{I}_V=I_V{\mathsf{\Gamma }}_{3,2}$: it is a supplied complex-linear product involution, not a complex structure. The complex structures of $V_c,V_w$ are inherited from the supplied carrier. Define

$$\begin{array}{cc}\hfill G_{3,2}& :=\mathrm{SU}(3,2)=\{g\in \mathrm{SL}(5,\mathbb{C}):g^{†}{\mathsf{\Gamma }}_{3,2}g={\mathsf{\Gamma }}_{3,2}\},\hfill \\ \hfill K_{\mathrm{SM}}& :=G_{3,2}\cap \mathrm{SU}\left(5\right)=\mathrm{S}(\mathrm{U}\left(3\right)\times \mathrm{U}\left(2\right)),\hfill \\ \hfill {\mathfrak{p}}_{\mathrm{SM}}& :=\left\{X_Z=\left(\begin{array}{cc}0& Z\\ Z^{†}& 0\end{array}\right):Z\in Hom(V_w,V_c)\right\},\hfill \\ \hfill {\mathcal{N}}_{\mathrm{SM}}& :=\{P\in {\mathcal{P}}_5^1:P{\mathsf{\Gamma }}_{3,2}P={\mathsf{\Gamma }}_{3,2}\}.\hfill \end{array}$$

(591)

The grading is distinct from the invariant tangent complex structure

$$I_{\mathrm{AIII}}\left(X_Z\right):=X_{\mathrm{i}Z}=-\frac{6}{5}[\mathrm{i}{Y}_H,X_Z],Y_H:=diag(-I_3/3,I_2/2).$$

(592)

The map $I_{\mathrm{AIII}}$ is induced by the supplied scalar multiplication on ${Hom}_{\mathbb{C}}(V_w,V_c)$; it is $K_{\mathrm{SM}}$-equivariant, squares to $-\mathrm{id}$, and extends to the $G_{3,2}$-invariant integrable complex structure on ${\mathcal{N}}_{\mathrm{SM}}$. It acts on $T{\mathcal{N}}_{\mathrm{SM}}$ and is distinct from $I_V$, every Born endomorphism $I_B$, and the positive metric normalizer J. Then ${\mathfrak{p}}_{\mathrm{SM}}$ is a Lie triple system,

$${\mathcal{N}}_{\mathrm{SM}}=exp\left({\mathfrak{p}}_{\mathrm{SM}}\right)\simeq G_{3,2}/K_{\mathrm{SM}}$$

(593)

is a totally geodesic AIII symmetric subspace of ${\mathcal{P}}_5^1$, and the ambient canonical connection restricts to the canonical $K_{\mathrm{SM}}$-connection $A_{\mathrm{SM}}$ on $G_{3,2}\to {\mathcal{N}}_{\mathrm{SM}}$. At the identity lift,

$${Hol}_e^0\left(A_{\mathrm{SM}}\right)={Hol}_e\left(A_{\mathrm{SM}}\right)=K_{\mathrm{SM}}.$$

(594)

Moreover, the faithful-state sector

$${\mathcal{S}}_{\mathrm{SM}}^{+}:=\left\{{\rho }_X=\frac{{\mathrm{e}}^{-X}}{Tr{\mathrm{e}}^{-X}}:X\in {\mathfrak{p}}_{\mathrm{SM}}\right\}=\{\rho \in {\mathcal{S}}_5^{+}:\mathsf{\Phi }\left(\rho \right){\mathsf{\Gamma }}_{3,2}\mathsf{\Phi }\left(\rho \right)={\mathsf{\Gamma }}_{3,2}\}$$

(595)

is mapped diffeomorphically to ${\mathcal{N}}_{\mathrm{SM}}$ by Φ and to ${\mathfrak{p}}_{\mathrm{SM}}$ by $\mathcal{X}=-log\mathsf{\Phi }$. Consequently the pulled-back principal connection on ${\mathcal{S}}_{\mathrm{SM}}^{+}$ has the same full $K_{\mathrm{SM}}$ holonomy. The proof is in Appendix D.7.

Corollary 37

(Global form and representation descent). The surjective homomorphism

$$\phi :\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)⟶K_{\mathrm{SM}},\phi (g_3,g_2,z)=diag(z^{-2}{g}_3,z^3{g}_2),$$

(596)

has kernel

$${\mathsf{\Gamma }}_6=\{(z^2{I}_3,z^{-3}{I}_2,z):z^6=1\}\simeq {\mathbb{Z}}_6.$$

(597)

Thus

$$K_{\mathrm{SM}}\simeq \frac{\mathrm{SU}{\left(3\right)}_c\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{U}{\left(1\right)}_Y}{{\mathbb{Z}}_6}.$$

(598)

If a covering-group representation has color triality $t\in {\mathbb{Z}}_3$, weak parity $s\in {\mathbb{Z}}_2$, and integer Abelian charge $q=6y$, where y is its numerical weak hypercharge, it descends to $K_{\mathrm{SM}}$ exactly when

$$q+2t+3s\equiv 0(mod6).$$

(599)

Block determinants give the surjectivity and kernel of (596); the generator $z=exp(\mathrm{i}\pi /3)$ acts with phase $exp\left[\mathrm{i}\pi \right(q+2t+3s)/3]$, giving precisely (599). This is the standard global-form and representation-descent computation, and the usual one-family fermion and Higgs representations satisfy it [92,93].

Theorem 21

(Carrier-generated exterior-family representation). Under the hypotheses of Theorem 20, let

$${\mathcal{F}}^{+}\left(V\right):={\mathsf{\Lambda }}^{\mathrm{even}}V={\mathsf{\Lambda }}^{0}V\oplus {\mathsf{\Lambda }}^{2}V\oplus {\mathsf{\Lambda }}^{4}V.$$

(600)

This is a functorially defined unitary $K_{\mathrm{SM}}$ module, and the associated bundle

$${\mathcal{E}}^{+}:=G_{3,2}{\times }_{K_{\mathrm{SM}}}{\mathcal{F}}^{+}\left(V\right)⟶{\mathcal{N}}_{\mathrm{SM}}$$

(601)

inherits the connection induced by $A_{\mathrm{SM}}$. Its restricted and full associated holonomy are the image of $K_{\mathrm{SM}}$ in $\mathrm{U}\left({\mathcal{F}}^{+}\left(V\right)\right)$, and this image is isomorphic to $K_{\mathrm{SM}}$.

For the covering map Equation (596), with $y=q/6$, the branching is

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill {\mathsf{\Lambda }}^{0}V& ={(\mathbf{1},\mathbf{1})}_0,\hfill \\ \hfill {\mathsf{\Lambda }}^{2}V& ={(\overline{\mathbf{3}},\mathbf{1})}_{-2/3}\oplus {(\mathbf{3},\mathbf{2})}_{1/6}\oplus {(\mathbf{1},\mathbf{1})}_1,\hfill \\ \hfill {\mathsf{\Lambda }}^{4}V& ={(\overline{\mathbf{3}},\mathbf{1})}_{1/3}\oplus {(\mathbf{1},\mathbf{2})}_{-1/2}.\hfill \end{array}\\ \hline\end{array}$$

(602)

Thus the nontrivial summands of ${\mathcal{F}}^{+}\left(V\right)$ have the representation labels conventionally assigned to one left-handed Standard-Model family $(Q,u^c,d^c,L,e^c)$ [34]; the additional trivial summand may be interpreted as a sterile ${\nu }^c$. Every summand descends to the global group $K_{\mathrm{SM}}$. This is a representation-theoretic statement; no spacetime chirality has yet been assigned.

Proof from exterior-power branching. 

From (596), $V={(\mathbf{3},\mathbf{1})}_{-1/3}\oplus {(\mathbf{1},\mathbf{2})}_{1/2}$. The standard direct-sum exterior-power rule, determinant duality, and ${\mathsf{\Lambda }}^2\mathbf{3}\simeq \overline{\mathbf{3}}$ give (602) [34,84]. The $V^{\ast }$ summand is faithful, so associated holonomy is the image of $K_{\mathrm{SM}}$; descent follows because the exterior family is the restriction of an $\mathrm{SU}\left(5\right)$ module. □

Corollary 38

(Anomaly cancellation and precise representation status). If the summands of Equation (602) are assigned to four-dimensional left-handed Weyl fields, all perturbative $K_{\mathrm{SM}}$ gauge anomalies and the mixed gauge-gravitational anomaly cancel, and the usual $\mathrm{SU}\left(2\right)$ global anomaly is absent. The odd exterior module ${\mathcal{F}}^{-}\left(V\right):={\mathsf{\Lambda }}^{\mathrm{odd}}V$ is the complex-conjugate family.

The exterior algebra and its parity grading are canonical functors of the supplied carrier. The entropy and holonomy constructions do not select even rather than odd parity as physical Weyl chirality, identify internal exterior parity with spacetime chirality, choose this functor over other Schur functors, select a generation multiplicity, or supply fermion kinetic or Yukawa data. Under the four-dimensional left-handed Weyl assignment stated above, the result is therefore a distinguished anomaly-cancelling associated-family candidate, not a uniqueness theorem for physical matter.

Proof from the standard one-family anomaly. 

The nontrivial module is the standard $\mathbf{10}\oplus \overline{\mathbf{5}}$ of $\mathrm{SU}\left(5\right)$, whose cubic and linear anomaly traces vanish; restriction gives the perturbative and mixed subgroup cancellations [34,92]. Its four weak doublets give an even mod-two index, excluding the global $\mathrm{SU}\left(2\right)$ anomaly [35]. Determinant duality identifies the odd exterior module with the conjugate family, proving the remaining claim. □

Corollary 39

(Inherited single-trace coupling normalization). Suppose a spacetime gauge completion is supplied and its three $K_{\mathrm{SM}}$ factors use the single carrier-trace kinetic form inherited from V, with one coefficient $g_5$. If $Tr\left(T_a{T}_b\right)={\delta }_{ab}/2$ for the non-Abelian Hermitian generators and $Y_H=diag(-I_3/3,I_2/2)$, canonical component normalization gives

$$\begin{array}{|c|}\hline g_3=g_2=g_5,g_Y^2=\frac{3}{5}{g}_5^2,g_1:=\sqrt{\frac{5}{3}}{g}_Y=g_5.\\ \hline\end{array}$$

(603)

This fixes relative normalization at the supplied matching scale, not the overall coupling, the scale itself, or its renormalization-group evolution. Without the single-trace hypothesis an $Ad\left(K_{\mathrm{SM}}\right)$-invariant kinetic form has three independent positive coefficients.

Proof from the standard $\mathrm{SU}\left(5\right)$ trace normalization. 

The non-Abelian blocks already have trace norm $1/2$, whereas $Tr\left(Y_H^2\right)=5/6$; their ratio is $5/3$. The standard single-trace $\mathrm{SU}\left(5\right)$ normalization therefore gives $g_Y^2=(3/5)g_5^2$ and $g_3=g_2=g_5$ [33,34], with exactly the qualifications in the statement. □

Corollary 40

(Legendre and compact-dual realizations). Choose a real basis $O_1,\dots ,O_{12}$ of ${\mathfrak{p}}_{\mathrm{SM}}$, put ${\sigma }_{\ast }=I/5$, and define

$${\lambda }_{\mathrm{SM}}\left(\theta \right):={\mathsf{\Lambda }}_{{\sigma }_{\ast }}\left({\theta }^a{O}_a\right),E_{\mathrm{SM}}\left(\theta \right):=E_{{\sigma }_{\ast }}\left({\theta }^a{O}_a\right),m_{\mathrm{SM}}\left(\theta \right):=\nabla {\lambda }_{\mathrm{SM}}\left(\theta \right).$$

(604)

The restricted partition is $K_{\mathrm{SM}}$-invariant, and Theorem 4 makes $m_{\mathrm{SM}}:{\mathbb{R}}^{12}\to int{\mathcal{C}}_O$ a diffeomorphism. If

$$F_{\mathrm{src}}:=\mathsf{\Phi }\circ E_{\mathrm{SM}},A_{\mathrm{src}}:=F_{\mathrm{src}}^{\ast }{A}_{\mathrm{SM}},A_{\mathrm{mom}}:={\left(m_{\mathrm{SM}}^{-1}\right)}^{\ast }{A}_{\mathrm{src}},$$

(605)

then both transported principal connections have full $K_{\mathrm{SM}}$ holonomy. These are coordinate changes, not identifications with the BKM Levi-Civita connection.

The compact AIII dual is the Grassmannian ${Gr}_3\left({\mathbb{C}}^5\right)=\mathrm{SU}\left(5\right)/K_{\mathrm{SM}}$. If

$${\rho }_{3,2}=r_3{P}_c+r_2{P}_w,r_3,r_2>0,3r_3+2r_2=1,r_3\ne r_2,$$

(606)

then, with $Y_H:=diag(-I_3/3,I_2/2)$,

$$\mathcal{X}\left({\rho }_{3,2}\right)=\frac{6}{5}log\left(\frac{r_3}{r_2}\right)Y_H.$$

(607)

Writing $\lambda =(6/5)log(r_3/r_2)$, equivariance of the Gibbs log gives a diffeomorphism

$${\mathcal{O}}_{\rho }:=\{U{\rho }_{3,2}{U}^{†}:U\in \mathrm{SU}\left(5\right)\}⟶{\mathcal{O}}_{\lambda Y}:=\{U\left(\lambda Y_H\right)U^{†}:U\in \mathrm{SU}\left(5\right)\}.$$

(608)

These are two embeddings of the same coset $\mathrm{SU}\left(5\right)/K_{\mathrm{SM}}$, so the map intertwines their common homogeneous principal bundle and canonical connection; it does not identify that connection with the BKM Levi-Civita connection. The canonical connection has full principal holonomy $K_{\mathrm{SM}}$. At $r_3=r_2=1/5$ the orbit collapses and the stabilizer enhances to $\mathrm{SU}\left(5\right)$; a nonzero source, constrained moment, or symmetry-breaking potential is therefore additional input.

Remark 16

(Global and physical status). The geometry also does not exclude the other conventional connected global forms. The observed local multiplets are compatible with the four possibilities described in [92,93]:

$$K_{\mathsf{\Gamma }}:=\frac{\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)}{\mathsf{\Gamma }},\mathsf{\Gamma }\in \{1,{\mathbb{Z}}_2,{\mathbb{Z}}_3,{\mathbb{Z}}_6\}.$$

(609)

For $\mathsf{\Gamma }<{\mathsf{\Gamma }}_6$, realizing $K_{\mathsf{\Gamma }}$ requires choosing a lift of the structure group and connection through $K_{\mathsf{\Gamma }}\to K_{\mathrm{SM}}$, whose kernel is ${\mathsf{\Gamma }}_6/\mathsf{\Gamma }$. On a paracompact base X, the obstruction to such a lift is the standard central-extension class in ${\check{H}}^2(X,{\mathsf{\Gamma }}_6/\mathsf{\Gamma })$ [94], Chapter I, Section 6. It vanishes on the contractible base ${\mathcal{N}}_{\mathrm{SM}}$; there the lifted connection has the same full curvature algebra and hence full $K_{\mathsf{\Gamma }}$ holonomy, although no lift is canonically selected. On a general spacetime its existence is additional topological data. The group in Equation (594) is principal holonomy. The isotropy action $Z\mapsto k_{3}Z{k}_2^{-1}$ has kernel $K_{\mathrm{SM}}\cap Z\left(\mathrm{SU}\left(5\right)\right)\simeq {\mathbb{Z}}_5$, so tangent holonomy is $K_{\mathrm{SM}}/{\mathbb{Z}}_5$. The induced tangent representation factors through this quotient and is not the Standard Model matter representation; the distinct exterior-family associated bundle is Equation (601). Assigning that bundle to physical spacetime fermions remains a separate step. The hypercharge $\mathrm{U}\left(1\right)$ here is the relative block phase generated by $\mathrm{i}{Y}_H$, not the scalar projective phase of Appendix H.3 and not the diagonal-qutrit Fisher $\mathrm{SO}\left(2\right)\simeq \mathrm{U}\left(1\right)$ mismatch in Equation (490).

Gauge structure group and connection holonomy must not be identified: the former is the chosen local redundancy, whereas the latter depends on the connection and may be a proper subgroup. The construction fixes the relative $3+2$ block generator only up to sign and overall physical normalization; it fixes neither the hypercharge coupling nor the other gauge couplings. Thus it establishes a gauge-group-compatible kinematic connection, not the Standard Model field theory. For a spacetime map f, one only has $Hol\left(f^{\ast }{A}_{\mathrm{SM}}\right)\subseteq K_{\mathrm{SM}}$. By Ambrose-Singer, equality holds precisely when parallel transport to one base point of all curvature values $F_{A_{\mathrm{SM}},f\left(x\right)}(\mathrm{d}{f}_{x}u,\mathrm{d}{f}_{x}v)$ spans ${\mathfrak{k}}_{\mathrm{SM}}$ as $x,u,v$ vary. The canonical field remains composite, $F_{A_{\mathrm{SM}}}=-E_{\mathrm{SM}}\wedge E_{\mathrm{SM}}$. Identifying its constrained pullback with a physical field is extra input. An unconstrained gauge field requires an independently supplied connection; on the same principal bundle it has the local form $A_{\mathrm{phys}}=f^{\ast }{A}_{\mathrm{SM}}+a$ with $a\in {\mathsf{\Omega }}^1(Ad{P}_{K_{\mathrm{SM}}})$. A Yang-Mills action on spacetime, the overall coupling and matching scale, physical chirality, Higgs and Yukawa sectors, and generation data remain inputs. On the statistical symmetric-space base, however, $A_{\mathrm{SM}}$ satisfies the local source-free Yang-Mills equation by Proposition 15; its unrenormalized total action on that noncompact base is infinite. Under the additional single-trace hypothesis, the relative coupling normalization is Corollary 39. Legendre duality only reparametrizes this composite connection; it does not identify the BKM Levi-Civita connection with $A_{\mathrm{SM}}$.

The same composite Cartan-connection/coframe qualification applies to the ambient $\mathrm{SU}\left(n\right)$ construction [80], Section 3. The analogous representation and topology qualifications follow from the preceding associated-bundle and lifting arguments. The optional compact-dual torsion construction uses the induced Cartan compact-dual map $\mathcal{I}:\mathfrak{k}\to \mathfrak{p}$ of Equation (A152) and additionally chooses its normalization; it is confined to Appendix D.8; it affects none of the channel-output or discrete-generator results.

7. Dynamical Input and Exact Relative-Information Balance

7.1. Information Metric, Velocities, and Dynamical Mobility

The preceding geometry is static. A flow requires an independently supplied clock and mobility mapping entropy cotangents to velocities. A positive-semidefinite Onsager map $\mathbb{M}:T^{\ast }M\to TM$ is such constitutive input; for $\mathcal{D}\left(\theta \right)=D_{\mathrm{U}}({\rho }_{\theta }\parallel {\rho }_{\mathrm{eq}})$ it gives

$$v_t=-{\mathbb{M}}_{\theta }\mathrm{d}\mathcal{D},\dot{\mathcal{D}}=-\langle \mathrm{d}\mathcal{D},{\mathbb{M}}_{\theta }\mathrm{d}\mathcal{D}\rangle \le 0.$$

(610)

Natural-gradient flow is the special choice $\mathbb{M}=g^{♯}$; general mobilities independently determine time scale, constraints, and degeneracies [95,96]. In particular, detailed-balance quantum Markov dynamics generally uses a generator-dependent transport metric rather than BKM [97].

Proposition 28

(BKM natural-gradient exponential relaxation). Fix $\sigma \in {\mathcal{S}}_n^{+}$ and independently supply a relaxation time $\tau >0$. For every ${\rho }_0\in {\mathcal{S}}_n^{+}$, consider $t\ge 0$ and put

$$K_{\rho }:=log\rho -log\sigma ,{\tilde{K}}_{\rho }:=K_{\rho }-Tr\left(\rho K_{\rho }\right)I.$$

(611)

The BKM natural-gradient equation for ${\mathcal{D}}_{\sigma }\left(\rho \right):=D_{\mathrm{U}}(\rho \parallel \sigma )$, with ${\rho }_{t=0}={\rho }_0$, is

$${\dot{\rho }}_t=-\frac{1}{\tau }{\mathsf{K}}_{{\rho }_t}\left({\tilde{K}}_{{\rho }_t}\right).$$

(612)

With the traceless score $H_0:=H_{{\rho }_0\mid \sigma }$, its full-state solution is the exponential-affine interpolation

$${\rho }_t=E_{\sigma }\left({\mathrm{e}}^{-t/\tau }{H}_0\right).$$

(613)

Thus the exact relaxation is the partition-response ray of Theorem 3 traversed exponentially in the supplied time; it is a nonlinear state flow, not automatically a CPTP semigroup. The flow obeys

$$\begin{array}{cc}\hfill \frac{\mathrm{d}}{\mathrm{d}t}{D}_{\mathrm{U}}({\rho }_t\parallel \sigma )& =-\frac{1}{\tau }{Cov}_{{\rho }_t}^{\mathrm{KM}}(K_{{\rho }_t},K_{{\rho }_t})=-\tau g_{{\rho }_t}^{\mathrm{BKM}}({\dot{\rho }}_t,{\dot{\rho }}_t),\hfill \end{array}$$

(614)

$$\begin{array}{cc}\hfill D_{\mathrm{U}}({\rho }_0\parallel \sigma )& =\tau {\int }_0^{\infty }{g}_{{\rho }_t}^{\mathrm{BKM}}({\dot{\rho }}_t,{\dot{\rho }}_t)\mathrm{d}t.\hfill \end{array}$$

(615)

Here the covariance has the meaning of Equation (61). This is an exact dissipative realization of the entropy endpoint, not conservative symmetric-space mechanics. The choice of BKM mobility and τ is constitutive input; on a constrained output image the BKM gradient must be projected and the closed exponential formula need not remain in the image.

Proof. 

The differential of ${\mathcal{D}}_{\sigma }$ on trace-zero tangents is $V\mapsto Tr\left(V{K}_{\rho }\right)$. By Equation (59), its BKM gradient is ${\mathsf{K}}_{\rho }\left({\tilde{K}}_{\rho }\right)$. Applying $D{log}_{\rho }={\mathsf{K}}_{\rho }^{-1}$ to Equation (612) gives, modulo scalar matrices,

$$\frac{\mathrm{d}}{\mathrm{d}t}[log{\rho }_t-log\sigma ]=-\frac{1}{\tau }[log{\rho }_t-log\sigma ],$$

(616)

whose normalized solution is Equation (613). The chain rule gives the first equality in Equation (614); the definition of the metric gives the second. Integration to the limit ${\rho }_t\to \sigma$ proves Equation (615). □

General entropic and Bregman proximal-point methods are standard [18,19,20]. The composition law below follows from the particular relative-log contraction in Equation (152), within the established core identified before Theorem 6; it is not a generic Bregman-resolvent semigroup law.

Corollary 41

(Entropic resolvent semigroup, Rényi value cocycle, and exact BKM sampling). For fixed $\sigma \in {\mathcal{S}}_n^{+}$, write $P_r^{\sigma }:={prox}_{r,\sigma }^{\mathrm{U}}$ for $r>0$ and set $P_0^{\sigma }=\mathrm{id}$ by continuous extension. Then, for $r,s\ge 0$,

$$\begin{array}{|c|}\hline P_r^{\sigma }\circ P_s^{\sigma }=P_{r\oplus s}^{\sigma },r\oplus s:=r+s+rs.\\ \hline\end{array}$$

(617)

Use $a={(1+r)}^{-1}={\mathrm{e}}^{-u}$ to define ${\mathsf{\Phi }}_u^{\sigma }:=P_{{\mathrm{e}}^u-1}^{\sigma }$ and $R_a^{\sigma }\left(\rho \right):=P_{(1-a)/a}^{\sigma }\left(\rho \right)={\gamma }_a$. Then, for $u,v\ge 0$ and $a,b\in (0,1]$,

$${\mathsf{\Phi }}_u^{\sigma }\circ {\mathsf{\Phi }}_v^{\sigma }={\mathsf{\Phi }}_{u+v}^{\sigma },{\mathsf{\Phi }}_u^{\sigma }\left(\rho \right)=E_{\sigma }\left({\mathrm{e}}^{-u}{H}_{\rho \mid \sigma }\right),$$

(618)

and

$$R_a^{\sigma }\circ R_b^{\sigma }=R_{ab}^{\sigma },H_{R_a^{\sigma }\left(\rho \right)\mid \sigma }=a{H}_{\rho \mid \sigma }.$$

(619)

The maps compose, whereas their barycentric values obey the distinct discounted cocycle

$$\begin{array}{|c|}\hline C_{ab}^{\mathrm{b}}(\rho ,\sigma )=C_a^{\mathrm{b}}(R_b^{\sigma }\left(\rho \right),\sigma )+a{C}_b^{\mathrm{b}}(\rho ,\sigma ).\\ \hline\end{array}$$

(620)

In logarithmic time, put ${\mathcal{C}}_u^{\sigma }\left(\rho \right):=C_{{\mathrm{e}}^{-u}}^{\mathrm{b}}(\rho ,\sigma )$. Then, for $u,v\ge 0$,

$$\begin{array}{cc}\hfill {\mathcal{C}}_{u+v}^{\sigma }\left(\rho \right)& ={\mathcal{C}}_u^{\sigma }\left({\mathsf{\Phi }}_v^{\sigma }\left(\rho \right)\right)+{\mathrm{e}}^{-u}{\mathcal{C}}_v^{\sigma }\left(\rho \right),\hfill \\ \hfill {\partial }_u{\mathcal{C}}_u^{\sigma }\left(\rho \right)+{\mathcal{C}}_u^{\sigma }\left(\rho \right)& =D_{\mathrm{U}}({\mathsf{\Phi }}_u^{\sigma }\left(\rho \right)\parallel \sigma ),\hfill \end{array}$$

(621)

$$\begin{array}{cc}\hfill {\mathcal{C}}_u^{\sigma }\left(\rho \right)& ={\int }_0^u{\mathrm{e}}^{-(u-v)}{D}_{\mathrm{U}}({\mathsf{\Phi }}_v^{\sigma }\left(\rho \right)\parallel \sigma )\mathrm{d}v.\hfill \end{array}$$

(622)

Thus $C_t^{\mathrm{b}}=(1-t)D_t^{\mathrm{b}}$ is a discounted dimensionless information value, not an ordinary Hopf-Lax value semigroup. Multiplication by the separately supplied ${\kappa }_A$ makes it action-valued.

The infinitesimal state-space generator is

$$\begin{array}{cc}\hfill Y_{\sigma }\left(\rho \right)& :=-{grad}_{\mathrm{BKM}}{D}_{\mathrm{U}}(\rho \parallel \sigma )=-{\mathsf{K}}_{\rho }\left(H_{\rho \mid \sigma }-Tr\left(\rho H_{\rho \mid \sigma }\right)I\right),\hfill \\ \hfill (1+r){\partial }_r{P}_r^{\sigma }\left(\rho \right)& =Y_{\sigma }\left(P_r^{\sigma }\left(\rho \right)\right).\hfill \end{array}$$

(623)

For $F\in C^1\left({\mathcal{S}}_n^{+}\right)$, its Koopman generator is $\left({\mathcal{L}}_{\sigma }F\right)\left(\rho \right):=D{F}_{\rho }\left[Y_{\sigma }\left(\rho \right)\right]$. In particular, every Hermitian observable A obeys

$${\mathcal{L}}_{\sigma }Tr\left(\rho A\right)=-{Cov}_{\rho }^{\mathrm{KM}}(A,H_{\rho \mid \sigma }).$$

(624)

For $p_r={\mathsf{\Phi }}_u^{\sigma }\left(\rho \right)$, under the affine identification of quotient cotangent spaces, the Fenchel optimizer in Equation (173) is precisely the instantaneous dissipative force:

$$\left[P_r^{★}\right]=[log{p}_r-log\sigma ],\frac{\mathrm{d}{p}_r}{\mathrm{d}u}=-{\left(P_r^{★}\right)}^{{♯}_{p_r}}.$$

(625)

Thus ${\mathsf{\Phi }}_{t/\tau }^{\sigma }$ is exactly the nonlinear BKM relaxation flow in Proposition 28; the interpretation of t as physical time still uses the independently supplied τ.

Let $r_k>0$ and define

$${\rho }_{k+1}:=P_{r_k}^{\sigma }\left({\rho }_k\right),H_k:=H_{{\rho }_k\mid \sigma },T_k:=\tau {\displaystyle \sum _{j=0}^{k-1}}log(1+r_j).$$

(626)

Then

$$\begin{array}{|c|}\hline H_k=a_k{H}_0,a_k:={\displaystyle \prod _{j=0}^{k-1}}{(1+r_j)}^{-1}={\mathrm{e}}^{-T_k/\tau },{\rho }_k=E_{\sigma }\left(a_k{H}_0\right)={\rho }_{T_k}.\\ \hline\end{array}$$

(627)

Every iterate is therefore an exact sample of the continuous relaxation. In particular, every bound and asymptotic expansion in Corollary 43 becomes an exact discrete estimate after substituting $t=T_k$ and ${\mathrm{e}}^{-t/\tau }=a_k$. Put

$${\mathcal{R}}_k:=\left(1+\frac{1}{r_k}\right)D_{\mathrm{U}}({\rho }_k\parallel {\rho }_{k+1})+\frac{1}{r_k}{D}_{\mathrm{U}}({\rho }_{k+1}\parallel {\rho }_k)\ge 0.$$

(628)

Each step and every finite prefix obey

$$\begin{array}{cc}\hfill D_{\mathrm{U}}({\rho }_k\parallel \sigma )-D_{\mathrm{U}}({\rho }_{k+1}\parallel \sigma )& ={\mathcal{R}}_k=\tau {\int }_{T_k}^{T_{k+1}}{g}_{{\rho }_t}^{\mathrm{BKM}}({\dot{\rho }}_t,{\dot{\rho }}_t)\mathrm{d}t,\hfill \end{array}$$

(629)

$$\begin{array}{cc}\hfill D_{\mathrm{U}}({\rho }_0\parallel \sigma )-D_{\mathrm{U}}({\rho }_N\parallel \sigma )& ={\displaystyle \sum _{k=0}^{N-1}}{\mathcal{R}}_k=\tau {\int }_0^{T_N}{g}_{{\rho }_t}^{\mathrm{BKM}}({\dot{\rho }}_t,{\dot{\rho }}_t)\mathrm{d}t.\hfill \end{array}$$

(630)

The iterates converge to σ for every faithful initial state if and only if $T_k\to +\infty$, equivalently ${\sum }_{j}log(1+r_j)=+\infty$. In that case

$$D_{\mathrm{U}}({\rho }_0\parallel \sigma )={\displaystyle \sum _{k=0}^{\infty }}{\mathcal{R}}_k=\tau {\int }_0^{\infty }{g}_{{\rho }_t}^{\mathrm{BKM}}({\dot{\rho }}_t,{\dot{\rho }}_t)\mathrm{d}t.$$

(631)

If $r_k=h_k/\tau$, then

$$\frac{H_{k+1}-H_k}{h_k}=-\frac{1}{\tau }{H}_{k+1},T_{k+1}-T_k=\tau log(1+h_k/\tau ).$$

(632)

This is backward Euler at nominal step $h_k$ in relative-log coordinates and the exact flow over the displayed shorter interval. Conversely, exact sampling at a prescribed interval $h_k$ uses $r_k={\mathrm{e}}^{h_k/\tau }-1$. None of these nonlinear state maps is thereby a linear CPTP dynamical semigroup.

Proof. 

By Equation (152),

$${\ell }_{\sigma }\left(P_r^{\sigma }{P}_s^{\sigma }\rho \right)=\frac{{\ell }_{\sigma }\left(\rho \right)}{(1+r)(1+s)}={\ell }_{\sigma }\left(P_{r+s+rs}^{\sigma }\rho \right).$$

(633)

Injectivity of the relative-log chart proves Equations (617) and (618). The same score contraction proves Equation (619). Substitution into the partition cocycle gives Equation (620); differentiating it at the identity, or differentiating $C_{{\mathrm{e}}^{-u}}^{\mathrm{b}}=-{\phi }_{\rho \mid \sigma }\left({\mathrm{e}}^{-u}\right)$, gives Equation (622). Differentiation of the score contraction and the BKM score-tangent identification prove Equations (623) and (625); the observable formula is trace duality. Iteration gives Equation (627). Putting $\omega ={\rho }_k$ in Equation (151) proves the first equality in Equation (629); the continuous dissipation identity on $[T_k,T_{k+1}]$ proves the second. Telescoping proves the finite and infinite balances. The remaining claims follow from $a_k={\mathrm{e}}^{-T_k/\tau }$ and direct rearrangement of the score recursion. □

Remark 17

(Canonical cotangent lift and distinct generators). The exact state vector field has the standard autonomous cotangent lift [98], Chapter 12. With the convention ${\iota }_{X_H}{\omega }_A=-\mathrm{d}H$ used below, define

$$J_{\sigma }(\rho ,\alpha ):=\alpha \left(Y_{\sigma }\left(\rho \right)\right)=-\alpha \left({grad}_{\mathrm{BKM}}{D}_{\mathrm{U}}(\rho \parallel \sigma )\right).$$

(634)

Its Hamiltonian flow projects exactly to ${\mathsf{\Phi }}_u^{\sigma }$; physical time $t_{\mathrm{phys}}=\tau u$ uses $J_{\sigma }/\tau$. This is a genuine canonical lift but is linear in cotangent momentum and is not a positive mechanical energy (see Table 1). In particular, the Hamilton-Jacobi characteristics of ${\mathcal{H}}_r^{\mathrm{U}}$ are not being identified with the fixed-input curve $r\mapsto P_r^{\sigma }\left(\rho \right)$.

Corollary 42

(Exact Helmholtz and uniform-reference balances). Let ${\mathcal{R}}_k$ be as in Equation (628). If $\sigma ={\tau }_{\beta }={\mathrm{e}}^{-\beta H_{\mathrm{sys}}}/Z_{\beta }$, then

$$\begin{array}{cc}\hfill F_{\beta }\left({\rho }_k\right)-F_{\beta }\left({\rho }_{k+1}\right)& ={\beta }^{-1}{\mathcal{R}}_k,\hfill \end{array}$$

(635)

$$\begin{array}{cc}\hfill F_{\beta }\left({\rho }_0\right)-F_{\beta }\left({\rho }_N\right)& ={\beta }^{-1}{\displaystyle \sum _{k=0}^{N-1}}{\mathcal{R}}_k.\hfill \end{array}$$

(636)

When $T_k\to \infty$,

$$F_{\beta }\left({\rho }_0\right)-F_{\beta }\left({\tau }_{\beta }\right)={\beta }^{-1}{\displaystyle \sum _{k=0}^{\infty }}{\mathcal{R}}_k.$$

(637)

This is the precise free-energy sense in which the prescribed BKM flow regains equilibrium. Heat, work, and bath-entropy interpretations still require a physical realization of that constitutive flow.

For $\sigma =I/n$,

$$P_r^{I/n}\left(\rho \right)=\frac{{\rho }^{1/(1+r)}}{Tr{\rho }^{1/(1+r)}},$$

(638)

and

$$\begin{array}{cc}\hfill S_{\mathrm{vN}}\left({\rho }_{k+1}\right)-S_{\mathrm{vN}}\left({\rho }_k\right)& ={\mathcal{R}}_k,\hfill \end{array}$$

(639)

$$\begin{array}{cc}\hfill S_{\mathrm{vN}}\left({\rho }_N\right)-S_{\mathrm{vN}}\left({\rho }_0\right)& ={\displaystyle \sum _{k=0}^{N-1}}{\mathcal{R}}_k.\hfill \end{array}$$

(640)

Under convergence,

$$logn-S_{\mathrm{vN}}\left({\rho }_0\right)={\displaystyle \sum _{k=0}^{\infty }}{\mathcal{R}}_k.$$

(641)

Equivalently, the identity $S_{\mathrm{vN}}\left(\rho \right)=logn-D_{\mathrm{U}}(\rho \parallel I/n)$ provides a relative-entropy formulation of the entropy information contrast that helped motivate this line of inquiry through the work of Vopson and Lepadatu [99]. Every nonstationary step is strict.

Proof. 

Substitute ${\beta }^{-1}{D}_{\mathrm{U}}(\rho \parallel {\tau }_{\beta })=F_{\beta }\left(\rho \right)-F_{\beta }\left({\tau }_{\beta }\right)$ and $D_{\mathrm{U}}(\rho \parallel I/n)=logn-S_{\mathrm{vN}}\left(\rho \right)$ in Equation (629). The power map is Equation (148) with $\sigma =I/n$. □

Remark 18

(Commuting flow and normalized entropy deficit). If $[{\rho }_t,\sigma ]=0$ with common-basis eigenvalues $p_i\left(t\right),{\sigma }_i$, the BKM flow becomes

$$\tau {\dot{p}}_i=-p_i\left[log\frac{p_i}{{\sigma }_i}-D(p\parallel \sigma )\right].$$

(642)

It is the Fisher-Rao/replicator-type relative-entropy gradient flow, not a generic linear detailed-balance master equation; the latter generally uses a generator-dependent transport metric [96,97]. The uniform-reference power map above is likewise nonlinear and is not a quantum channel.

Separately, if a CPTP map $\mathsf{\Lambda }:{\mathcal{S}}_n\to {\mathcal{S}}_m$ satisfies $\mathsf{\Lambda }(I_n/n)=I_m/m$, data processing gives

$$\begin{array}{|c|}\hline logm-S_{\mathrm{vN}}\left(\mathsf{\Lambda }\rho \right)\le logn-S_{\mathrm{vN}}\left(\rho \right),S_{\mathrm{vN}}\left(\mathsf{\Lambda }\rho \right)-S_{\mathrm{vN}}\left(\rho \right)\ge log\frac{m}{n}.\\ \hline\end{array}$$

(643)

Thus raw entropy is nondecreasing for $m\ge n$ and strictly increasing for $m>n$; for $m<n$, nondecrease is not guaranteed. For example, $\mathsf{\Lambda }\left(\rho \right)=\rho \otimes I_k/k$ has $m=nk$ and entropy gain $logk$. The dimension-independent statement is contraction of the normalized entropy deficit, not unqualified “increased mixing.”

Corollary 43

(Quantitative BKM relaxation rates). For the flow in Proposition 28, put

$$a_t:={\mathrm{e}}^{-t/\tau },\lambda \left(a\right):={\mathsf{\Lambda }}_{\sigma }\left(a{H}_0\right),q\left(a\right):={Cov}_{E_{\sigma }\left(a{H}_0\right)}^{\mathrm{KM}}(H_0,H_0),$$

(644)

and define

$$D_{+}\left(t\right):=D_{\mathrm{U}}({\rho }_t\parallel \sigma ),D_{-}\left(t\right):=D_{\mathrm{U}}(\sigma \parallel {\rho }_t),\mathcal{J}\left(t\right):=D_{+}\left(t\right)+D_{-}\left(t\right).$$

(645)

Then

$$\begin{array}{cccc}\hfill D_{+}\left(t\right)& =a_t{\lambda }^{\prime }\left(a_t\right)-\lambda \left(a_t\right),\hfill & \hfill D_{-}\left(t\right)& =\lambda \left(a_t\right)-a_t{\lambda }^{\prime }\left(0\right),\hfill \\ \hfill {\dot{D}}_{+}\left(t\right)& =-\frac{a_t^2}{\tau }q\left(a_t\right),\hfill & \hfill {\dot{D}}_{-}\left(t\right)& =-\frac{1}{\tau }\mathcal{J}\left(t\right),\hfill \\ \hfill \dot{\mathcal{J}}\left(t\right)& =-\frac{1}{\tau }[\mathcal{J}\left(t\right)+a_t^{2}q\left(a_t\right)]\le -\frac{1}{\tau }\mathcal{J}\left(t\right).\hfill \end{array}$$

(646)

Consequently,

$$D_{\pm }\left(t\right)\le \mathcal{J}\left(t\right)\le {\mathrm{e}}^{-t/\tau }\mathcal{J}\left(0\right),$$

(647)

while the source-dependent bounds are quadratically sharper:

$$\begin{array}{cccc}\hfill D_{\pm }\left(t\right)& \le \frac{{\mathrm{e}}^{-2t/\tau }}{8}osc{\left(H_0\right)}^2,\hfill & \hfill \mathcal{J}\left(t\right)& \le \frac{{\mathrm{e}}^{-2t/\tau }}{4}osc{\left(H_0\right)}^2,\hfill \end{array}$$

(648)

$$\begin{array}{cccc}\hfill {∥{\rho }_t-\sigma ∥}_1& \le \frac{{\mathrm{e}}^{-t/\tau }}{2}osc\left(H_0\right),\hfill & \hfill \left|Tr\left[({\rho }_t-\sigma )A\right]\right|& \le \frac{{\mathrm{e}}^{-t/\tau }}{4}osc\left(H_0\right)osc\left(A\right).\hfill \end{array}$$

(649)

As $t\to \infty$,

$$\begin{array}{cc}\hfill D_{\pm }\left(t\right)& =\frac{1}{2}{\mathrm{e}}^{-2t/\tau }{Cov}_{\sigma }^{\mathrm{KM}}(H_0,H_0)+O\left({\mathrm{e}}^{-3t/\tau }\right),\hfill \\ \hfill \mathcal{J}\left(t\right)& ={\mathrm{e}}^{-2t/\tau }{Cov}_{\sigma }^{\mathrm{KM}}(H_0,H_0)+O\left({\mathrm{e}}^{-3t/\tau }\right).\hfill \end{array}$$

(650)

For every Hermitian A, first-order partition response also gives

$$Tr\left[({\rho }_t-\sigma )A\right]={\mathrm{e}}^{-t/\tau }{Cov}_{\sigma }^{\mathrm{KM}}(A,H_0)+O\left({\mathrm{e}}^{-2t/\tau }\right).$$

(651)

Thus all bounded observable deviations are bounded at rate $1/\tau$, and this is their leading rate when the displayed covariance is nonzero. If ${\rho }_0\ne \sigma$, the information gap is second order and has asymptotic rate $2/\tau$; when σ is a supplied Gibbs state, the same statement applies to its proportional free-energy gap.

For $0\le t_0<t_1$ the exact dissipation also gives

$$D_{+}\left(t_0\right)-D_{+}\left(t_1\right)=\tau {\int }_{t_0}^{t_1}{g}_{{\rho }_t}^{\mathrm{BKM}}({\dot{\rho }}_t,{\dot{\rho }}_t)\mathrm{d}t\ge \frac{\tau }{t_1-t_0}{d}_{\mathrm{BKM}}{({\rho }_{t_0},{\rho }_{t_1})}^2.$$

(652)

This is a bound for the supplied BKM mobility and relaxation scale, not a universal quantum speed limit or a statement about arbitrary CPTP semigroups.

Proof. 

The Bregman formulas along the ray give the first line of Equation (646); differentiating with ${\dot{a}}_t=-a_t/\tau$ gives the remaining lines. Gronwall proves Equation (647), while Equations (100) and (101) give Equation (648). Quantum Pinsker’s inequality gives the trace-norm estimate. Centering A at the midpoint of its spectral range and using trace duality gives the observable estimate. Taylor expansion of $\lambda$ at zero proves Equation (650); first-order differentiation of $E_{\sigma }\left(a{H}_0\right)$ at $a=0$ gives Equation (651). Finally, Equation (614), Cauchy-Schwarz for path length, and the definition of Riemannian distance prove Equation (652). □

Corollary 44

(Asymptotic bridge suppression under uniform-reference relaxation). Set $n=3$ and $\sigma =I_3/3$ in Proposition 28. If $X_t$ is the traceless Gibbs coordinate of ${\rho }_t$, then $X_t={\mathrm{e}}^{-t/\tau }{X}_0$. Consequently, as $t\to \infty$,

$$\begin{array}{|c|}\hline {\mathcal{V}}_{\mathrm{br}}\left(X_t\right)=\frac{5}{128}{\mathrm{e}}^{-2t/\tau }{∥X_0∥}_{g_0}^2+O\left({\mathrm{e}}^{-3t/\tau }\right).\\ \hline\end{array}$$

(653)

For the proximal iteration in Equation (627), the exact sampled expansion replaces t by $T_k$ and ${\mathrm{e}}^{-t/\tau }$ by $a_k$, as $a_k\to 0$. Thus the canonical bridge diagnostic is suppressed quadratically near the tracial equilibrium. It does not drive this flow: BKM mobility and τ remain the independently supplied constitutive data.

Proof. 

For the uniform reference, the score representative in Equation (613) is $-X_0$, so normalized exponentiation gives ${\rho }_t={\rho }_{{\mathrm{e}}^{-t/\tau }{X}_0}$. Substitution into Equation (534) proves Equation (653); Equation (627) gives the discrete formula. □

For a smooth faithful curve with $V_t={\dot{\rho }}_t$,

$$D_{\mathrm{U}}({\rho }_t\parallel {\rho }_{t+\mathrm{d}t})=\frac{\mathrm{d}{t}^2}{2}{g}_{{\rho }_t}^{\mathrm{BKM}}(V_t,V_t)+O\left(\mathrm{d}{t}^3\right).$$

(654)

Hence $\parallel V_t{\parallel }_{\mathrm{BKM}}$ is a second-order local distinguishability speed, whereas $-\mathrm{d}{D}_{\mathrm{U}}({\rho }_t\parallel \tau )/\mathrm{d}t$ is a first-order relaxation rate. The endpoint divergence is generally neither path length nor unweighted path energy; Equation (109) is specific to an exponential-affine segment. A moving reference adds a calibration-drift term, and linearization separates the BKM Hessian from the mobility; both formulas are collected in Appendix F.2.

7.2. Conditional Unresolved Dynamics and GENERIC Placement

The compression tensors in Propositions 19 and 20 are static. Temporal memory requires an independently supplied generator. The following finite-dimensional statement fixes the precise relation.

Proposition 29

(Mori-Zwanzig kernel from a supplied linear dynamics). Let $P^2=P=P^{†}$ on a finite-dimensional Hilbert space, $Q=I-P$, and let u solve $\dot{u}=Au$. Put $v:=Pu$, $w:=Qu$, and

$$A_{RS}{:=RAS|}_{imS},R,S\in \{P,Q\},A_Q:=A_{QQ}.$$

(655)

Then the resolved variable obeys the exact equation

$$\begin{array}{cc}\hfill \dot{v}\left(t\right)& =A_{PP}v\left(t\right)+A_{PQ}{\mathrm{e}}^{t{A}_Q}w\left(0\right)+{\int }_0^t\mathcal{K}(t-s)v\left(s\right)\mathrm{d}s,\hfill \end{array}$$

(656)

$$\begin{array}{cc}\hfill \mathcal{K}\left(t\right)& :=A_{PQ}{\mathrm{e}}^{t{A}_Q}{A}_{QP}.\hfill \end{array}$$

(657)

Under reciprocal coupling $A_{PQ}=A_{QP}^{†}$, writing $\mathsf{U}:=A_{QP}$ gives $\mathcal{K}\left(t\right)={\mathsf{U}}^{†}{\mathrm{e}}^{t{A}_Q}\mathsf{U}$ and $\mathcal{K}\left(0\right)={\mathsf{U}}^{†}\mathsf{U}$. Positivity or Hermiticity at $t>0$ nevertheless requires corresponding properties of the unresolved propagator; the zero-lag Gram response alone does not supply them.

More generally, direction-dependent couplings give the cross-response

$${\mathfrak{G}}_P(t;X,Y):={\mathsf{U}}_X^{†}{\mathrm{e}}^{t{A}_Q}{\mathsf{U}}_Y={\displaystyle \sum _{r=0}^{\infty }}\frac{t^r}{r!}{\mathsf{U}}_X^{†}{A}_Q^r{\mathsf{U}}_Y.$$

(658)

For the static couplings ${\mathsf{U}}_X=Q{L}_X{P|}_V$ with $L_X=L_X^{†}$ from Proposition 20, twice its zero-lag antisymmetric part is the static defect in Equation (449). For $Rez$ beyond the semigroup growth bound,

$${\widehat{\mathfrak{G}}}_P(z;X,Y)={\mathsf{U}}_X^{†}{(zI-A_Q)}^{-1}{\mathsf{U}}_Y.$$

(659)

There is no universal identification of the moments ${\mathsf{U}}_X^{†}{A}_Q^r{\mathsf{U}}_Y$ with covariant derivatives of static curvature; such an identification requires an additional evolution law relating $A_Q\mathsf{U}$ to the state derivative of $\mathsf{U}$.

Proof from the Mori-Zwanzig identity. 

Variation of constants for the unresolved block is the standard finite-dimensional Mori-Zwanzig/Duhamel identity and gives (656); exponential expansion and Laplace transformation give the response moments and resolvent [100,101,102,103]. The zero-lag and finite-time qualifications then follow from the displayed block factors, not from an additional dynamical assumption. □

Proposition 30

(Zero-frequency reversible-dissipative response). Suppose the unresolved generator has the strictly dissipative decomposition

$$A_Q=J_Q-{\mathsf{\Gamma }}_Q,J_Q^{†}=-J_Q,{\mathsf{\Gamma }}_Q^{†}={\mathsf{\Gamma }}_Q\succ 0,$$

(660)

so that $-A_Q$ is strictly accretive, and define

$$B_Q:={(-A_Q)}^{-1}={({\mathsf{\Gamma }}_Q-J_Q)}^{-1}.$$

(661)

Then

$$\begin{array}{|c|}\hline Sym{B}_Q=B_Q^{†}{\mathsf{\Gamma }}_Q{B}_Q⪰0,Alt{B}_Q=B_Q^{†}{J}_Q{B}_Q.\\ \hline\end{array}$$

(662)

For a force coupling $\mathcal{C}:\mathcal{F}\to imQ$, the effective zero-frequency response

$${\mathsf{\Sigma }}_{\mathrm{eff}}:={\mathcal{C}}^{†}{B}_Q\mathcal{C}$$

(663)

therefore has

$$\begin{array}{|c|}\hline Sym{\mathsf{\Sigma }}_{\mathrm{eff}}={\left({\mathsf{\Gamma }}_Q^{1/2}{B}_Q\mathcal{C}\right)}^{†}\left({\mathsf{\Gamma }}_Q^{1/2}{B}_Q\mathcal{C}\right)⪰0,Alt{\mathsf{\Sigma }}_{\mathrm{eff}}={\mathcal{C}}^{†}{B}_Q^{†}{J}_Q{B}_Q\mathcal{C}.\\ \hline\end{array}$$

(664)

This is a positive/reactive response split, not by itself a GENERIC structure.

Proof. 

Since $B_Q^{-1}={\mathsf{\Gamma }}_Q-J_Q$ and ${\left(B_Q^{†}\right)}^{-1}={\mathsf{\Gamma }}_Q+J_Q$, multiplication on the left by $B_Q^{†}$ and on the right by $B_Q$ gives respectively the symmetric and antisymmetric identities. Composition with $\mathcal{C}$ preserves adjoints and positivity. □

For comparison, a genuine GENERIC evolution has

$$\dot{z}=\mathbb{J}\left(z\right)\mathrm{d}E\left(z\right)+\mathbb{K}\left(z\right)\mathrm{d}S\left(z\right),{\mathbb{J}}^{†}=-\mathbb{J},{\mathbb{K}}^{†}=\mathbb{K}⪰0,$$

(665)

with the Jacobi identity and degeneracy conditions

$${[\mathbb{J},\mathbb{J}]}_{\mathrm{Sch}}=0,\mathbb{J}\mathrm{d}S=0,\mathbb{K}\mathrm{d}E=0$$

(666)

[104,105,106]. Equations (662)- (664) have the required algebraic symmetries, but Jacobi, both degeneracies, smooth state dependence, physical units, a time scale, conservation laws, and quantum CPTP admissibility are additional requirements. In particular, the endomorphism-valued two-form ${\mathsf{\Omega }}_P(X,Y)$ and the force-to-velocity maps in GENERIC have different tensor types.

There are two useful exact placement results. First, the nondegenerate Born/cotangent Poisson tensor cannot satisfy ${\mathbb{J}}_{\mathrm{can}}\mathrm{d}S=0$ for a nonconstant entropy, because a symplectic Poisson tensor has no nonconstant Casimirs. On density matrices the natural degenerate alternative is the Lie-Poisson tensor

$${\mathbb{J}}_{\rho }\left(A\right):=-\frac{\mathrm{i}}{\hslash }[A,\rho ].$$

(667)

It satisfies Jacobi and ${\mathbb{J}}_{\rho }\mathrm{d}{S}_{\mathrm{vN}}=0$, since $\mathrm{d}{S}_{\mathrm{vN}}=-log\rho -I$ commutes with $\rho$.

Second, writing $G:=g_{\mathrm{BKM}}^{-1}:T^{\ast }M\to TM$ and $u_E:=G\mathrm{d}E$, the energy-projected BKM cometric

$$\begin{array}{|c|}\hline {\mathbb{K}}_E\left(\alpha \right):=G\alpha -\frac{\alpha \left(u_E\right)}{\mathrm{d}E\left(u_E\right)}{u}_E\\ \hline\end{array}$$

(668)

is well defined on the open set $\mathrm{d}E\ne 0$ and obeys

$${\mathbb{K}}_E^{†}={\mathbb{K}}_E,{\mathbb{K}}_E⪰0,{\mathbb{K}}_E\mathrm{d}E=0.$$

(669)

The positivity is Cauchy-Schwarz for the cometric. This is a natural constitutive candidate, not a mobility derived from static information geometry, and it can fail to extend smoothly through critical points of E.

7.3. Reversible Non-Abelian Transport and Magnus Parity

Continuous composition uses a time-ordered exponential and the Magnus series [107]; a modern review is [78], Sections 2-3. A horizontal lift $\tilde{\gamma }\left(t\right)\in \mathrm{SL}(n,\mathbb{C})$ identifies velocities with the fixed Lie algebra:

$${\tilde{\gamma }}^{-1}\dot{\tilde{\gamma }}=X\left(t\right)\in \mathfrak{p}.$$

(670)

On an interval of convergence, $\tilde{\gamma }\left(T\right)=\tilde{\gamma }\left(0\right)exp{\sum }_{r\ge 1}{\mathsf{\Omega }}_r\left(T\right)$, with leading compact term

$${\mathsf{\Omega }}_2\left(T\right)=-\frac{1}{2}{\int }_0^T\mathrm{d}{t}_1{\int }_0^{t_1}\mathrm{d}{t}_2[X\left(t_1\right),X\left(t_2\right)].$$

(671)

Proposition 31

(Symmetric-space parity of the Magnus expansion). If $X\left(t\right)\in \mathfrak{p}$ for all t, then every homogeneous Magnus term obeys

$${\mathsf{\Omega }}_{2m+1}\left(T\right)\in \mathfrak{p},{\mathsf{\Omega }}_{2m}\left(T\right)\in \mathfrak{k}.$$

(672)

Thus odd orders generate noncompact displacement and even orders compact transport.

For any path, ${\mathsf{\Omega }}_2$ is the quadratic compact contribution to the Magnus logarithm and vanishes for commuting lifted velocities. It is the leading logarithm of canonical holonomy for a perturbatively small loop family whose first-order noncompact displacement closes; for a finite loop, no truncation is implied and higher Magnus terms contribute. The proof, soldered-gradient formula, and $n=2$ Thomas-Wigner specialization are in Appendix F.3. This lift is reversible. Within the finite-dimensional construction, irreversible effective evolution requires additional dynamical or coarse-graining input; discarding a correlated record is one example, whereas gauge projection is not.

7.4. Gauge-Covariant Random Transport and Information Balance

The principal $\mathrm{SU}\left(n\right)$ fiber is gauge redundancy, not a reservoir. A reduced channel requires an additional physical coarse-graining or averaging mechanism applied to represented transport. The finite model below makes the stronger, separately supplied assumption of an orthogonally distinguishable flag register.

For framed endpoints and a path $\gamma$, a unitary representation R gives

$$U_{\gamma }:=R\left[\mathcal{P}exp\left(-{\int }_{\gamma }A\right)\right]:{\mathcal{H}}_{\mathrm{in}}⟶{\mathcal{H}}_{\mathrm{out}}.$$

(673)

For a based loop this is holonomy. Endpoint frame changes act by

$$U_{\gamma }⟼R{\left(k_{\mathrm{out}}\right)}^{-1}{U}_{\gamma }R\left(k_{\mathrm{in}}\right),$$

(674)

which reduces to conjugation for based loops. The finite flagged ensemble below is the only path-register structure used in the balance.

The next result combines Donald’s finite-ensemble identity-the finite-dimensional cq relative-entropy chain rule-with an explicit reference-change term for a chosen cq extension [14,108]; it is not a representation theorem for arbitrary channels.

Proposition 32

(Reference-corrected cq balance for a finite flagged extension). Let ${\mathcal{H}}_{\mathrm{in}}$ and ${\mathcal{H}}_{\mathrm{out}}$ be finite-dimensional Hilbert spaces of the same dimension. Let $\mathcal{R}$ be finite and nonempty, let $p_r>0$ with ${\sum }_{r\in \mathcal{R}}{p}_r=1$ be fixed independently of the input, let $\left\{\right|r\rangle :r\in \mathcal{R}\}$ be orthonormal in a supplied flag-register Hilbert space ${\mathcal{H}}_{\mathsf{\Gamma }}$, and let $U_r:{\mathcal{H}}_{\mathrm{in}}\to {\mathcal{H}}_{\mathrm{out}}$ be unitary isomorphisms. For any density operator ρ on ${\mathcal{H}}_{\mathrm{in}}$, define

$$\begin{array}{cc}\hfill {\rho }_r& :=U_r\rho U_r^{†},{\widehat{\mathcal{E}}}_{\mathrm{flag}}\left(\rho \right):={\mathsf{\Omega }}_{\mathsf{\Gamma }S}:={\displaystyle \sum _{r\in \mathcal{R}}}{p}_r|r\rangle \langle r|\otimes {\rho }_r,\hfill \\ \hfill \mathcal{E}\left(\rho \right)& :={Tr}_{\mathsf{\Gamma }}{\mathsf{\Omega }}_{\mathsf{\Gamma }S}={\displaystyle \sum _{r\in \mathcal{R}}}{p}_r{\rho }_r.\hfill \end{array}$$

(675)

Both ${\widehat{\mathcal{E}}}_{\mathrm{flag}}$ and $\mathcal{E}$ are CPTP. Their marginals are ${\mathsf{\Omega }}_{\mathsf{\Gamma }}={\sum }_r{p}_r|r\rangle \langle r|$ and ${\mathsf{\Omega }}_S=\mathcal{E}\left(\rho \right)$. For arbitrary faithful density operators ${\tau }_{\mathrm{in}}$ on ${\mathcal{H}}_{\mathrm{in}}$ and ${\tau }_{\mathrm{out}}$ on ${\mathcal{H}}_{\mathrm{out}}$, set

$${\mathfrak{M}}_{{\tau }_{\mathrm{in}},{\tau }_{\mathrm{out}}}:=\sum _{r\in \mathcal{R}}{p}_r{U}_r^{†}(log{\tau }_{\mathrm{out}})U_r-log{\tau }_{\mathrm{in}}.$$

(676)

Then

$$\begin{array}{|c|}\hline D_{\mathrm{U}}(\rho \parallel {\tau }_{\mathrm{in}})-D_{\mathrm{U}}(\mathcal{E}\left(\rho \right)\parallel {\tau }_{\mathrm{out}})=I{(\mathsf{\Gamma }:S)}_{\mathsf{\Omega }}+Tr\left(\rho {\mathfrak{M}}_{{\tau }_{\mathrm{in}},{\tau }_{\mathrm{out}}}\right),\\ \hline\end{array}$$

(677)

where

$$I{(\mathsf{\Gamma }:S)}_{\mathsf{\Omega }}:=D_{\mathrm{U}}({\mathsf{\Omega }}_{\mathsf{\Gamma }S}\parallel {\mathsf{\Omega }}_{\mathsf{\Gamma }}\otimes {\mathsf{\Omega }}_S)=\sum _{r\in \mathcal{R}}{p}_r{D}_{\mathrm{U}}({\rho }_r\parallel \mathcal{E}\left(\rho \right)).$$

(678)

For this cq state, $I{(\mathsf{\Gamma }:S)}_{\mathsf{\Omega }}$ is the Holevo quantity $\chi \left(\{p_r,{\rho }_r\}\right)$ [108]. For a specified input ρ, the uncorrected balance holds if and only if $Tr\left(\rho {\mathfrak{M}}_{{\tau }_{\mathrm{in}},{\tau }_{\mathrm{out}}}\right)=0$. The uncorrected balance holds for every input state-and its validity at $\rho ={\tau }_{\mathrm{in}}$ alone already forces this input-uniform condition-if and only if

$$U_r{\tau }_{\mathrm{in}}{U}_r^{†}={\tau }_{\mathrm{out}}\mathit{for}\mathit{every}r\in \mathcal{R}.$$

(679)

Under these equivalent input-uniform conditions, $\mathcal{E}\left({\tau }_{\mathrm{in}}\right)={\tau }_{\mathrm{out}}$ and

$$D_{\mathrm{U}}(\rho \parallel {\tau }_{\mathrm{in}})=D_{\mathrm{U}}({\mathsf{\Omega }}_{\mathsf{\Gamma }S}\parallel {\mathsf{\Omega }}_{\mathsf{\Gamma }}\otimes {\tau }_{\mathrm{out}})=D_{\mathrm{U}}(\mathcal{E}\left(\rho \right)\parallel {\tau }_{\mathrm{out}})+I{(\mathsf{\Gamma }:S)}_{\mathsf{\Omega }}.$$

(680)

The Kraus-Stinespring theorem shows that the map in Equation (675) is CPTP and is the dephased cq output of a coherent dilation [71]. The orthogonal flag and its classical observable are therefore additional choices not selected by $\mathcal{E}$. For arbitrary references, neither ${\mathfrak{M}}_{{\tau }_{\mathrm{in}},{\tau }_{\mathrm{out}}}$ nor its expectation has a fixed sign; even ${\tau }_{\mathrm{out}}=\mathcal{E}\left({\tau }_{\mathrm{in}}\right)$ alone does not remove the correction. The proof, including the sharp equality criterion, is in Appendix F.4.

For based loops with ${\tau }_{\mathrm{in}}={\tau }_{\mathrm{out}}=\tau$, put $G_{\mathrm{ens}}:=\overline{\langle U_r:r\in \mathcal{R}\rangle }$. Then Equation (679) is exactly $\tau \in G_{\mathrm{ens}}^{\prime }$. If the representation of this selected branch group is irreducible, Schur’s lemma yields $\tau =I/d$. Hence the maximally mixed conclusion follows when $G_{\mathrm{ens}}$ is irreducible-in particular, when it is dense in the defining $\mathrm{SU}\left(n\right)$ holonomy group-or when reference invariance is imposed for every canonical holonomy. Full holonomy of the ambient connection alone does not imply that a chosen finite ensemble generates such a subgroup. A nontrivial Gibbs reference therefore requires a restricted commuting subgroup, a reducible carrier, distinct endpoint references, or a branch ensemble that does not sample the full holonomy.

For equal d-dimensional endpoint fibers with ${\tau }_{\mathrm{in}}={\tau }_{\mathrm{out}}=I/d$,

$$I{(\mathsf{\Gamma }:S)}_{\mathsf{\Omega }}=S_{\mathrm{vN}}\left(\mathcal{E}\left(\rho \right)\right)-S_{\mathrm{vN}}\left(\rho \right)={\mathsf{\Delta }}_{\mathcal{E}}\left(\rho ,\frac{I}{d}\right).$$

(681)

If ${\mathcal{H}}_{\mathsf{\Gamma }}$ is supplied as an actual ancillary record and is discarded, ${\mathsf{\Omega }}_{\mathsf{\Gamma }S}\mapsto {Tr}_{\mathsf{\Gamma }}{\mathsf{\Omega }}_{\mathsf{\Gamma }S}$ is a physical partial trace; otherwise it is only marginalization of the chosen cq extension. In neither case is it gauge projection. Equation (680) is an equality refinement of DPI under the additional extension and branchwise-reference hypotheses, not a channel-intrinsic decomposition. Different paths may induce the same transport, so compact residue need not encode the full history.

The semigroup, logarithmic-Sobolev, and isotropic holonomy-diffusion calculations are deferred to Appendix F.

8. Action-Scaled Endpoint Functionals, Discrete Generators, and Quantization Obstructions

Remark 19

(Fixed endpoint scale versus mechanical step calibration). The paper uses two inequivalent coefficients with action units. The fixed ${\kappa }_A$ is independent of endpoints and of any time step. It converts $B_{\psi }$ into the exact endpoint primitive ${\kappa }_A{B}_{\psi }$ and scales the corresponding Type-I/II canonical relation. Here “exact” refers to the primitive and generated relation, not to the exact discrete Lagrangian of a continuous mechanical system.

For the latter comparison one must instead supply an inertial coefficient ${\mu }_{\ast }$ and a clock step $h>0$, and use ${\kappa }_h={\mu }_{\ast }/h$, where $\left[{\mu }_{\ast }\right]=\mathrm{action}·\mathrm{time}$. Indeed, if $\eta \left(q_1\right)-\eta \left(q_0\right)=hv$, then

$$B_{\psi }(q_1,q_0)=\frac{h^2}{2}{g}_{q_0}(v,v)+O\left(h^3\right),{\kappa }_A{B}_{\psi }=O\left(h^2\right),{\kappa }_h{B}_{\psi }=\frac{{\mu }_{\ast }h}{2}{g}_{q_0}(v,v)+O\left(h^2\right).$$

(682)

Only the last expression has the $O\left(h\right)$ scaling of kinetic principal action; likewise its endpoint momentum tends to ${\mu }_{\ast }{g}_{q_0}(v,·)$, whereas the fixed-scale momentum is $O\left(h\right)$. Thus all mechanical consistency and error claims below use ${\kappa }_h$, not ${\kappa }_A$. Equality ${\kappa }_A={\mu }_{\ast }/h_0$ at one selected step merely matches two coefficients at that resolution and has no continuum or quantization significance. Neither calibration determines ℏ. For a non-dimensionless $B_{\psi }$, every statement is unchanged after assigning $\left[\kappa \right]=\mathrm{action}/\left[B_{\psi }\right]$.

8.1. The Exact Action-Scaled Cotangent Lift

Let $(M,\eta ,\psi )$ be Hessian on a convex affine domain and fix $p\in M$. Removing the affine gauge freedom $\psi \mapsto \psi +a_b{\eta }^b+b$ gives the invariant centered potential and one-form

$$\begin{array}{cc}\hfill {\psi }^{\left[p\right]}\left(\eta \right)& :=\psi \left(\eta \right)-\psi \left(p\right)-{\partial }_a\psi \left(p\right)\left({\eta }^a-{\eta }^a\left(p\right)\right),\hfill \end{array}$$

(683)

$$\begin{array}{cc}\hfill {\alpha }_p& :=\mathrm{d}{\psi }^{\left[p\right]}=\left({\partial }_a\psi \left(\eta \right)-{\partial }_a\psi \left(p\right)\right)\mathrm{d}{\eta }^a.\hfill \end{array}$$

(684)

Thus ${\psi }^{\left[p\right]}\left(q\right)=B_{\psi }(q,p)$ and $Hess{\psi }^{\left[p\right]}=g$. Supply ${\kappa }_A>0$ with action units and, on $T^{\ast }M$, write

$${\lambda }_A:={\mathsf{A}}_a\mathrm{d}{\eta }^a,{\omega }_A:=\mathrm{d}{\lambda }_A=\mathrm{d}{\mathsf{A}}_a\wedge \mathrm{d}{\eta }^a.$$

(685)

Throughout the action-scaled cotangent and discrete-mechanical constructions, we use the Hamiltonian convention

$${\iota }_{X_H}{\omega }_A=-\mathrm{d}H.$$

(686)

All phase-flow and momentum-map signs below are understood with this convention. The scaled section

$${\iota }_{{\kappa }_A,p}\left(\eta \right):=\left(\eta ,{\mathsf{A}}_a={\kappa }_A[{\partial }_a\psi \left(\eta \right)-{\partial }_a\psi \left(p\right)]\right),$$

(687)

has pullbacks

$${\iota }_{{\kappa }_A,p}^{\ast }{\lambda }_A={\kappa }_A{\alpha }_p={\kappa }_A\mathrm{d}{\psi }^{\left[p\right]},{\iota }_{{\kappa }_A,p}^{\ast }{\omega }_A=0.$$

(688)

and is an exact Lagrangian embedding [109], Chapter 9, with the fixed endpoint calibration of Remark 19.

Corollary 45

(Reference-centered action-scaled Bregman endpoint). For every piecewise smooth path c from p to q,

$$\begin{array}{|c|}\hline {\mathcal{A}}_{\mathrm{rel}}^{\left({\kappa }_A\right)}[c;p]:={\int }_{{\iota }_{{\kappa }_A,p}\circ c}{\lambda }_A={\kappa }_A{B}_{\psi }(q,p).\\ \hline\end{array}$$

(689)

The functional is path independent, directed, affine-gauge invariant, and nonnegative; for positive-definite g it vanishes exactly when $p=q$.

This follows by integrating Equation (688); strict convexity gives the equality condition.

For faithful states in mixture coordinates, $\mathcal{F}\left(\rho \right)=Tr(\rho log\rho )$ and trace-zero tangents give

$$\mathrm{d}{\mathcal{F}}_{\rho }-\mathrm{d}{\mathcal{F}}_{\tau }\equiv log\rho -log\tau ,{\mathcal{A}}_{\tau }^{\left({\kappa }_A\right)}\left(\rho \right)={\kappa }_A{D}_{\mathrm{U}}(\rho \parallel \tau ).$$

(690)

Equivalently, $D_{\mathrm{U}}({\rho }_X\parallel {\rho }_Y)=B_{\mathsf{\Psi }}(Y,X)$ in Gibbs coordinates. Thus Umegaki divergence, not an affine-gauge-dependent potential difference, is the exact centered primitive. On ${\mathcal{Q}}_{\mathsf{\Lambda }}$ its pullback is represented by ${\mathsf{\Lambda }}^{\ast }(log\mathsf{\Lambda }\left(\rho \right)-log\mathsf{\Lambda }\left(\tau \right))$ and annihilates $kerL$.

Corollary 46

(Full, accessible, and conditionally recorded action-scaled endpoints). For a CPTP channel Λ and faithful reference τ, the centered construction on the operational quotient gives

$${\mathcal{A}}_{\mathsf{\Lambda },\tau }^{\left({\kappa }_A\right)}\left(\left[\rho \right]\right):={\kappa }_A{D}_{\mathrm{U}}(\mathsf{\Lambda }\left(\rho \right)\parallel \mathsf{\Lambda }\left(\tau \right)),{\mathcal{A}}_{\tau }^{\left({\kappa }_A\right)}\left(\rho \right)-{\mathcal{A}}_{\mathsf{\Lambda },\tau }^{\left({\kappa }_A\right)}\left(\left[\rho \right]\right)={\kappa }_A{\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\tau ).$$

(691)

If $\mathsf{\Lambda }=\mathcal{E}$ is equipped with the finite cq flagged extension of Proposition 32, put ${\mathfrak{M}}_{\mathcal{E},\tau }:={\mathfrak{M}}_{\tau ,\mathcal{E}\left(\tau \right)}$. Then

$${\mathcal{A}}_{\tau }^{\left({\kappa }_A\right)}\left(\rho \right)-{\mathcal{A}}_{\mathcal{E},\tau }^{\left({\kappa }_A\right)}\left(\left[\rho \right]\right)={\kappa }_A\left[I{(\mathsf{\Gamma }:S)}_{\mathsf{\Omega }}+Tr\left(\rho {\mathfrak{M}}_{\mathcal{E},\tau }\right)\right].$$

(692)

This equals ${\kappa }_{A}I{(\mathsf{\Gamma }:S)}_{\mathsf{\Omega }}$ for every input if and only if $U_r\tau U_r^{†}=\mathcal{E}\left(\tau \right)$ for every active branch. Thus the correlation interpretation is conditional on the supplied extension and branchwise reference condition; it is not information provided by the reduced map.

The bridge in Corollary 12 also separates the two action calibrations at the local level. For a tangent V, a velocity v, and sufficiently small $ϵ$ and $h>0$ such that the displayed states remain faithful,

$$\begin{array}{cc}\hfill {\mathcal{A}}_{\tau }^{\left({\kappa }_A\right)}(\tau +ϵV)-{\mathcal{A}}_{\mathsf{\Lambda },\tau }^{\left({\kappa }_A\right)}\left([\tau +ϵV]\right)& =\frac{{\kappa }_A{ϵ}^2}{2}{g}_{\mathsf{\Lambda },\tau }^{\mathrm{loss}}(V,V)+{\kappa }_{A}O\left({ϵ}^3\right),\hfill \end{array}$$

(693)

$$\begin{array}{cc}\hfill \frac{{\mu }_{\ast }}{h}{\mathsf{\Delta }}_{\mathsf{\Lambda }}(\tau +hv,\tau )& =\frac{{\mu }_{\ast }h}{2}{g}_{\mathsf{\Lambda },\tau }^{\mathrm{loss}}(v,v)+{\mu }_{\ast }O\left(h^2\right).\hfill \end{array}$$

(694)

The second line concerns only the divergence contribution to a mechanically calibrated generator; comparison of full mechanical actions also requires compatible input and output potential data.

8.2. Variational Endpoint Families, Exact Canonical Reconstruction, and the Mechanical Limit

The fixed-reference graph is Lagrangian and does not itself carry Hamiltonian dynamics. Its independently chosen relaxation vector field does have the standard cotangent lift of Remark 17, but that construction is linear in momentum and is not a positive mechanical energy. Promoting the reference to the preceding endpoint instead makes a Bregman divergence on $M\times M$ a Type-I discrete generating function [36], Section 1.3; see also [37], Sections 5.1-5.2. The scalar value is not momentum, but its two endpoint derivatives generate incoming and outgoing momenta.

Lemma 4

(A complete endpoint integral generates its Hamiltonian characteristics). Let $I\subset (0,\infty )$ be an interval and let $S\in C^3(I\times U_0\times U_1)$ have nondegenerate mixed endpoint Hessian $D_0{D}_1{S}_h$. After shrinking the interval and coordinate neighborhoods if necessary, assume there are open sets $\mathcal{E}\subset I\times U_0\times U_1$ and $\mathcal{D}\subset T^{\ast }{U}_1$ on which the right-Legendre map is a $C^2$ diffeomorphism onto the common product chart:

$$\begin{array}{cccc}\hfill \mathcal{R}:\mathcal{E}& \stackrel{\simeq }{\to }I\times \mathcal{D},\hfill & \hfill (h,q_0,q)& ⟼\left(h,q,D_1{S}_h(q_0,q)\right),\hfill \\ \hfill {\mathcal{R}}^{-1}(h,q,p)& =\left(h,Q(h,q,p),q\right),\hfill & \hfill Q& \in C^2(I\times \mathcal{D},U_0).\hfill \end{array}$$

For fixed h, define the endpoint Legendre maps

$$\begin{array}{cc}\hfill {\mathbb{F}}_h^{-}S(q_0,q)& :=\left(q_0,-D_0{S}_h(q_0,q)\right),\hfill \\ \hfill {\mathbb{F}}_h^{+}S(q_0,q)& :=\left(q,D_1{S}_h(q_0,q)\right).\hfill \end{array}$$

The mixed-Hessian hypothesis makes both maps local diffeomorphisms. On each selected component on which ${\mathbb{F}}_h^{-}S$ is invertible, let

$${\mathsf{\Phi }}_h:={\mathbb{F}}_h^{+}S\circ {\left({\mathbb{F}}_h^{-}S\right)}^{-1}$$

be the generated local symplectic map. With $Q_h(q,p):=Q(h,q,p)$, define

$${\mathcal{H}}_h(q,p):=-{\partial }_h{S}_h(Q_h(q,p),q),(q,p)\in \mathcal{D}.$$

(695)

Define the extended-graph parametrization

$$\begin{array}{cc}\hfill {\widehat{j}}_S:\mathcal{E}& ⟶\overline{T^{\ast }{U}_0}\times T^{\ast }{U}_1\times I,\hfill \\ \hfill {\widehat{j}}_S(h,q_0,q)& :=\left(q_0,-D_0{S}_h(q_0,q);q,D_1{S}_h(q_0,q);h\right).\hfill \end{array}$$

On $\mathcal{E}$, the common Hamilton-Jacobi and extended-exactness identities are

$$\begin{array}{cc}\hfill {\partial }_h{S}_h(q_0,q)+{\mathcal{H}}_h\left(q,D_1{S}_h(q_0,q)\right)& =0,\hfill \end{array}$$

(696)

$$\begin{array}{cc}\hfill {\widehat{j}}_S^{\ast }\left(-{\lambda }_{A,0}+{\lambda }_{A,1}-{\mathcal{H}}_h\mathrm{d}h\right)& =\mathrm{d}S.\hfill \end{array}$$

(697)

For fixed incoming data $(q_0,p_0)$, on every subinterval $J\subset I$ for which the selected generated outgoing branch $(q_h,p_h)$ exists and remains in $\mathcal{D}$, the isotopy and its characteristics obey

$$\begin{array}{cc}\hfill {\partial }_h{\mathsf{\Phi }}_h& =X_{{\mathcal{H}}_h}\circ {\mathsf{\Phi }}_h,\hfill \end{array}$$

(698)

$$\begin{array}{cc}\hfill {\dot{q}}_h& =D_p{\mathcal{H}}_h(q_h,p_h),{\dot{p}}_h=-D_q{\mathcal{H}}_h(q_h,p_h),\hfill \end{array}$$

(699)

and

$$S_{h_1}(q_0,q_{h_1})-S_{h_0}(q_0,q_{h_0})={\int }_{h_0}^{h_1}\left[p_h\left({\dot{q}}_h\right)-{\mathcal{H}}_h(q_h,p_h)\right]\mathrm{d}h.$$

(700)

If $U_0$ and $U_1$ are identified and the branch extends to $h=0$ with ${\mathsf{\Phi }}_0=\mathrm{id}$ and $S_h(q_0,q_h)\to 0$, the lower limit may be set to zero. In general, the Hamiltonian is explicitly h-dependent and the endpoint maps need not form an autonomous semigroup. Writing ${\mathcal{U}}_h:=Dom{\mathsf{\Phi }}_h$, the two-time propagator is

$${\mathsf{\Phi }}_{h_1,h_0}:={\mathsf{\Phi }}_{h_1}\circ {\mathsf{\Phi }}_{h_0}^{-1}$$

(701)

only on ${\mathsf{\Phi }}_{h_0}({\mathcal{U}}_{h_0}\cap {\mathcal{U}}_{h_1})$, restricted to the component on which both maps use the same selected right-Legendre branch. On each connected common right-Legendre domain, any other Hamiltonian generating the same isotopy is ${\mathcal{H}}_h+c\left(h\right)$. Replacing $S_h$ by $S_h+a\left(h\right)$ leaves its endpoint relations and canonical maps unchanged and replaces ${\mathcal{H}}_h$ by ${\mathcal{H}}_h-a^{\prime }\left(h\right)$. If $h=f\left(\tau \right)$ is a monotone $C^1$ reparameterization and ${\tilde{S}}_{\tau }=S_{f\left(\tau \right)}$, then the reparameterized generator is

$${\tilde{\mathcal{H}}}_{\tau }=f^{\prime }\left(\tau \right){\mathcal{H}}_{f\left(\tau \right)}.$$

(702)

The proof is deferred to Appendix G.1. The applications below specify their inverse endpoints, natural domains, and generators and invoke these common identities. Fiberwise hyperregularity and an instantaneous Lagrangian require separate verification; mixed-endpoint nondegeneracy alone does not supply them.

• Status of the Hamiltonian reconstruction. By Lemma 4, every specified $C^3$ endpoint family below satisfying the nondegenerate mixed-Hessian and selected common right-Legendre-chart hypotheses generates an exact branch-local, generally nonautonomous Hamiltonian isotopy on that chart. The isotopy determines its generator only up to an h-dependent scalar; the displayed endpoint primitive fixes that gauge, while clock reparameterization rescales the generator. For the free Bregman family the chosen normalization gives ${\mathcal{H}}_h(q,0)=0$. The calibrations and independent physical inputs are those of Remark 19 and Table 1. Exactness concerns classical cotangent dynamics and its on-shell endpoint action. It does not by itself select a physical clock, a microscopic quantum Hamiltonian, or von Neumann evolution, and it does not imply autonomy.

The weighted dual-coordinate minimizer and conjugate Jensen-gap identity used next are the standard sided Bregman-centroid formulas of [23]. The proof is retained to fix orientation. What is developed here is the normalized mixed-Hessian domain, endpoint canonical generator, and its Umegaki/BKM scale-response specialization, using the established envelope core of Theorem 6.

Theorem 22

(Normalized dual-Jensen Bregman-Moreau generator). Let M be a finite-dimensional open convex affine domain and let $\psi \in C^{\infty }\left(M\right)$ be strictly convex, with positive-definite Hessian, such that $\mathrm{d}\psi :M\to M^{\ast }:=\mathrm{d}\psi \left(M\right)$ is a diffeomorphism onto an open convex dual domain. Write $g=Hess\psi$, let ${\psi }^{\ast }$ be the dual potential, identify tangent spaces by the affine trivialization, and put ${\vartheta }_q:=\mathrm{d}{\psi }_q$. For $q_0,q_1\in M$ and $0<t<1$, let

$${\vartheta }_{z_t}:=(1-t){\vartheta }_{q_0}+t{\vartheta }_{q_1}.$$

(703)

Then the endpoint-balanced cost

$$\begin{array}{cc}\hfill {\mathfrak{B}}_t^{\psi }(q_1,q_0)& :=\underset{z\in M}{min}\left\{\frac{1}{t}{B}_{\psi }(z,q_0)+\frac{1}{1-t}{B}_{\psi }(z,q_1)\right\}\hfill \end{array}$$

(704)

$$\begin{array}{cc}& =\frac{(1-t){\psi }^{\ast }\left({\vartheta }_{q_0}\right)+t{\psi }^{\ast }\left({\vartheta }_{q_1}\right)-{\psi }^{\ast }((1-t){\vartheta }_{q_0}+t{\vartheta }_{q_1})}{t(1-t)}\hfill \end{array}$$

(705)

has the unique minimizer $z_t$. Its endpoint metric gradients and exact two-sided deformation rule are

$$\begin{array}{cccc}\hfill {grad}_{q_0}{\mathfrak{B}}_t^{\psi }& =\frac{q_0-z_t}{t},\hfill & \hfill {grad}_{q_1}{\mathfrak{B}}_t^{\psi }& =\frac{q_1-z_t}{1-t},\hfill \\ \hfill z_t& =q_0-t{grad}_{q_0}{\mathfrak{B}}_t^{\psi }=q_1-(1-t){grad}_{q_1}{\mathfrak{B}}_t^{\psi }.\hfill \end{array}$$

(706)

If $g_q^{♭}$ denotes the metric musical map, then

$$\begin{array}{cc}\hfill D_0{z}_t& =(1-t){\left(g_{z_t}^{♭}\right)}^{-1}{g}_{q_0}^{♭},\hfill \\ \hfill D_0{D}_1{\mathfrak{B}}_t^{\psi }[W,V]& =-g_{q_1}\left({\left(g_{z_t}^{♭}\right)}^{-1}{g}_{q_0}^{♭}W,V\right).\hfill \end{array}$$

(707)

The mixed Hessian is therefore nondegenerate everywhere. For every fixed action conversion ${\kappa }_A>0$, ${\kappa }_A{\mathfrak{B}}_t^{\psi }$ is a regular Type-I generator and defines an exact Lagrangian canonical relation with local symplectic Legendre maps. Moreover,

$${\mathfrak{B}}_t^{\psi }(q_1,q_0)={\mathfrak{B}}_{1-t}^{\psi }(q_0,q_1),{\mathfrak{B}}_0^{\psi }=B_{\psi }(q_0,q_1),{\mathfrak{B}}_1^{\psi }=B_{\psi }(q_1,q_0),$$

(708)

where the endpoints are continuous limits.

For $\psi \left(\rho \right)=Tr(\rho log\rho )$ on faithful states,

$$\begin{array}{|c|}\hline {\mathfrak{B}}_t^{\mathrm{RM}}(\rho ,\sigma ):={\mathfrak{B}}_t^{\psi }(\rho ,\sigma )=\frac{1}{t}{D}_t^{\mathrm{b}}(\rho \parallel \sigma ),z_t={\gamma }_t.\\ \hline\end{array}$$

(709)

Hence the exact Rényi-Moreau identity yields a regular endpoint generator, not merely a scalar value. With $r=(1-t)/t$, ${\mathcal{M}}_r^{\sigma }=t{\mathfrak{B}}_t^{\mathrm{RM}}$; consequently, after supplying ${\kappa }_A$,

$$S_{r,{\kappa }_A}^{\sigma }:={\kappa }_A{\mathcal{M}}_r^{\sigma },{\mathcal{H}}_{r,{\kappa }_A}^{\mathrm{U}}(\rho ,\alpha ):={\kappa }_A{\mathcal{H}}_r^{\mathrm{U}}(\rho ,\alpha /{\kappa }_A)$$

(710)

give a complete local integral ${\partial }_r{S}_{r,{\kappa }_A}^{\sigma }+{\mathcal{H}}_{r,{\kappa }_A}^{\mathrm{U}}(\rho ,D_{\rho }{S}_{r,{\kappa }_A}^{\sigma })=0$. Completeness here means nondegeneracy in the endpoint parameter σ; it does not identify regularization scale with physical time.

Proof from sided Bregman-centroid theory. 

The weighted dual-coordinate minimizer and conjugate Jensen-gap identity are the standard sided Bregman-centroid formulas [23]. Specializing them to the weights $(1-t,t)$ gives $\mathrm{d}{\psi }_{z_t}=(1-t){\vartheta }_{q_0}+t{\vartheta }_{q_1}$, the unique minimizer, and Equation (705) with the manuscript’s endpoint orientation. The envelope theorem gives the two endpoint gradients. Differentiating Equation (703) and then the second endpoint gradient gives Equation (707); invertibility of the metric musical maps proves nondegeneracy and hence the regular Type-I canonical relation. Endpoint exchange and the two continuous limits follow from the normalized Jensen gap. Finally, Equation (147) gives the Umegaki specialization, while Equation (157) multiplied by ${\kappa }_A$ gives the displayed complete integral. □

Theorem 23

(Regular Bregman Type-I discrete generator). Under the Hessian hypotheses of 22, let $V\in C^2\left(M\right)$ be an energy-valued potential, let $\mathsf{\Delta }t>0$, and let ${\kappa }_{\mathsf{\Delta }t}>0$ depend only on the step and be such that ${\kappa }_{\mathsf{\Delta }t}{B}_{\psi }$ has action units. The subscript permits step dependence but imposes no scaling law at this algebraic stage. Define

$$L_{d,\mathsf{\Delta }t}^{\to }(q_0,q_1):={\kappa }_{\mathsf{\Delta }t}{B}_{\psi }(q_1,q_0)-\frac{\mathsf{\Delta }t}{2}\left[V\left(q_0\right)+V\left(q_1\right)\right].$$

(711)

Here $D_0$ and $D_1$ denote differentiation in the first and second endpoint, respectively. Using the global affine trivialization and $\mathsf{\Delta }\eta :=\eta \left(q_1\right)-\eta \left(q_0\right)$,

$$\begin{array}{cc}\hfill D_0{D}_1{L}_{d,\mathsf{\Delta }t}^{\to }& =-{\kappa }_{\mathsf{\Delta }t}{g}_{q_0},\hfill \end{array}$$

(712)

$$\begin{array}{cc}\hfill {\mathsf{A}}_0:=-D_0{L}_{d,\mathsf{\Delta }t}^{\to }& ={\kappa }_{\mathsf{\Delta }t}{g}_{q_0}(\mathsf{\Delta }\eta ,·)+\frac{\mathsf{\Delta }t}{2}\mathrm{d}{V}_{q_0},\hfill \end{array}$$

(713)

$$\begin{array}{cc}\hfill {\mathsf{A}}_1:=D_1{L}_{d,\mathsf{\Delta }t}^{\to }& ={\kappa }_{\mathsf{\Delta }t}(\mathrm{d}{\psi }_{q_1}-\mathrm{d}{\psi }_{q_0})-\frac{\mathsf{\Delta }t}{2}\mathrm{d}{V}_{q_1}.\hfill \end{array}$$

(714)

Hence $L_{d,\mathsf{\Delta }t}^{\to }$ is regular. Its discrete Legendre maps

$${\mathbb{F}}^{-}{L}_d=(q_0,{\mathsf{A}}_0),{\mathbb{F}}^{+}{L}_d=(q_1,{\mathsf{A}}_1)$$

(715)

are local diffeomorphisms and, for strictly convex ψ, diffeomorphisms onto their natural open images. With the manuscript convention ${\omega }_A=\mathrm{d}{\lambda }_A$, set

$${\mathsf{\Theta }}_{d,\mathsf{\Delta }t}^{+}:=D_1{L}_{d,\mathsf{\Delta }t}^{\to }·\mathrm{d}{q}_1,{\omega }_{d,\mathsf{\Delta }t}^{\to }:=\mathrm{d}{\mathsf{\Theta }}_{d,\mathsf{\Delta }t}^{+}=-{\kappa }_{\mathsf{\Delta }t}{g}_{ab}\left(q_0\right)\mathrm{d}{\eta }_0^a\wedge \mathrm{d}{\eta }_1^b.$$

(716)

This exact action-scaled two-form is symplectic on $M\times M$, and

$${\mathsf{\Phi }}_{L_d}:={\mathbb{F}}^{+}{L}_d\circ {\left({\mathbb{F}}^{-}{L}_d\right)}^{-1}$$

(717)

is a symplectomorphism between the two Legendre images. Equivalently, the associated canonical relation is

$${\mathsf{\Gamma }}_{L_d}:=\{(q_0,{\mathsf{A}}_0;q_1,{\mathsf{A}}_1):{\mathsf{A}}_0=-D_0{L}_d,{\mathsf{A}}_1=D_1{L}_d\}\subset \overline{T^{\ast }M}\times T^{\ast }M.$$

(718)

It is exact Lagrangian [36,37]. Stationarity of ${\sum }_k{L}_d(q_k,q_{k+1})$ gives

$$D_1{L}_d(q_{k-1},q_k)+D_0{L}_d(q_k,q_{k+1})=0.$$

(719)

More precisely, for fixed exterior endpoints define

$${\mathbb{S}}_{N,\mathsf{\Delta }t}(q_0,\dots ,q_N):={\displaystyle \sum _{k=0}^{N-1}}{L}_{d,\mathsf{\Delta }t}^{\to }(q_k,q_{k+1}).$$

(720)

Its endpoint-fixed critical points are exactly the solutions of Equation (719), equivalently the momentum-matched iterates of ${\mathsf{\Phi }}_{L_d}$. On any smooth critical branch on which the internal endpoint equations are nondegenerate, its on-shell value obeys

$$\mathrm{d}{\mathbb{S}}_{N,\mathsf{\Delta }t}^{\mathrm{on}}=-{\mathsf{A}}_0·\mathrm{d}{q}_0+{\mathsf{A}}_N·\mathrm{d}{q}_N.$$

(721)

Hence stationary composition gives the exact Lagrangian relation ${\mathsf{\Gamma }}_{L_d}^{\circ N}$ rather than only a recurrence formula.

Theorem 24

(Exact local Hamiltonian reconstruction of the directed free endpoint family). Assume the Legendre-Hessian hypotheses of Theorem 22, fix ${\mu }_{\ast }>0$, and, for $h>0$, put

$$S_h^{\to }(q_0,q):=\frac{{\mu }_{\ast }}{h}{B}_{\psi }(q,q_0).$$

(722)

In the affine cotangent trivialization, define the natural right Legendre domain and reconstructed initial endpoint by

$$\begin{array}{cc}\hfill {\mathcal{D}}_h^{\mathrm{free}}& :=\left\{(q,p)\in T^{\ast }M:{\vartheta }_q-\frac{h}{{\mu }_{\ast }}p\in M^{\ast }\right\},\hfill \end{array}$$

(723)

$$\begin{array}{cc}\hfill Q_h(q,p)& :={\left(\mathrm{d}\psi \right)}^{-1}\left({\vartheta }_q-\frac{h}{{\mu }_{\ast }}p\right),{\mathcal{H}}_h^{\to }(q,p):=\frac{{\mu }_{\ast }}{h^2}{B}_{\psi }\left(q,Q_h(q,p)\right).\hfill \end{array}$$

(724)

Then ${\mathcal{H}}_h^{\to }$ is fiberwise strictly convex and hyperregular:

$$D_p{\mathcal{H}}_h^{\to }(q,p)=\frac{q-Q_h(q,p)}{h},D_p^2{\mathcal{H}}_h^{\to }(q,p)=\frac{1}{{\mu }_{\ast }}{g}_{Q_h(q,p)}^{-1}>0.$$

(725)

For each q, its fiber derivative is a diffeomorphism from ${\mathcal{D}}_h^{\mathrm{free}}\cap T_q^{\ast }M$ onto the natural velocity domain $\{v\in T_{q}M:q-hv\in M\}$; “hyperregular” is meant on these open domains, not as a claim that they equal the entire fibers. Its exact fiber Legendre transform is

$$\begin{array}{|c|}\hline {\mathcal{L}}_h^{\to }(q,v)=\frac{{\mu }_{\ast }}{h^2}{B}_{\psi }(q-hv,q),q-hv\in M,\\ \hline\end{array}$$

(726)

and both sides extend on compact subsets through $h=0$ with

$${\mathcal{H}}_0(q,p)=\frac{1}{2{\mu }_{\ast }}{g}_q^{-1}(p,p),{\mathcal{L}}_0(q,v)=\frac{{\mu }_{\ast }}{2}{g}_q(v,v).$$

(727)

The mixed Hessian is $D_0{D}_1{S}_h^{\to }=-({\mu }_{\ast }/h)g_{q_0}$, and Equation (724) supplies the smooth right-Legendre inverse. Hence Lemma 4 applies on every common local subdomain of ${\mathcal{D}}_h^{\mathrm{free}}$; the generated maps extend with ${\mathsf{\Phi }}_0^{\to }=\mathrm{id}$.

More explicitly, for $q_0,q_1\in M$, let $v=(q_1-q_0)/h$ in affine coordinates and set, for $0<t\le h$,

$$q_t=q_0+tv,p_t=\frac{{\mu }_{\ast }}{t}\left({\vartheta }_{q_t}-{\vartheta }_{q_0}\right).$$

(728)

This curve obeys Hamilton’s equations for ${\mathcal{H}}_t^{\to }$ and the Euler-Lagrange equations for ${\mathcal{L}}_t^{\to }$, and the endpoint family is its exact on-shell action:

$$\begin{array}{|c|}\hline \frac{{\mu }_{\ast }}{h}{B}_{\psi }(q_1,q_0)={\int }_0^h{\mathcal{L}}_t^{\to }(q_t,{\dot{q}}_t)\mathrm{d}t.\\ \hline\end{array}$$

(729)

Thus the forward Bregman orientation is Hamilton’s principal endpoint function for the reconstructed $({\mathcal{H}}_h^{\to },{\mathcal{L}}_h^{\to })$ system on the stated branch, whereas the reverse orientation is its instantaneous Lagrangian along the characteristic. This does not identify the finite-h principal function with that of the autonomous BKM kinetic Hamiltonian: exact agreement holds for a quadratic Hessian potential, while in general the proved relation is the compact-local $h\downarrow 0$ limit. Interpreting h as physical time requires the independent clock calibration.

The family-specific inverse, Legendre transform, characteristic, and on-shell action are proved in Appendix G.4.

Corollary 47

(Trapezoidal potential and reconstructed endpoint mechanics). In Theorem 23, make the mechanical choice ${\kappa }_h={\mu }_{\ast }/h$ and assume $V\in C^3\left(M\right)$. Set ${\pi }_h:=p+(h/2)\mathrm{d}{V}_q$. On the natural right Legendre domain

$${\mathcal{D}}_h^V:=\left\{(q,p)\in T^{\ast }M:{\vartheta }_q-\frac{h}{{\mu }_{\ast }}{\pi }_h\in M^{\ast }\right\},$$

(730)

put

$$\begin{array}{cc}\hfill Q_h^V(q,p)& :={\left(\mathrm{d}\psi \right)}^{-1}\left({\vartheta }_q-\frac{h}{{\mu }_{\ast }}{\pi }_h\right)=Q_h(q,{\pi }_h),\hfill \end{array}$$

(731)

$$\begin{array}{cc}\hfill {\mathcal{H}}_h^{\to ,V}(q,p)& :={\mathcal{H}}_h^{\to }(q,{\pi }_h)+\frac{1}{2}\left[V\left(Q_h^V(q,p)\right)+V\left(q\right)\right].\hfill \end{array}$$

(732)

The conclusions of Lemma 4 apply with $S_h=L_{d,h}^{\to }$ and ${\mathcal{H}}_h={\mathcal{H}}_h^{\to ,V}$ on every common local subdomain of ${\mathcal{D}}_h^V$. On compact phase sets and for sufficiently small h,

$${\mathcal{H}}_h^{\to ,V}(q,p)=\frac{1}{2{\mu }_{\ast }}{g}_q^{-1}(p,p)+V\left(q\right)+O\left(h\right),{\mathsf{\Phi }}_0^{\to }=\mathrm{id}.$$

(733)

Thus the endpoint map exactly integrates its reconstructed nonautonomous Hamiltonian and, separately, is the variational approximation to the autonomous mechanical Hamiltonian quantified in Corollary 52.

Proof. 

The outgoing momentum in Equation (714) is equivalent to Equation (731). Taking the negative partial h-derivative of $L_{d,h}^{\to }$ at fixed endpoints gives Equation (732). Its explicit inverse and the nondegenerate mixed Hessian Equation (712) verify the lemma’s hypotheses after restriction to a common chart. Finally, $Q_h^V=q-(h/{\mu }_{\ast })g_q^{♯}p+O\left(h^2\right)$ on compact sets, and Taylor expansion gives the Hamiltonian limit. The incoming relation gives $q_h=q_0+(h/{\mu }_{\ast })g_{q_0}^{♯}(p_0-(h/2)\mathrm{d}{V}_{q_0})$; substitution in the outgoing relation gives $p_h=p_0+O\left(h\right)$, proving the phase-space limit in Equation (733). □

Corollary 48

(Bridge potential in the endpoint mechanics). On the qutrit state manifold, independently supply an energy coefficient ${\kappa }_{\mathrm{br}}>0$ and use $V={\kappa }_{\mathrm{br}}{\mathcal{V}}_{\mathrm{br}}$ in Corollary 47. The autonomous mechanical limit is

$$H_{\mathrm{br}}(q,p)=\frac{1}{2{\mu }_{\ast }}{g}_q^{-1}(p,p)+{\kappa }_{\mathrm{br}}{\mathcal{V}}_{\mathrm{br}}\left(q\right).$$

(734)

The tracial point $(0,0)$ is an equilibrium, and all eight linearized state-space modes have the common frequency

$$\begin{array}{|c|}\hline {\omega }_{\mathrm{br}}^2=\frac{5{\kappa }_{\mathrm{br}}}{64{\mu }_{\ast }}.\\ \hline\end{array}$$

(735)

The corresponding Bregman endpoint generator remains regular and exactly generates its reconstructed nonautonomous Hamiltonian; its comparison with Equation (734) has the orders stated in Corollary 52. Here ${\kappa }_{\mathrm{br}}$, ${\mu }_{\ast }$, and physical time are independent supplied calibrations. Thus Equation (735) is a finite-dimensional mechanical frequency, not a dynamically generated gauge-field mass or quantum spectral gap.

Proof. 

Substitution into Equations (732) and (733) proves the endpoint statements. At zero momentum, variation of the inverse metric contributes no linear term. Hamilton’s equations and Equation (534) therefore give

$$\ddot{q}+\frac{5{\kappa }_{\mathrm{br}}}{64{\mu }_{\ast }}q=0$$

(736)

in $g_0$-orthonormal linearized coordinates. □

The explicit global Legendre images, canonical-relation calculation, and Noether proof for Theorem 23 are collected in Appendix G; the short-time expansion and derivative-controlled error transfer are in Appendix G.8. Regularity alone gives a discrete symplectic map, not an approximation order or automatically the time-$\mathsf{\Delta }t$ map of an autonomous Hamiltonian; the required comparison with the exact discrete Lagrangian is supplied in Corollary 52 and Proposition A8.

Corollary 49

(Global Umegaki Type-I and Type-II generators). Choose the fixed endpoint specialization of Theorem 23, namely $V=0$ and ${\kappa }_{\mathsf{\Delta }t}\equiv {\kappa }_A$ independently of any step. On the faithful-state manifold in mixture coordinates, the pure endpoint generator

$$L_{d,\mathrm{U}}^{\to }({\rho }_0,{\rho }_1):={\kappa }_A{D}_{\mathrm{U}}({\rho }_1\parallel {\rho }_0)$$

(737)

is regular, with

$$D_0{D}_1{L}_{d,\mathrm{U}}^{\to }=-{\kappa }_A{g}_{{\rho }_0}^{\mathrm{BKM}},{\mathsf{A}}_1\equiv {\kappa }_A(log{\rho }_1-log{\rho }_0)(mod\mathbb{R}I).$$

(738)

By Theorem 2, the outgoing Legendre relation and Type-II generating function for the traceless Hermitian representative $P_1$ of ${\mathsf{A}}_1$ are

$$\begin{array}{cc}\hfill {\rho }_1& =\frac{exp(log{\rho }_0+P_1/{\kappa }_A)}{Trexp(log{\rho }_0+P_1/{\kappa }_A)},\hfill \end{array}$$

(739)

$$\begin{array}{cc}\hfill H_{d,\mathrm{U}}^{+}({\rho }_0,P_1)& ={\kappa }_{A}logTrexp(log{\rho }_0+P_1/{\kappa }_A)={\kappa }_A{\mathsf{\Lambda }}_{{\rho }_0}(P_1/{\kappa }_A).\hfill \end{array}$$

(740)

The Fenchel certificate of Theorem 3 yields both exact endpoint orientations:

$$\begin{array}{|c|}\hline H_{d,\mathrm{U}}^{+}({\rho }_0,P_1)=Tr\left({\rho }_0{P}_1\right)+{\kappa }_A{D}_{\mathrm{U}}({\rho }_0\parallel {\rho }_1)=Tr\left({\rho }_1{P}_1\right)-{\kappa }_A{D}_{\mathrm{U}}({\rho }_1\parallel {\rho }_0).\\ \hline\end{array}$$

(741)

and determinant normalization of the reconstructed endpoint gives the global Lie-log translation

$$\mathsf{\Phi }\left({\rho }_1\right)=exp\left(log\mathsf{\Phi }\left({\rho }_0\right)+P_1/{\kappa }_A\right),\mathcal{X}\left({\rho }_1\right)=\mathcal{X}\left({\rho }_0\right)-P_1/{\kappa }_A.$$

(742)

The traceless convention fixes both the normal-covector gauge and the affine origin; nontraceless representatives use the affine lift below. Thus the full-state Type-II generator is the same global log-partition construction that underlies the Gibbs chart. It is action-scaled, not an energy-valued continuous Hamiltonian. The exact Type-I construction restricts to the channel-output manifold ${\mathcal{Q}}_{\mathsf{\Lambda }}$, and the difference of the two pure endpoint generators is

$${\kappa }_A{\mathsf{\Delta }}_{\mathsf{\Lambda }}({\rho }_1,{\rho }_0).$$

(743)

For every output image, its precise Type-II domain and restricted conjugate are given in 33. The ambient reduced-carrier log-partition formula applies directly when the output image is the full faithful-state manifold. More generally, call the output imagecompatibly exponential-autoparallelwhen its traceless log-coordinate image is relatively open in an affine subspace A of the reduced-carrier traceless Hermitian operators and the trace pairing identifies the direction space of A isomorphically with $E_{\mathsf{\Lambda }}^{\ast }$. Then every admissible restricted Legendre increment has a unique representative tangent to A, and the ambient log-partition formula restricts on the corresponding Legendre image. Without this hypothesis, the intrinsic conjugate $f_{\mathsf{\Lambda }}^{\ast }$ below, rather than the ambient formula, is the Type-II generator.

The same Type-II partition generator contains the entire proximal bridge on one momentum ray. To evaluate nontraceless representatives, introduce its affine lift

$$\begin{array}{cc}\hfill {\tilde{H}}_{d,\mathrm{U}}^{+}(\sigma ,P)& :={\kappa }_A{\mathsf{\Lambda }}_{\sigma }(P/{\kappa }_A),P\in Herm\left(n\right),\hfill \\ \hfill {\tilde{H}}_{d,\mathrm{U}}^{+}(\sigma ,P+cI)& ={\tilde{H}}_{d,\mathrm{U}}^{+}(\sigma ,P)+c.\hfill \end{array}$$

(744)

It agrees with $H_{d,\mathrm{U}}^{+}$ in the traceless gauge. For faithful $\rho ,\sigma$, choose the full-log representative

$$P_t:={\kappa }_{A}t{L}_{\rho \mid \sigma }={\kappa }_{A}t(log\rho -log\sigma ),0\le t\le 1.$$

(745)

Then

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill D_P{\tilde{H}}_{d,\mathrm{U}}^{+}(\sigma ,P_t)\left[A\right]& =Tr\left({\gamma }_{t}A\right),\hfill \\ \hfill {\tilde{H}}_{d,\mathrm{U}}^{+}(\sigma ,P_t)& ={\kappa }_A{\phi }_{\rho \mid \sigma }\left(t\right)=-{\kappa }_A(1-t)D_t^{\mathrm{b}}(\rho \parallel \sigma ),\hfill \\ \hfill \frac{\mathrm{d}}{\mathrm{d}t}{\tilde{H}}_{d,\mathrm{U}}^{+}(\sigma ,P_t)& ={\kappa }_{A}Tr\left({\gamma }_t{L}_{\rho \mid \sigma }\right),\hfill \\ \hfill \frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}{\tilde{H}}_{d,\mathrm{U}}^{+}(\sigma ,P_t)& ={\kappa }_A{Cov}_{{\gamma }_t}^{\mathrm{KM}}(L_{\rho \mid \sigma },L_{\rho \mid \sigma }).\hfill \end{array}\\ \hline\end{array}$$

(746)

In the first line A may be restricted to the trace-zero tangent space when the derivative is read on the cotangent quotient. Thus the momentum derivative produces the proximal deformation, while the on-ray value is exactly $-{\kappa }_A{C}_t^{\mathrm{b}}=-{\kappa }_A(1-t){\mathcal{M}}_{(1-t)/t}^{\sigma }\left(\rho \right)$. Replacing the full-log representative by the traceless one adds the known scalar gauge term. Equation (746) is an affine-lift evaluation of the Umegaki Type-II generator, not a claim that this log-partition function is the Type-II conjugate of ${\mathfrak{B}}_t^{\mathrm{RM}}$.

Corollary 50

(Mechanical scaling and exact response of the partition generator). Choose the distinct mechanical calibration ${\kappa }_h={\mu }_{\ast }/h$, and represent the outgoing cotangent class by its traceless Hermitian representative P. Using the cotangent covariance in Equation (61), the Type-II generator is

$$G_h^{+}(\rho ,P)=\frac{{\mu }_{\ast }}{h}{\mathsf{\Lambda }}_{\rho }\left(\frac{h}{{\mu }_{\ast }}P\right)$$

(747)

Define the positive-source mirror segment ${\rho }_s^{+}:=E_{\rho }(shP/{\mu }_{\ast })$. The nonlinear kinetic remainder has the exact representations

$$\begin{array}{|c|}\hline {\mathcal{K}}_h^{+}(\rho ,\left[P\right]):=\frac{G_h^{+}(\rho ,P)-Tr\left(\rho P\right)}{h}=\frac{{\mu }_{\ast }}{h^2}{D}_{\mathrm{U}}(\rho \parallel {\rho }_1^{+})=\frac{1}{{\mu }_{\ast }}{\int }_0^1(1-s){Cov}_{{\rho }_s^{+}}^{\mathrm{KM}}(P,P)\mathrm{d}s.\\ \hline\end{array}$$

(748)

The negative-momentum remainder is exactly the Hamiltonian reconstructed from the mechanically scaled Type-I endpoint family in Theorem 24. Put ${\overline{P}}_h:=hP/{\mu }_{\ast }=P/{\kappa }_h$, the dimensionless source, and let ${\mathcal{H}}_h^{\mathrm{U}}$ denote the Umegaki specialization of Equation (724). Then

$$\begin{array}{|c|}\hline {\mathcal{H}}_h^{\mathrm{U}}(\rho ,\left[P\right])=\frac{Tr\left(\rho P\right)+G_h^{+}(\rho ,-P)}{h}={\mathcal{K}}_h^{+}(\rho ,[-P])=\frac{{\mu }_{\ast }}{h^2}{\mathcal{K}}_{1,\rho }^{-}\left(\left[{\overline{P}}_h\right]\right).\\ \hline\end{array}$$

(749)

Thus the Type-I Hamilton-Jacobi Hamiltonian, the reversed Type-II kinetic remainder, and the centered negative-source spectral log-Laplace cumulant are one energy-valued function after calibration. This equality concerns the functions; ${\overline{P}}_h$ is a source rescaling and does not identify a Moreau regularization scale with physical time. Consequently, the generator has the compact-uniform expansion

$$\begin{array}{|c|}\hline G_h^{+}(\rho ,P)=Tr\left(\rho P\right)+h{H}_{\mathrm{BKM}}(\rho ,\left[P\right])+O\left(h^2\right),\\ \hline\end{array}{H}_{\mathrm{BKM}}(\rho ,\left[P\right]):=\frac{1}{2{\mu }_{\ast }}{Cov}_{\rho }^{\mathrm{KM}}(P,P).$$

(750)

The remainder is uniform for ρ in a compact subset of ${\mathcal{S}}_n^{+}$, P in a bounded subset of $\mathfrak{p}$, and $0<h\le h_0$. Other Hermitian representatives follow the affine-gauge rule Equation (744) with ${\kappa }_A$ replaced by ${\kappa }_h$. The covariance is invariant under $P\mapsto P+cI$ and represents ${\left(g_{\rho }^{\mathrm{BKM}}\right)}^{-1}(\left[P\right],\left[P\right])$. Thus the energy-valued remainder ${\mathcal{K}}_h^{+}$, rather than the action-valued Type-II generator $G_h^{+}$ with its affine leading term, is a finite-step nonlinear completion of the BKM cotangent kinetic Hamiltonian. It is not a finite-step completion of Equation (557); the two tangent metrics agree only at the calibrated center described by Equations (348) and (569).

Proof. 

The reverse-orientation identity in 3 gives the relative-entropy formula in Equation (748); its weighted susceptibility integral gives the last expression. Taking $h\to 0$ yields $H_{\mathrm{BKM}}$ and, one order further, the displayed generator expansion with the stated compact-uniform remainder. Finally, substituting $-P$ into the definition of ${\mathcal{K}}_h^{+}$, substituting ${\overline{P}}_h=hP/{\mu }_{\ast }$, and setting $r=1$ in Equation (170) proves every equality in Equation (749). □

Proposition 33

(Channel-output Type-II domain). Put $E_{\mathsf{\Lambda }}:=imL$, choose $q_{\ast }\in {\mathcal{Q}}_{\mathsf{\Lambda }}$, and set

$${\mathsf{\Omega }}_{\mathsf{\Lambda }}:={\mathcal{Q}}_{\mathsf{\Lambda }}-q_{\ast }\subset E_{\mathsf{\Lambda }},f_{\mathsf{\Lambda }}\left(x\right):={\mathcal{F}}_{\mathsf{\Lambda }}(q_{\ast }+x),f_{\mathsf{\Lambda }}^{\ast }\left(\eta \right):=\underset{x\in {\mathsf{\Omega }}_{\mathsf{\Lambda }}}{sup}\{\eta \left(x\right)-f_{\mathsf{\Lambda }}\left(x\right)\}.$$

(751)

For $L_{d,\mathsf{\Lambda }}^{\to }(q_0,q_1):={\kappa }_A{B}_{{\mathcal{F}}_{\mathsf{\Lambda }}}(q_1,q_0)$, the attained regular Type-II generator is defined precisely on

$$\begin{array}{cc}\hfill {\mathcal{D}}_{\mathsf{\Lambda }}^{+}:=\{(q_0,p_1)\in {\mathcal{Q}}_{\mathsf{\Lambda }}\times E_{\mathsf{\Lambda }}^{\ast }:& \mathrm{d}{f}_{\mathsf{\Lambda }}\left(x_0\right)+p_1/{\kappa }_A\in \mathrm{d}{f}_{\mathsf{\Lambda }}\left({\mathsf{\Omega }}_{\mathsf{\Lambda }}\right)\},\hfill \\ & x_0:=q_0-q_{\ast },\hfill \end{array}$$

(752)

and there equals

$$H_{d,\mathsf{\Lambda }}^{+}(q_0,p_1)={\kappa }_A\left[f_{\mathsf{\Lambda }}^{\ast }\left(\mathrm{d}{f}_{\mathsf{\Lambda }}\left(x_0\right)+p_1/{\kappa }_A\right)-f_{\mathsf{\Lambda }}^{\ast }\left(\mathrm{d}{f}_{\mathsf{\Lambda }}\left(x_0\right)\right)\right].$$

(753)

This generator describes the exact canonical relation between its natural Legendre images; it is not asserted to cover all of $T^{\ast }{\mathcal{Q}}_{\mathsf{\Lambda }}$. The restricted Fenchel value can exist outside this attainment domain.

Proof. 

Writing $x_i=q_i-q_{\ast }$ gives $p_1={\kappa }_A[\mathrm{d}{f}_{\mathsf{\Lambda }}\left(x_1\right)-\mathrm{d}{f}_{\mathsf{\Lambda }}\left(x_0\right)]$. Strict convexity makes $\mathrm{d}{f}_{\mathsf{\Lambda }}$ injective, so this relation has a solution exactly on Equation (752). At this solution, Fenchel equality evaluates the affine objective $p_1\left(x_1\right)-{\kappa }_A{B}_{f_{\mathsf{\Lambda }}}(x_1,x_0)$ as Equation (753). □

Injective proper-image complement.

The strict-depolarization calculation in Appendix F.5.2 complements dephasing by giving an injective channel with a proper output image. It also shows that the analytic Type-II formula is attained only on its natural Legendre domain Equation (A257).

8.3. Cubic Defect, Symmetrization, and Phase-Map Accuracy

Corollary 51

(Cubic action defect and adjoint symmetrization). Return to the general Hessian setting of 23 and now choose its distinct mechanical specialization. Assume that $\psi \in C^4$, $V\in C^3$, and work in a no-caustic neighborhood where, for $\eta \left(q_1\right)=\eta \left(q_0\right)+\mathsf{\Delta }tv$, the unique time-rescaled extremal depends smoothly on $(q_0,v,\mathsf{\Delta }t)$ through $\mathsf{\Delta }t=0$. Choose an inertial scale ${\mu }_{\ast }>0$, set

$${\kappa }_{\mathsf{\Delta }t}:=\frac{{\mu }_{\ast }}{\mathsf{\Delta }t},L(q,\dot{q}):=\frac{{\mu }_{\ast }}{2}{g}_q(\dot{q},\dot{q})-V\left(q\right).$$

(754)

Its Legendre-transform Hamiltonian is

$$H(q,p):=\frac{1}{2{\mu }_{\ast }}{g}_q^{-1}(p,p)+V\left(q\right).$$

(755)

Let $q_{01}^E:[0,\mathsf{\Delta }t]\to M$ be that unique Euler-Lagrange extremal with $q_{01}^E\left(0\right)=q_0$ and $q_{01}^E\left(\mathsf{\Delta }t\right)=q_1$, and define the exact discrete Lagrangian by

$$L_{d,\mathsf{\Delta }t}^E(q_0,q_1):={\int }_0^{\mathsf{\Delta }t}L\left(q_{01}^E\left(t\right),{\dot{q}}_{01}^E\left(t\right)\right)\mathrm{d}t.$$

(756)

The exact-discrete-Lagrangian theorem identifies the endpoint Legendre transforms with $p^E\left(0\right)$ and $p^E\left(\mathsf{\Delta }t\right)$; hence its Type-I relation is exactly the time-$\mathsf{\Delta }t$ Hamiltonian flow of Equation (755) on this no-caustic branch [36], Lemma 1.6.2 and Theorem 1.6.3. When the affine coordinates and g are dimensionless, $\left[{\mu }_{\ast }\right]=\mathrm{energy}·{\mathrm{time}}^2=\mathrm{action}·\mathrm{time}$, so ${\kappa }_{\mathsf{\Delta }t}$ has action units and L has energy units. This is the second calibration in Remark 19; every approximation-order statement below uses it. Uniformly for $q_0$ in a fixed compact subset of the no-caustic neighborhood and v in a bounded set,

$$L_{d,\mathsf{\Delta }t}^{\to }-L_{d,\mathsf{\Delta }t}^E=-\frac{{\mu }_{\ast }\mathsf{\Delta }{t}^2}{12}{\mathcal{C}}_{q_0}(v,v,v)+O\left(\mathsf{\Delta }{t}^3\right),\mathcal{C}=\nabla g=D^3\psi .$$

(757)

This is an action-level defect; the derivative-controlled passage to the induced phase map is stated in Corollary 52. For the adjoint statement, extend the mechanically calibrated formula to signed $h\ne 0$ with coefficient ${\mu }_{\ast }/h$ and define

$${\left(L_{d,h}^{\to }\right)}^{\ast }(q_0,q_1):=-L_{d,-h}^{\to }(q_1,q_0).$$

(758)

The adjoint average is

$$L_{d,h}^{\mathrm{sym}}(q_0,q_1):=\frac{1}{2}\left[L_{d,h}^{\to }+{\left(L_{d,h}^{\to }\right)}^{\ast }\right]=\frac{{\mu }_{\ast }}{2h}\left[B_{\psi }(q_1,q_0)+B_{\psi }(q_0,q_1)\right]-\frac{h}{2}\left[V\left(q_0\right)+V\left(q_1\right)\right]$$

(759)

and obeys

$$D_0{D}_1{L}_{d,h}^{\mathrm{sym}}=-\frac{{\mu }_{\ast }}{2h}(g_{q_0}+g_{q_1}),L_{d,h}^{\mathrm{sym}}-L_{d,h}^E=O\left(h^3\right).$$

(760)

It is regular and self-adjoint (time-symmetric), and its induced maps satisfy ${\mathsf{\Phi }}_{-h}^{\mathrm{sym}}={\left({\mathsf{\Phi }}_h^{\mathrm{sym}}\right)}^{-1}$ on their common local domains [36], Theorem 2.4.1 and Section 2.4.2.

The normalized dual-Jensen family in Theorem 22 gives a second, exact adjoint interpolation:

$$L_{d,h}^{\left(\theta \right)}(q_0,q_1):=\frac{{\mu }_{\ast }}{h}{\mathfrak{B}}_{\theta }^{\psi }(q_1,q_0)-\frac{h}{2}\left[V\left(q_0\right)+V\left(q_1\right)\right],0<\theta <1.$$

(761)

If $z_{\theta }$ is the unique variational barycenter in Equation (703), then

$$D_0{D}_1{L}_{d,h}^{\left(\theta \right)}[W,U]=-\frac{{\mu }_{\ast }}{h}{g}_{q_1}\left({\left(g_{z_{\theta }}^{♭}\right)}^{-1}{g}_{q_0}^{♭}W,U\right),$$

(762)

so every member is a regular Type-I generator. For fixed θ, on any right-Legendre chart common to an h-interval, let $Q_{h,\theta }(q,p)$ denote the corresponding single smooth local branch satisfying

$$p=D_1{L}_{d,h}^{\left(\theta \right)}\left(Q_{h,\theta }(q,p),q\right)$$

(763)

and define

$${\mathcal{H}}_{h,\theta }^V(q,p):=\frac{{\mu }_{\ast }}{h^2}{\mathfrak{B}}_{\theta }^{\psi }\left(q,Q_{h,\theta }(q,p)\right)+\frac{1}{2}\left[V\left(Q_{h,\theta }(q,p)\right)+V\left(q\right)\right].$$

(764)

Fixed-endpoint differentiation identifies this with Equation (695); the nondegenerate mixed Hessian verifies the lemma’s remaining hypothesis on this common chart. For fixed incoming data $(q_0,p_0)$ and $h_0,h_1>0$ on the selected characteristic branch, the Hamiltonian on-shell integral in Equation (700) has the explicit value

$${\mathcal{A}}_{h_0,h_1}^{\left(\theta \right),V}={\left[\frac{{\mu }_{\ast }}{h}{\mathfrak{B}}_{\theta }^{\psi }(q_h,q_0)-\frac{h}{2}\left[V\left(q_0\right)+V\left(q_h\right)\right]\right]}_{h=h_0}^{h=h_1}.$$

(765)

For faithful states and $V=0$, the reconstruction is explicit in Proposition A7. This exact nonautonomous mechanics and the autonomous approximation order below are distinct claims. Its signed-step adjoint is ${\left(L_{d,h}^{\left(\theta \right)}\right)}^{\ast }=L_{d,h}^{(1-\theta )}$, and uniformly on the scaled endpoint sets above,

$$\begin{array}{cc}\hfill {\mathfrak{B}}_{\theta }^{\psi }(q_1,q_0)& =\frac{h^2}{2}{g}_{q_0}(v,v)+\frac{(2-\theta )h^3}{6}{\mathcal{C}}_{q_0}(v,v,v)+O\left(h^4\right),\hfill \end{array}$$

(766)

$$\begin{array}{cc}\hfill L_{d,h}^{\left(\theta \right)}-L_{d,h}^E& =\frac{{\mu }_{\ast }{h}^2}{12}(1-2\theta ){\mathcal{C}}_{q_0}(v,v,v)+O\left(h^3\right).\hfill \end{array}$$

(767)

Thus the continuous limits $\theta \uparrow 1$ and $\theta \downarrow 0$ recover the directed Umegaki/Bregman generator and its adjoint, respectively, while $\theta =1/2$ cancels the cubic action defect. In the Umegaki specialization the midpoint generator is

$$L_{d,h}^{\mathrm{RM},\mathrm{mid}}({\rho }_0,{\rho }_1)=\frac{2{\mu }_{\ast }}{h}{D}_{1/2}^{\mathrm{b}}({\rho }_1\parallel {\rho }_0)-\frac{h}{2}\left[V\left({\rho }_0\right)+V\left({\rho }_1\right)\right].$$

(768)

It is regular, exactly self-adjoint, and second order under the derivative and stability hypotheses of Corollary 52. It is generally different from the Jeffreys adjoint average, although the two agree through cubic order. Its Rényi-Moreau information term retains the exact variational representation and, for CPTP maps, data processing; the full mechanical generator has no channel-monotonicity claim unless the separately supplied potential is channel compatible.

For $h>0$, all three mechanically scaled endpoint constructions use ${\kappa }_h={\mu }_{\ast }/h$; negative h is only the signed continuation needed to define an adjoint. The estimates require $\eta \left(q_1\right)-\eta \left(q_0\right)=O\left(h\right)$; fixed separated endpoints and caustics are excluded. If ψ is quadratic, the directed and symmetric generators coincide and are at least second order for general smooth V under the next corollary’s hypotheses; they equal the exact free discrete Lagrangian when $V=0$.

Corollary 52

(One-step and fixed-time phase-map error). Strengthen the preceding hypotheses to $\psi \in C^6$, $V\in C^4$, and $C^4$ compact-uniform dependence of the rescaled no-caustic extremal on $(q_0,v,h)$. Let $0<h\le h_0$ and identify tangent vectors with their components in the global affine trivialization. In scaled endpoint variables, set

$${\mathsf{\Sigma }}_h(q,v):=\left(q,{\eta }^{-1}(\eta \left(q\right)+hv)\right).$$

(769)

On every compact scaled-endpoint set K, the remainders in Equations (757), (760) and (767) then hold in $C^2\left(K\right)$: the remainder and all $(q,v)$ derivatives through order two obey the displayed power of h. Let ${\phi }_h$ be the Legendre-transformed Euler-Lagrange phase flow of L (equivalently, the Hamiltonian flow of H with the manuscript convention ${\omega }_A=\mathrm{d}{\lambda }_A$). If ${\mathsf{\Phi }}_h^{\to }$ and ${\mathsf{\Phi }}_h^{\mathrm{sym}}$ are the phase maps induced by the directed and symmetric discrete Legendre transforms, and ${\mathsf{\Phi }}_h^{\mathrm{RM},\mathrm{mid}}$ is the midpoint Rényi-Moreau phase map in the Umegaki specialization, then, in any fixed local phase-space chart and on every compact phase set $\mathcal{K}$ whose incoming Legendre inverses remain in K,

$$\begin{array}{cc}\hfill {∥{\mathsf{\Phi }}_h^{\to }-{\phi }_h∥}_{C^1\left(\mathcal{K}\right)}& \le C_{\mathcal{K}}{h}^2,\hfill \\ \hfill {∥{\mathsf{\Phi }}_h^{\mathrm{sym}}-{\phi }_h∥}_{C^1\left(\mathcal{K}\right)}+{∥{\mathsf{\Phi }}_h^{\mathrm{RM},\mathrm{mid}}-{\phi }_h∥}_{C^1\left(\mathcal{K}\right)}& \le C_{\mathcal{K}}{h}^3.\hfill \end{array}$$

(770)

In the standard variational-integrator terminology of [36], Theorem 2.3.1, and [37], Section 5.4, the manuscript-derived one-step bounds imply order at least one for the directed method and order at least two for the self-adjoint methods. The coefficient below shows that the directed order is exactly one wherever it is nonzero; special data can raise the order.

More sharply, let ${\pi }_M:T^{\ast }M\to M$, put $u:={\mu }_{\ast }^{-1}{g}_{q_0}^{♯}{p}_0$, and define ${\mathcal{C}}^{♯}$ by

$$g_{q_0}\left({\mathcal{C}}_{q_0}^{♯}(u,u),w\right)={\mathcal{C}}_{q_0}(u,u,w).$$

(771)

Then the directed configuration defect is

$$\eta \left({\pi }_M{\mathsf{\Phi }}_h^{\to }(q_0,p_0)\right)-\eta \left({\pi }_M{\phi }_h(q_0,p_0)\right)=\frac{h^2}{4}{\mathcal{C}}_{q_0}^{♯}(u,u)+O_{\mathcal{K}}\left(h^3\right).$$

(772)

Thus first order is sharp wherever ${\mathcal{C}}_{q_0}^{♯}(u,u)\ne 0$.

For every $z\in \mathcal{K}$ whose exact and all three numerical iterates remain in a common compact regular phase tube for $0\le kh\le T$,

$$\begin{array}{cc}\hfill \underset{0\le kh\le T}{max}∥{\left({\mathsf{\Phi }}_h^{\to }\right)}^k\left(z\right)-{\phi }_{kh}\left(z\right)∥& \le C_{\mathcal{K},T}h,\hfill \\ \hfill \underset{0\le kh\le T}{max}∥{\left({\mathsf{\Phi }}_h^{\mathrm{sym}}\right)}^k\left(z\right)-{\phi }_{kh}\left(z\right)∥& \le C_{\mathcal{K},T}{h}^2,\hfill \\ \hfill \underset{0\le kh\le T}{max}∥{\left({\mathsf{\Phi }}_h^{\mathrm{RM},\mathrm{mid}}\right)}^k\left(z\right)-{\phi }_{kh}\left(z\right)∥& \le C_{\mathcal{K},T}{h}^2.\hfill \end{array}$$

(773)

These are fixed-time global bounds. They imply neither invariance of a global phase domain, global-in-time nonlinear stability, nor a long-time energy-error bound. On a faithful-state manifold, the compact tube must remain uniformly separated from the rank-deficient boundary.

The role of divergences as discrete generating functions is standard [37], Sections 5.1-5.2. On the scaled endpoint pairs considered here, $B_{\psi }(q_1,q_0)=O\left(h^2\right)$, so the mechanical normalization ${\mu }_{\ast }{B}_{\psi }/h$ is action-valued and $O\left(h\right)$. The approximation orders in Corollary 52 follow from the fixed-gauge, derivative-controlled comparison with the exact discrete Lagrangian: a $C^2$-controlled $O\left(h^{p+1}\right)$ action remainder in scaled variables yields order at least p under the hypotheses of Proposition A8. For the Umegaki specialization, symmetrization cancels the cubic action defect. It loses the single oriented Stein exponent and, in general, representation by one directed Bregman divergence, but retains data processing term by term. Indeed, for every CPTP map $\mathsf{\Lambda }$,

$$\begin{array}{c}{J}_{\mathrm{U}}(\rho ,\sigma ):=D_{\mathrm{U}}(\rho \parallel \sigma )+D_{\mathrm{U}}(\sigma \parallel \rho ),\\ J_{\mathrm{U}}(\rho ,\sigma )-J_{\mathrm{U}}(\mathsf{\Lambda }\left(\rho \right),\mathsf{\Lambda }\left(\sigma \right))={\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma )+{\mathsf{\Delta }}_{\mathsf{\Lambda }}(\sigma ,\rho )\ge 0.\end{array}$$

(774)

Thus, for state endpoints, the divergence contribution to $L_{d,h}^{\mathrm{sym}}$ retains an exact channel-contraction interpretation with prefactor ${\mu }_{\ast }/\left(2h\right)$. The mechanical potential term has no such interpretation unless compatible output-potential data are supplied.

Corollary 53

(Discrete Noether charge). Suppose a Lie group G acts ∇-affinely on M, $\psi \circ {\mathsf{\Phi }}_g=\psi +{\ell }_g$ with ${\ell }_g$ affine, and $V\circ {\mathsf{\Phi }}_g=V$. Then Equation (711) is diagonally G-invariant and

$$\langle J_d(q_0,q_1),\xi \rangle :=D_1{L}_d(q_0,q_1)\left[{\xi }_M\left(q_1\right)\right]=-D_0{L}_d(q_0,q_1)\left[{\xi }_M\left(q_0\right)\right]$$

(775)

is equivariant. With the present sign convention,

$$\mathrm{d}\langle J_d,\xi \rangle =-{\iota }_{{\xi }_{M\times M}}{\omega }_{d,\mathsf{\Delta }t}^{\to },J_d(q_{k-1},q_k)=J_d(q_k,q_{k+1})$$

(776)

along every solution of Equation (719).

The affine term in $\psi \circ {\mathsf{\Phi }}_g$ cancels from the Bregman divergence; the standard discrete Noether theorem then gives the endpoint expressions, equivariance, and conservation in Equations (775) and (776) ([36], Theorem 1.3.3). The following trace representative is the manuscript-specific specialization.

For simultaneous unitary conjugation of faithful states and the pairing $\langle H,\xi \rangle =\mathrm{i}Tr\left(H\xi \right)$, the Hermitian representative is

$$J_d({\rho }_0,{\rho }_1)\widehat{=}-\mathrm{i}[{\rho }_1,{\mathsf{A}}_1].$$

(777)

8.4. Conditional Thermal and Continuous Comparison

If the reference is independently calibrated as ${\tau }_{\beta }={\mathrm{e}}^{-\beta H_{\mathrm{sys}}}/Z_{\beta }$, then

$$-log{\tau }_{\beta }=\beta H_{\mathrm{sys}}+log{Z}_{\beta }I,\mathcal{X}\left({\tau }_{\beta }\right)=\beta \left(H_{\mathrm{sys}}-\frac{Tr{H}_{\mathrm{sys}}}{n}I\right).$$

(778)

Thus the global log chart gives the dimensionless traceless energy generator, while the separately supplied ${\beta }^{-1}$ fixes energy units. With $t_A:=\beta {\kappa }_A$,

$$V_{\beta }\left(\rho \right):={\beta }^{-1}{D}_{\mathrm{U}}(\rho \parallel {\tau }_{\beta })=\frac{{\mathcal{A}}_{\tau }^{\left({\kappa }_A\right)}\left(\rho \right)}{t_A}$$

(779)

is nonequilibrium Helmholtz free-energy excess, not internal-energy displacement. The derived $t_A$ merely has time units; it is neither the numerical step h nor a mechanically selected evolution or relaxation time. The exact internal/free-energy and channel-availability balances are in Appendix G.5.

The state ${\tau }_{\beta }$ alone fixes neither $\beta$ nor the energy zero and does not identify $H_{\mathrm{sys}}$ with a state-space Hamiltonian. After an independent positive mechanical tensor $\mathsf{m}$ is chosen, one may define

$${\mathcal{H}}_{\beta ,\mathsf{m}}(q,\mathsf{A})=\frac{1}{2}{\mathsf{m}}_q^{-1}(\mathsf{A},\mathsf{A})+V_{\beta }\left(q\right).$$

(780)

This is reversible cotangent mechanics, not von Neumann or CPTP dynamics. For $\xi \in \mathfrak{su}\left(n\right)$, let ${\xi }_M$ denote the infinitesimal conjugation field and define the canonical cotangent momentum-map component

$$J_{\xi }(q,\mathsf{A}):=\mathsf{A}\left({\xi }_M\left(q\right)\right)$$

along a solution of ${\mathcal{H}}_{\beta ,\mathsf{m}}$. Standard cotangent Noether theory gives conservation when both $\mathsf{m}$ and $V_{\beta }$ are invariant [109]; it does not identify the fixed-reference action-scaled entropy functional with Hamilton’s principal function. For invariant kinetic data, the fixed thermal potential gives the exact source law

$$\begin{array}{cc}\hfill {\dot{J}}_{\xi }& ={\beta }^{-1}Tr\left([\xi ,\rho ]log{\tau }_{\beta }\right)\hfill \\ & =Tr\left(\rho [\xi ,H_{\mathrm{sys}}]\right).\hfill \end{array}$$

(781)

Thus precisely the stabilizer charges $[\xi ,{\tau }_{\beta }]=0$ are conserved for all states; varying $\beta$ at fixed $H_{\mathrm{sys}}$ does not change that stabilizer. A promoted reference contributes to an enlarged conserved charge only if it receives its own phase-space sector and the full Hamiltonian is invariant. The thermal discrete balance and $n=2$ internal charge are given in Appendix G; the positive Hamilton-Jacobi obstruction is proved in Appendix G.7.

8.5. Conditional Rank-One Comparison with Poincaré Rapidity

Let $T_i:={\sigma }_i/2$, $B_i:=T_i\in \mathfrak{p}$, and $J_i:=-\mathrm{i}{T}_i\in \mathfrak{k}$. Then

$$\begin{array}{cccccc}\hfill [J_i,J_j]& ={{ϵ}_{ij}}^k{J}_k,\hfill & \hfill [J_i,B_j]& ={{ϵ}_{ij}}^k{B}_k,\hfill & \hfill [B_i,B_j]& =-{{ϵ}_{ij}}^k{J}_k.\hfill \end{array}$$

(782)

The $B_i$ are hyperbolic transvections and, in the Lorentz realization, boosts; they are not the commuting spacetime translations $P^{\mu }$ supplied by the Poincaré extension [26,110].

For $m>0$, the positive sheet $p^0>0$, ${\left(p^0\right)}^2-{\left|\mathit{p}\right|}^2=m^2{c}^2$, is identified with ${\mathcal{P}}_2^1$ by

$$Q\left(p\right):=\frac{p^{0}I+\mathit{p}·\sigma }{mc},Q\left(p\right)>0,detQ\left(p\right)=\frac{{\left(p^0\right)}^2-{\left|\mathit{p}\right|}^2}{m^2{c}^2}=1.$$

(783)

The inverse is $p^0=(mc/2)TrQ$ and $p_i=(mc/2)Tr\left(Q{\sigma }_i\right)$. For a unit vector $\left|\mathit{n}\right|=1$, set $p^0=mccosh\eta$ and $\mathit{p}=mcsinh\eta \mathit{n}$; then

$$Q\left(p\right)={\mathrm{e}}^{\eta \mathit{n}·\mathbf{\sigma }},X=-logQ=-2\eta n^i{B}_i,\eta =-\frac{1}{2}Tr\left[(\mathit{n}·\sigma )X\right].$$

(784)

This is the factor of two in Equation (344). Thus X is minus the radial rapidity matrix, and the displayed scalar is its collinear principal-log rapidity [110].

Hyperbolic transvections and spacetime translations must be separated. The former are boosts and obey Equation (782); the latter commute with one another but enter nontrivially through the mixed Poincaré brackets. On a single boost orbit with coordinate $\eta$, the cotangent-lifted Liouville form is ${\mathcal{K}}_{B_{\mathit{n}}}\mathrm{d}\eta$, where ${\mathcal{K}}_{B_{\mathit{n}}}$ has action units. This is a classical boost-generator-rapidity pair, not a quantization of the entropy endpoint functional.

After a positive-mass unitary Poincaré representation and an independent ℏ are supplied, the mixed boost-translation brackets imply, by joint spectral calculus for ${\widehat{\eta }}_{\mathit{n}}=artanh({\widehat{P}}_{\mathit{n}}/{\widehat{P}}^0)$, the boost-covariant shift and hence, on the dense Weyl-smooth domain of Theorem A4,

$$U_{\mathit{n}}{\left(s\right)}^{†}{\widehat{\eta }}_{\mathit{n}}{U}_{\mathit{n}}\left(s\right)={\widehat{\eta }}_{\mathit{n}}+sI⟹[{\widehat{\eta }}_{\mathit{n}},{\widehat{\mathsf{K}}}_{\mathit{n}}]=\mathrm{i}\hslash I,\mathsf{\Delta }{\widehat{\eta }}_{\mathit{n}}\mathsf{\Delta }{\widehat{\mathsf{K}}}_{\mathit{n}}\ge \frac{\hslash }{2}.$$

(785)

The derivative is taken on the dense Weyl-smooth domain constructed in Theorem A4; the full spectral statement and proof are in Appendix H.1 [111,112,113]. The hyperbolic geometry supplies the boost action and principal-log rapidity, but the translation sector, operator realization, and action scale are independent inputs. In particular, ${\widehat{\mathsf{K}}}_{\mathit{n}}$ is not the scalar ${\mathcal{A}}_{\tau }^{\left({\kappa }_A\right)}={\kappa }_A{D}_{\mathrm{U}}(\rho \parallel \tau )$. On a separately supplied right-moving chiral sector of the longitudinal $1+1$ theory, the same affine Weyl covariance pairs the boost generator with logarithmic light-cone momentum, ${\widehat{\eta }}_{+}=log({\widehat{P}}_{+}/p_{★})$, as proved in Corollary A7; this is a boundary-sector spectral logarithm, not the determinant-one matrix logarithm on ${\mathcal{P}}_2^1$.

8.6. Quantization Obstruction and an Optional Compact Phase Sector

Proposition 34

(Exact-cotangent obstruction). The canonical cotangent lift and a globally exact statistical graph neither determine a numerical value of ℏ nor produce a nonzero cohomological action period.

By Kostant-Souriau integrality, a nonzero cyclic period group $Per\left(\omega \right)=a_0\mathbb{Z}$ would allow $\hslash =a_0/\left(2\pi k\right)$, $k\in {\mathbb{Z}}_{>0}$. Here the state manifold and its cotangent phase space are contractible and the relevant forms are exact, so $Per\left(\omega \right)=\left\{0\right\}$: every supplied ℏ is admissible and none is selected. The canonical principal quotient bundle is also topologically trivial despite nonzero curvature and full holonomy. See Appendices H.2 and H.3 and [114,115,116].

Proposition 35

(Action-time rescaling covariance). Let $a,b>0$ rescale physical time and action units, respectively. Write ${\tau }_{\mathrm{rel}}$ for any supplied relaxation time and S for any action-valued endpoint or path functional, and set

$$\begin{array}{c}{t}^{\prime }=at,h^{\prime }=ah,{\tau }_{\mathrm{rel}}^{\prime }=a{\tau }_{\mathrm{rel}},\\ p^{\prime }=bp,{\lambda }_A^{\prime }=b{\lambda }_A,{\omega }_A^{\prime }=b{\omega }_A,S^{\prime }=bS,{\kappa }_A^{\prime }=b{\kappa }_A,{\hslash }^{\prime }=b\hslash ,\\ {\mu }_{\ast }^{\prime }=ab{\mu }_{\ast },H^{\prime }=\frac{b}{a}H,{\beta }^{\prime }=\frac{a}{b}\beta .\end{array}$$

(786)

Then the dimensionless state, BKM, Born, and compact-transport geometry is unchanged, and

$${\beta }^{\prime }{H}^{\prime }=\beta H,\frac{t^{\prime }{H}^{\prime }}{{\hslash }^{\prime }}=\frac{tH}{\hslash },\frac{S^{\prime }}{{\hslash }^{\prime }}=\frac{S}{\hslash },\frac{h^{\prime }{p}^{\prime }}{{\mu }_{\ast }^{\prime }}=\frac{hp}{{\mu }_{\ast }},\frac{t^{\prime }}{{\tau }_{\mathrm{rel}}^{\prime }}=\frac{t}{{\tau }_{\mathrm{rel}}}.$$

(787)

For the reconstructed Bregman mechanics,

$$S_{h^{\prime }}^{\prime }=b{S}_h,Q_{h^{\prime }}^{\prime }(q,p^{\prime })=Q_h(q,p),{\mathcal{H}}_{h^{\prime }}^{\prime }(q,p^{\prime })=\frac{b}{a}{\mathcal{H}}_h(q,p).$$

(788)

Consequently $q^{\prime }\left(t^{\prime }\right)=q\left(t\right)$ and $p^{\prime }\left(t^{\prime }\right)=bp\left(t\right)$ obey the primed Hamilton equations whenever $\left(q\right(t),p(t\left)\right)$ obey the unprimed equations. No construction natural from only the dimensionless entropy/BKM/Born data can therefore select a positive absolute action factor b or physical-time factor a.

If an independently action-normalized symplectic sector has $Per\left(\omega \right)=a_0\mathbb{Z}$, $a_0>0$, level k prequantization fixes $\hslash =a_0/\left(2\pi k\right)$; the primitive level has $k=1$. The exact contractible cotangent branch has $Per\left(\omega \right)=\left\{0\right\}$, so that mechanism is unavailable there.

Proof. 

Every identity in Equations (787) and (788) follows by substitution into the Gibbs exponent, normalized quantum phase, endpoint generator, relaxation ratio, and Equations (722) and (724). The chain rule then transforms the two Hamilton equations by the common factor $1/a$. The period statement is exactly the primitive version of the integrality condition in Equation (A359). □

The supplied Hermitian carrier $(V_{\mathbb{R}},I_V,{\langle ·,·\rangle }_V)$ already determines the compact ray orbit and its standard projective complex structure

$$\mathbb{CP}\left(V\right)\simeq {\mathbb{CP}}^{n-1}\cong \frac{\mathrm{SU}\left(n\right)}{S\left(\mathrm{U}\right(1)\times \mathrm{U}(n-1\left)\right)}.$$

(789)

For a ray $\ell \subset V$,

$$T_{\ell }\mathbb{CP}\left(V\right)\simeq {Hom}_{\mathbb{C}}(\ell ,V/\ell ),I_{\mathrm{proj},\ell }\left(A\right)=\mathrm{i}A.$$

(790)

Thus $I_{\mathrm{proj}}$ is induced functorially from the supplied $I_V$ but is neither the endomorphism $I_V$ on $V_{\mathbb{R}}$ nor a Born structure. The ray space is the outcome space of Proposition 11. Promoting it to an action-valued phase sector is an additional identification. With primitive Fubini-Study normalization

$$\left[\frac{{\mathsf{\Omega }}_{\mathrm{FS}}}{2\pi }\right]=c_1\left(\mathcal{O}\left(1\right)\right),{\int }_{{\mathbb{CP}}^1}\hslash {\mathsf{\Omega }}_{\mathrm{FS}}=2\pi \hslash =h_{\mathrm{P}}.$$

(791)

the ray orbit realizes a nonzero period once ℏ is supplied, but does not select its absolute value. The level, polarization, and coupling of this ray sector to the mixed-state cotangent branch remain additional choices [114,117,118]. The projective conventions are summarized in Appendix H.3.

9. Relative-Entropy Reconstruction, Defect Tower, and Controlled Completion

9.1. Exact Typed Reconstruction and Defects

Theorem 25

(Typed relative-entropy reconstruction and defect tower). Fix $n\ge 2$, a supplied complex Hermitian carrier $(V_{\mathbb{R}},I_V,{\langle ·,·\rangle }_V)$ with $V\simeq {\mathbb{C}}^n$, equip $Herm\left(V\right)$ with its trace pairing, and let ${\mathcal{S}}_n^{+}$ be the manifold of faithful trace-one states. Let $D_{\mathrm{U}}$ be Umegaki relative entropy. If a channel is invoked, additionally fix an integer $m\ge 1$ and a supplied complex Hermitian output carrier $(W_{\mathbb{R}},I_W,{\langle ·,·\rangle }_W)$ with $W\simeq {\mathbb{C}}^m$, let $\mathsf{\Lambda }:{End}_{\mathbb{C}}\left(V\right)\to {End}_{\mathbb{C}}\left(W\right)$ be CPTP, and set $P_{\mathsf{\Lambda }}:=supp\mathsf{\Lambda }\left({\mathrm{id}}_V\right)$, $W_{\mathsf{\Lambda }}:=P_{\mathsf{\Lambda }}W$, and $I_{W_{\mathsf{\Lambda }}}:=I_W{|}_{{\left(W_{\mathsf{\Lambda }}\right)}_{\mathbb{R}}}$. Then the following typed tower is exact.

(R) 

Relative-entropy reconstruction and regularization. As a two-point function on the supplied trace-one affine carrier, $D_{\mathrm{U}}$ determines the entropy potential $\mathcal{F}\left(\rho \right)=Tr(\rho log\rho )$ up to affine gauge and determines the relative logarithmic cotangent class $[log\rho -log\sigma ]\in Herm\left(V\right)/\mathbb{R}{\mathrm{id}}_V$. Its diagonal second- and third-order slots are determined by $g_{\mathrm{BKM}}$ and ${\mathcal{C}}^{\left(\mathrm{m}\right)}$, with the signs, zero slots, factor of two, and Eguchi conventions stated in Proposition 5. Hence these jets determine the mixture/exponential dual pair, their Levi-Civita midpoint, and two strongly integrable Sakamoto tangent-bundle Born tuples. Their constructed complex structures act on $T\left(T{\mathcal{S}}_n^{+}\right)$ and are distinct from the supplied $I_V$; the exact shear between them is Equation (23). The Hessian associator gives $R^{\mathrm{LC}}(X,Y)=-[K_X,K_Y]$; for $n\ge 3$ the resulting BKM tangent holonomy is $\mathrm{SO}(n^2-1)$. Determinant normalization and the principal logarithm, which are choices on the supplied matrix carrier rather than consequences of the contrast alone, select the global traceless Gibbs representative.

For every faithful reference σ, the same relative logarithm is the unique traceless Fenchel source for the exact log-partition duality. Its exponential tilts form an additive partition cocycle, and the two orientations of relative entropy are complementary weighted integrals of one Kubo-Mori susceptibility.

The state-space entropic Bregman-Moreau family and the moment-space quadratic Moreau-Yosida family are two different exact continuations of this Legendre data. The former gives the Rényi resolvent and, only after a mobility and relaxation scale are supplied, BKM natural-gradient relaxation; the latter gives the moment Hopf-Lax semigroup and scale Hamilton-Jacobi/Yosida response. Their scale parameters are not identified.

(Q) 

Quotient/channel geometry and defects. The channel produces the intrinsic support-reduced Hessian quotient ${\mathcal{Q}}_{\mathsf{\Lambda }}$, and $T{\mathcal{Q}}_{\mathsf{\Lambda }}$ carries its constructed output Born tuple with $I_{B,\mathsf{\Lambda }}^{\mathrm{out}}\in \mathsf{\Gamma }(End\left(T\left(T{\mathcal{Q}}_{\mathsf{\Lambda }}\right)\right))$. This endomorphism is not the inherited supplied scalar multiplication $I_{W_{\mathsf{\Lambda }}}$ on the reduced output amplitude carrier. With

$${\mathcal{H}}_{\mathsf{\Lambda }}:=\mathcal{F}-{\mathcal{F}}_{\mathsf{\Lambda }}\circ \mathsf{\Lambda },{\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma ):=D_{\mathrm{U}}(\rho \parallel \sigma )-D_{\mathrm{U}}(\mathsf{\Lambda }\rho \parallel \mathsf{\Lambda }\sigma ),$$

(792)

one has

$$\begin{array}{c}{\mathsf{\Delta }}_{\mathsf{\Lambda }}=B_{{\mathcal{H}}_{\mathsf{\Lambda }}},g_{\mathsf{\Lambda }}^{\mathrm{loss}}={Hess}_{{\nabla }^{\left(\mathrm{m}\right)}}{\mathcal{H}}_{\mathsf{\Lambda }}=g_{\mathrm{BKM}}^{\mathrm{in}}-{\mathsf{\Lambda }}^{\ast }{g}_{\mathrm{BKM}}^{\mathrm{out}}⪰0,{\mathcal{C}}_{\mathsf{\Lambda }}^{\mathrm{loss}}={\left({\nabla }^{\left(\mathrm{m}\right)}\right)}^3{\mathcal{H}}_{\mathsf{\Lambda }}\\ ={\mathcal{C}}^{\mathrm{in},\left(\mathrm{m}\right)}-{\mathsf{\Lambda }}^{\ast }{\mathcal{C}}^{\mathrm{out},\left(\mathrm{m}\right)}.\end{array}$$

(793)

For fixed composable affine channels these potentials, global defects, and their jets of every fixed order $r\ge 2$ obey the exact additive contravariant cocycle of Corollary 13. For faithful $\rho ,\sigma$, ${\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma )=0$ exactly when the fixed-reference Petz map ${\mathcal{R}}_{\sigma ,\mathsf{\Lambda }}$ recovers both ρ and σ. On a smooth embedded faithful model $\mathcal{S}$, with $\mathcal{N}=\mathsf{\Lambda }\left(\mathcal{S}\right)$ embedded, preservation of the full intrinsic contrast, connections, curvature, transport, and holonomy follows if one CPTP map $\mathcal{R}$ satisfies ${\mathcal{R}\mathsf{\Lambda }|}_{\mathcal{S}}={\mathrm{id}}_{\mathcal{S}}$. Although the loss cubic is symmetric, it has no positivity property, and $g_{\mathsf{\Lambda }}^{\mathrm{loss}}$ may be degenerate. On the input manifold it defines a Hessian/Born geometry only on a declared open region where $g_{\mathsf{\Lambda }}^{\mathrm{loss}}>0$. Descent to a quotient would instead require separate constant-rank, integrability, and projectability hypotheses and a declared carrier; even then, no bare input-minus-output curvature identity follows.

The partition/proximal and channel branches are joined by exact defects: the output-versus-pulled-back partition deficit is the sum of finite mirror-intertwining error and DPI loss; its Hessian at zero source is the squared norm of the direct-sum defect comprising infinitesimal mirror-response mismatch and tangent BKM loss; and, for $0<t<1$, the balanced Rényi-Moreau channel deficit is the stated weighted sum of two endpoint DPI losses and finite bridge-intertwining error.

(C) 

Compact comparison. Supplying the determinant-one positive cone with its affine-invariant metric gives the canonical principal $\mathrm{SU}\left(n\right)$ connection and the associated $\mathrm{PSU}\left(n\right)$ tangent connection, with their stated full holonomies. Positive metric normalization J compares the tangent connection with the BKM Levi-Civita connection by an exact relative transporter, connection-difference $End\left(T\right)$-valued one-form, and curvature defect. This is a comparison, not a conjugacy of the BKM and affine-invariant curvatures. Through the global Gibbs chart, the normalized comparison pulls back naturally to faithful state space and retains principal $\mathrm{SU}\left(n\right)$ and tangent $\mathrm{PSU}\left(n\right)$ holonomy. If $n=3$, then at the tracial qutrit state the oriented BKM metric-cubic pair identifies the embedded $Ad\left(\mathrm{SU}\right(3\left)\right)$ subgroup, while the BKM curvature resolves the pointwise split $\mathfrak{so}\left(8\right)=ad\mathfrak{su}\left(3\right)\oplus {\mathfrak{m}}_{20}$. A chosen nonzero ${\widehat{D}}^{\mathrm{AI}}$-parallel Cartan form supplies the global reduction of the oriented BKM frame bundle. After central holonomy is aligned with the embedded subgroup, the resulting parallel comparison cubic, not the state-dependent BKM cubic away from the center, defines the same reduction. Its adapted tangent connection has $\mathrm{PSU}\left(3\right)$ holonomy, and only the retained canonical Cartan principal lift has $\mathrm{SU}\left(3\right)$ holonomy. No qutrit reduction is asserted by this clause when $n\ne 3$.

(H) 

Hamiltonian endpoint reconstruction. A supplied fixed conversion ${\kappa }_A>0$ makes ${\kappa }_A{D}_{\mathrm{U}}$ a regular Type-I/II endpoint generating family whose endpoint derivatives define an exact Lagrangian cotangent relation. This variational statement is canonical at the endpoint-family level; it is not yet a time-parametrized mechanical flow. Separately, any $C^3$ endpoint family with nondegenerate mixed Hessian and a selected common right-Legendre product chart determines a branch-local, generally nonautonomous scale-Hamiltonian isotopy. The ${\kappa }_A$-scaled regular dual-Jensen/Rényi-Moreau family analyzed here is one concrete instance: its centroid/envelope core is standard, whereas the normalized endpoint-Hamiltonian realization and verified domain hypotheses are the manuscript-specific step. Interpreting the h-family mechanically and comparing it with an autonomous variational target require the separate Hessian-domain, inertia, and clock data $(M,\psi ,{\mu }_{\ast },h)$, with ${\kappa }_h={\mu }_{\ast }/h$. No theorem identifies ${\kappa }_A$ with ${\kappa }_h$.

Across branches(R),(C), and(H), shifted LogDet and Cartan spectral log-Laplace formulas supply exact resolvent-average and center/spectral-carrier bridges. The endpoint functionals have the stated quadratic contact and local proxy relations, but are not globally equal and these formulas establish no conjugacy between their Hamiltonian flows.

These interfaces identify neither $I_V$ nor $I_{W_{\mathsf{\Lambda }}}$ with a constructed Born endomorphism. They compare the stated structures through typed maps and defects without deriving their supplied carriers or physical calibrations.

Proof. 

The input/Born, quotient, and loss/cocycle statements reuse Theorem 10, with $W=V$ and $\mathsf{\Lambda }=\mathrm{id}$ when no channel is invoked. The entropy-jet and remaining (R) statements are Propositions 5 and 7, Theorems 2, 3, 6 and 7, and Corollaries 1, 27 and 41. The additional response and recovery statements in (Q) follow from Propositions 9 and 10, Theorem 9, and Corollaries 14–16; the stated limitations are part of the hypotheses and conclusions of those results. The general-n part of Clause (C) is 12–14, Corollaries 24 and 25, and Proposition 16; when $n=3$, its qutrit specialization is Theorems 15 and 17. Clause (H) is Corollaries 45 and 49, Theorems 22–24, Lemma 4, and Remark 19. The cross-branch comparison is Theorems 11 and 19, Corollary 22, and Proposition 27. □

9.2. Controlled-Completion Audit

Theorem 26

(Dimension-stratified controlled completion of the indexed Umegaki reconstruction). For each integer $n\ge 2$, use the supplied complex Hermitian carrier ${\mathcal{H}}_n\simeq {\mathbb{C}}^n$, with $I_{{\mathcal{H}}_n}v=\mathrm{i}v$, trace pairing, faithful trace-one domain ${\mathcal{S}}_n^{+}$, and Umegaki contrast in the base tuple

$${\mathfrak{I}}_0^{\left(n\right)}=(n,{\mathcal{H}}_{n,\mathbb{R}},I_{{\mathcal{H}}_n},{\langle ·,·\rangle }_n,Tr,{\mathcal{S}}_n^{+},D_{\mathrm{U}}),J\left(n\right)=J_{\mathrm{R}}\left(n\right)\bigsqcup J_{\mathrm{Q}}\left(n\right)\bigsqcup J_{\mathrm{C}}\left(n\right)\bigsqcup J_{\mathrm{H}}\left(n\right).$$

(794)

The branch labels are those of Equation (1). Split the dimension-sensitive parts by

$$\begin{array}{cc}\hfill J_{\mathrm{R}}\left(n\right)& =J_{\mathrm{R}}^{\mathrm{gen}}\left(n\right)\bigsqcup J_{\mathrm{R}}^{\mathrm{hol}}\left(n\right),\hfill \\ \hfill J_{\mathrm{C}}\left(n\right)& =J_{\mathrm{C}}^{\mathrm{gen}}\left(n\right)\bigsqcup J_{\mathrm{C}}^{\mathrm{cub}}\left(n\right)\bigsqcup J_{\mathrm{C}}^{\left(3\right)}\left(n\right)\bigsqcup J_{\mathrm{C}}^{\left(5\right)}\left(n\right),\hfill \\ \hfill J_{\mathrm{H}}\left(n\right)& =J_{\mathrm{H}}^{\mathrm{gen}}\left(n\right)\bigsqcup J_{\mathrm{H}}^{\left(3\right)}\left(n\right),\hfill \end{array}$$

(795)

where

$$\begin{array}{cc}\hfill J_{\mathrm{R}}^{\mathrm{hol}}\left(n\right)& =\left\{\begin{array}{cc}\left\{j_{\mathrm{BKMhol}}^{\left(n\right)}\right\},\hfill & n\ge 3,\hfill \\ ⌀,\hfill & n=2,\hfill \end{array}\right.\hfill \\ \hfill J_{\mathrm{C}}^{\mathrm{cub}}\left(n\right)& =\left\{\begin{array}{cc}\left\{j_{\mathrm{cubstab}}^{\left(n\right)}\right\},\hfill & n\ge 3,\hfill \\ ⌀,\hfill & n=2,\hfill \end{array}\right.\hfill \\ \hfill J_{\mathrm{C}}^{\left(3\right)}\left(n\right)& =\left\{\begin{array}{cc}{J}_{\mathrm{C}}^{\mathrm{qut}},\hfill & n=3,\hfill \\ ⌀,\hfill & n\ne 3,\hfill \end{array}\right.J_{\mathrm{C}}^{\left(5\right)}\left(n\right)=\left\{\begin{array}{cc}{J}_{\mathrm{C}}^{\mathrm{AIII}},\hfill & n=5,\hfill \\ ⌀,\hfill & n\ne 5.\hfill \end{array}\right.\hfill \\ \hfill J_{\mathrm{H}}^{\left(3\right)}\left(n\right)& =\left\{\begin{array}{cc}{J}_{\mathrm{H}}^{\mathrm{qut}},\hfill & n=3,\hfill \\ ⌀,\hfill & n\ne 3.\hfill \end{array}\right.\hfill \end{array}$$

(796)

Thus $J\left(n\right)$ contains only signatures whose dimension predicates are satisfied. Alternative or conditional clauses are separate indices, and each index denotes one distinct displayed hypothesis signature.

Concretely, $J_{\mathrm{R}}^{\mathrm{gen}}\left(n\right)$ indexes the entropy-jet, Gibbs/Born, partition, and two proximal constructions in Propositions 2, 5 and 7, Corollaries 1–3 and 41, Remark 1, and Theorems 2, 3, 6 and 7; $J_{\mathrm{R}}^{\mathrm{hol}}\left(n\right)$ contains the signature of Corollary 27 precisely when $n\ge 3$. $J_{\mathrm{Q}}\left(n\right)$ indexes the channel, cocycle, recovery, and measurement signatures in Theorems 8 and 9, Corollaries 12–17, and Propositions 9–11. $J_{\mathrm{C}}^{\mathrm{gen}}\left(n\right)$ indexes the general-n affine-invariant, normalized-comparison, shifted, Cartan, and Jordan signatures in Theorems 11–14, 16 and 19, Corollaries 24 and 25, and Propositions 16 and 25, with only the general-n clauses of the last theorem included there. The cubic-stabilizer set $J_{\mathrm{C}}^{\mathrm{cub}}\left(n\right)$ contains the signature of Proposition 22 precisely when $n\ge 3$. The qutrit set $J_{\mathrm{C}}^{\mathrm{qut}}$ indexes Theorems 15–18, Corollary 32, and Remark 12, using only the $n=3$ clauses of the Jordan theorem. The five-dimensional AIII set $J_{\mathrm{C}}^{\mathrm{AIII}}$ indexes Theorems 20 and 21 and Corollaries 37–39. Finally, $J_{\mathrm{H}}^{\mathrm{gen}}\left(n\right)$ indexes the fixed endpoint, scale-Hamiltonian, mechanical, potential, error, and modular signatures in Corollaries 22, 45, 47, 49 and 52, Lemma 4, Theorems 22–24, and Propositions 8, 27 and 35. The qutrit endpoint set $J_{\mathrm{H}}^{\mathrm{qut}}$ indexes Corollary 48.

For each $j\in J\left(n\right)$, let ${\mathfrak{a}}_j^{\left(n\right)}$ be the exact tuple of raw auxiliary data and previously constructed typed carriers required beyond ${\mathfrak{I}}_0^{\left(n\right)}$ by that signature; let $F_j^{\left(n\right)}$ be its displayed construction, composed along the preceding dependency arrows when necessary; and let ${\mathfrak{O}}_j^{\left(n\right)}$ be its exact displayed output tuple on the stated carrier. Because every $J\left(n\right)$ indexes a finite cited list, it is finite. The following grouped inventory summarizes the additional inputs; each exact tuple is the displayed hypothesis signature of its cited result, and a conclusion is omitted when its dimension or auxiliary signature is absent.

(R) 

Relative-entropy reconstruction and regularization. Branch(R)of Theorem 25 needs no auxiliary datum beyond ${\mathfrak{I}}_0^{\left(n\right)}$ for its entropy-jet and global Gibbs/Born conclusions. The supplied $I_{{\mathcal{H}}_n}$ remains input; the mixture and exponential Born complex structures on the doubled statistical carrier are outputs. The full BKM tangent-holonomy index is active only for $n\ge 3$; no such conclusion is asserted here at $n=2$. The partition/Fenchel and response signatures require a faithful reference σ. The state-space proximal extension requires a faithful center σ and $r>0$; interpreting the log-parameter as physical time additionally requires a supplied BKM mobility law and $\tau >0$. The alternative moment-space extension requires a faithful reference σ, an integer $1\le d\le n^2-1$, Hermitian observables ${\left(O_a\right)}_{a=1}^d$ minimal modulo the identity, equivalently a full-rank centered Gram matrix, and $\epsilon >0$; its Gram matrix is induced by the trace pairing. These are distinct families and their scales are not identified (Theorems 6 and 7, Corollary 41, and Proposition 7).

(Q) 

Quotient and channel geometry. The channel-dependent signatures in Branch(Q)require an integer $m\ge 1$, a supplied complex Hermitian output carrier $(W_{\mathbb{R}},I_W,{\langle ·,·\rangle }_W)$ with $W\simeq {\mathbb{C}}^m$, and a CPTP map $\mathsf{\Lambda }:{End}_{\mathbb{C}}\left({\mathcal{H}}_n\right)\to {End}_{\mathbb{C}}\left(W\right)$. Put

$$\begin{array}{cccc}\hfill P_{\mathsf{\Lambda }}& :=supp\mathsf{\Lambda }\left({\mathrm{id}}_{{\mathcal{H}}_n}\right),\hfill & \hfill W_{\mathsf{\Lambda }}& :=P_{\mathsf{\Lambda }}W,\hfill \\ \hfill r_{\mathsf{\Lambda }}& :={dim}_{\mathbb{C}}{W}_{\mathsf{\Lambda }}=rank{P}_{\mathsf{\Lambda }},\hfill & \hfill L_{\mathsf{\Lambda }}& {:=\mathsf{\Lambda }|}_{{Herm}_0\left({\mathcal{H}}_n\right)},\hfill \\ \hfill q_{\mathsf{\Lambda }}& :={rank}_{\mathbb{R}}{L}_{\mathsf{\Lambda }}={dim}_{\mathbb{R}}{\mathcal{Q}}_{\mathsf{\Lambda }}.\hfill \end{array}$$

(797)

Then

$$1\le r_{\mathsf{\Lambda }}\le m,0\le q_{\mathsf{\Lambda }}\le min\{n^2-1,r_{\mathsf{\Lambda }}^2-1\}.$$

(798)

For every finite-dimensional channel signature, the output logarithms and tensors live on $W_{\mathsf{\Lambda }}\simeq {\mathbb{C}}^{r_{\mathsf{\Lambda }}}$, which inherits the supplied $I_{W_{\mathsf{\Lambda }}}:=I_W{|}_{{\left(W_{\mathsf{\Lambda }}\right)}_{\mathbb{R}}}$. The intrinsic quotient has real dimension $q_{\mathsf{\Lambda }}$, while its Born carrier is the $2q_{\mathsf{\Lambda }}$-dimensional manifold $T{\mathcal{Q}}_{\mathsf{\Lambda }}$; the constructed $I_{B,\mathsf{\Lambda }}^{\mathrm{out}}\in \mathsf{\Gamma }(End\left(T\left(T{\mathcal{Q}}_{\mathsf{\Lambda }}\right)\right))$ acts on its tangent bundle and is not $I_{W_{\mathsf{\Lambda }}}$. The case $q_{\mathsf{\Lambda }}=0$ is the legitimate point quotient with its trivial zero-dimensional Born tuple; a nontrivial output geometry requires the additional condition $q_{\mathsf{\Lambda }}>0$. The pointwise loss amplitude is always defined. Its range is indexed as a smooth defect subbundle only on a declared stratum where its rank is locally constant. A positive-definite loss Hessian/Born geometry is instead indexed on an open set where $g_{\mathsf{\Lambda }}^{\mathrm{loss}}>0$. A quotient loss geometry is a third, alternative signature requiring constant rank of the radical, integrability, projectability, and a declared quotient carrier. A second composable channel is additional input only for the cocycle, and a finite POVM is optional measurement data. The continuous covariant ray readout is a separate continuous-outcome signature and is not assigned the finite-output rank $r_{\mathsf{\Lambda }}$. Model-wide preservation further requires a specified smooth embedded faithful model, an embedded output image, and one CPTP recovery map that is a common left inverse there. Intertwining the Born tuples additionally requires the stated mixture-affine Hessian hypotheses; no general noninjective channel identifies the ambient input and output Born manifolds.

(C) 

Compact comparison and dimension-specific reductions. The general affine-invariant comparison signatures are defined for every $n\ge 2$ and require a scale $\alpha >0$. Their central BKM identity calibration is $\alpha =1/n$; $\alpha =1/3$ is only the qutrit specialization. The central spectral bridge also chooses nonzero $Z\in \mathfrak{p}$. The separate cubic-stabilizer signature active for $n\ge 3$ uses only the central BKM metric and cubic reconstructed in Branch(R) (and the stated orientation for its $\mathrm{SO}$ stabilizer); it requires no affine-invariant scale. The entropy cubic itself is not qutrit-specific: it is defined for every $n\ge 2$, its central d-symbol vanishes for $n=2$ and is nonzero for $n\ge 3$. For $n\ge 3$, its general central metric-cubic stabilizer is given by Proposition 22. What is qutrit-specific below is its determinant form, exact oriented $\mathrm{PSU}\left(3\right)$ stabilizer, and the $\mathfrak{so}\left(8\right)={\mathfrak{h}}_8\oplus {\mathfrak{m}}_{20}$ consequences, where ${\mathfrak{h}}_8=ad\mathfrak{su}\left(3\right)$ and the summands have representation dimensions 8 and 20.

If $n=3$, the qutrit index is active. The chosen nonzero ${\widehat{D}}^{\mathrm{AI}}$-parallel Cartan form defines the oriented $\mathrm{PSU}\left(3\right)$ tangent reduction. Its sign has the same stabilizer and holonomy type; for a fixed model form ${\phi }_0$ one has the distinct, canonically isomorphic central translate $P_{-\phi }=P_{\phi }·(-\mathrm{id})$, while simultaneously replacing $(\phi ,{\phi }_0)$ by $(-\phi ,-{\phi }_0)$ leaves the frame-reduction set unchanged. In an aligned central adjoint frame, the oriented tracial BKM metric-Jordan-cubic pair and the aligned Cartan form have the common stabilizer $\mathrm{PSU}\left(3\right)$; they are fixed by the full normalized affine-invariant holonomy. The cubic therefore seeds the unique parallel comparison cubic ${\mathcal{C}}^J$ on that reduction. In the tracial fiber the oriented metric-cubic data determine the subgroup, split, and $J_{\perp }^{\mathrm{or}}$ through the invariant Cartan line. The supplied parallel reduction globalizes this complex structure; no principal lift is needed. Identifying its $+\mathrm{i}$-eigenspace with ${Sym}^3{\mathcal{H}}_3$ or its conjugate uses a specified defining-triplet identification and the orientation/sign convention of Remark 12. This representation labeling is not used by the curvature or holonomy results; $J_{\perp }^{\mathrm{or}}$ acts on the 20-dimensional value bundle, not the amplitude carrier or Born tangent bundle. The reduction and holonomy signatures allow arbitrary $\alpha >0$, whereas the displayed coefficient-exact central curvature split uses the matched calibration $\alpha =1/3$. The principal $\mathrm{SU}\left(3\right)$ conclusion additionally requires retaining the canonical Cartan lift $\tilde{P}\to {Herm}_0\left(3\right)$ pulled back along $X\mapsto P_X$; via the Gibbs chart this is the corresponding lift over ${\mathcal{S}}_3^{+}$. The projected tangent connection then has the uniquely induced principal lift $A^{\mathrm{col}}$. For this $n=3$ construction, Theorem 17 and Corollary 32 distinguish BKM $\mathrm{SO}\left(8\right)$, adapted tangent $\mathrm{PSU}\left(3\right)$, and lifted principal $\mathrm{SU}\left(3\right)$ holonomy.

If $n=5$, the AIII index is active only after independently supplying an orthogonal grading ${\mathcal{H}}_5=V_c\oplus V_w$ with $({dim}_{\mathbb{C}}{V}_c,{dim}_{\mathbb{C}}{V}_w)=(3,2)$. The summands inherit $I_{{\mathcal{H}}_5}$; the complex-linear grading involution ${\mathsf{\Gamma }}_{3,2}:=P_c-P_w$ is not a complex structure. It induces the distinct integrable AIII tangent structure $I_{\mathrm{AIII}}$ of Equation (592). This grading gives the AIII holonomy, its global form, and the functorial exterior module. The four-dimensional anomaly clause additionally requires the stated assignment to left-handed Weyl fields; the coupling clause instead requires a supplied spacetime gauge action and the single carrier-trace kinetic hypothesis. In particular, neither the value $n=5$ nor the BKM cubic selects this grading: at the tracial five-level state the oriented central metric-cubic stabilizer is $\mathrm{PSU}\left(5\right)⋊{\mathbb{Z}}_2$ in $\mathrm{SO}({Herm}_0\left(5\right),g_0)$. After a grading is supplied, the tangent image $Ad\left(K_{\mathrm{SM}}\right)\simeq K_{\mathrm{SM}}/{\mathbb{Z}}_5$ is a proper subgroup of $\mathrm{PSU}\left(5\right)$; the central pair selects neither that subgroup nor the grading. Under the supplied grading, the real 12-dimensional AIII base has principal holonomy $K_{\mathrm{SM}}$, tangent holonomy isomorphic to $K_{\mathrm{SM}}/{\mathbb{Z}}_5$, and a complex 16-dimensional even-exterior associated bundle with holonomy isomorphic to $K_{\mathrm{SM}}$. These are the principal, isotropy, and even-exterior transports induced by $A_{\mathrm{SM}}$ in Equation (594), Theorem 21, and Remark 16; full and restricted holonomy agree in each case. The qutrit and AIII packages therefore occupy different dimension fibers. An abstract group isomorphism needs no carrier data. A carrier-level identification between the defining triplet of the retained qutrit principal $\mathrm{SU}\left(3\right)$ lift and the AIII color triplet $V_c$ requires a supplied unitary ${\mathcal{H}}_3\to V_c$; the tangent $\mathrm{PSU}\left(3\right)$ reduction alone has no fundamental triplet. Identifying principal bundles and connections additionally requires a base map and a connection-preserving principal-bundle intertwiner. None of these cross-dimensional data is supplied here.

(H) 

Hamiltonian endpoint reconstruction. The general endpoint-characteristics signature requires $S\in C^3(I\times U_0\times U_1)$, equal finite dimensions $dim{U}_0=dim{U}_1$, nondegenerate mixed endpoint Hessian, and a selected $C^2$ product right-Legendre chart $\mathcal{E}\simeq I\times \mathcal{D}$; its flow is branch-local. The fixed Umegaki and Rényi-Moreau endpoint signatures on ${\mathcal{S}}_n^{+}$ require an action calibration ${\kappa }_A>0$. The generic dual-Jensen signature also requires its stated Legendre-Hessian model; the Umegaki specialization derives that model from ${\mathfrak{I}}_0^{\left(n\right)}$. The generic discrete signature separately requires a finite-dimensional open convex affine domain M, a strictly convex potential ψ with positive-definite Hessian, and a Legendre map $\mathrm{d}\psi :M\to \mathrm{d}\psi \left(M\right)$ that is a diffeomorphism onto an open convex dual domain; it also requires a step $\mathsf{\Delta }t>0$, a coefficient ${\kappa }_{\mathsf{\Delta }t}>0$ for which ${\kappa }_{\mathsf{\Delta }t}{B}_{\psi }$ has action units, and an energy-valued $V\in C^2\left(M\right)$. The LogDet Type-II signature separately requires $(P_0,\mathsf{\Pi },\kappa )$ in its strict positivity domain; free affine-invariant mechanics requires $({\mu }_{\mathrm{AI}},h)$. The Bregman mechanical signatures require $(M,\psi ,{\mu }_{\ast },h)$, with $V\in C^3\left(M\right)$ for the potential-bearing clause, and the stated regularity, no-caustic, compact-inverse, and common-compact-phase-tube hypotheses. If $n=3$, the separately indexed qutrit bridge-potential specialization additionally requires ${\kappa }_{\mathrm{br}}>0$ and uses $V={\kappa }_{\mathrm{br}}{\mathcal{V}}_{\mathrm{br}}$; its eight-mode equilibrium frequency is the output stated in Corollary 48. No qutrit bridge endpoint is indexed when $n\ne 3$. No theorem identifies these coefficients unless an indexed result explicitly imposes a specialization; in particular, the mechanical comparison sets ${\kappa }_{\mathsf{\Delta }t}={\kappa }_h={\mu }_{\ast }/h$. No theorem identifies ${\kappa }_A$ with ${\kappa }_h$ or supplies a universal equality among the calibrations. Modular and Bisognano-Wichmann readings require the corresponding faithful Gibbs or regulated modular data, while any phase normalization ℏ is external.

For every fixed $n\ge 2$, the active partial maps

$$F_j^{\left(n\right)}:({\mathfrak{I}}_0^{\left(n\right)},{\mathfrak{a}}_j^{\left(n\right)})⟶{\mathfrak{O}}_j^{\left(n\right)},j\in J\left(n\right),$$

(799)

form a controlled completion in the sense of 1. The collection over n is a dimension-stratified family of finite controlled completions, not one controlled completion of a single fixed carrier. In particular, no branch supplies an absent clock, action unit, measurement instrument, spacetime dynamics, any principal lift beyond the explicitly constructed canonical Cartan lift, matter representation, cross-dimensional intertwiner, or physical complex-bundle map Ξ of Equation (25) identifying a supplied amplitude structure with a constructed Born structure. Every asserted relation between active branches is one of the proved identities, inequalities, defect formulas, or obstructions in Theorem 25 or the indexed signature-level results.

Proof. 

Fix $n\ge 2$. The case definitions in Equation (796) remove every signature whose dimension predicate fails: the full BKM tangent-holonomy and general central metric-cubic stabilizer results are present only for $n\ge 3$, the qutrit package only for $n=3$, and the AIII package only for $n=5$. Splitting Theorem 16 into its general-n and qutrit signatures likewise prevents one theorem label from hiding two domains. The channel ranks in Equation (797) distinguish the reduced Hilbert carrier from the quotient tangent dimension, including the point quotient.

Each active indexed statement explicitly gives its carrier, hypothesis domain, and output codomain, proving (C1); the dimension-filtered tuples above and the grouped inventory give (C2). The relations and nonidentifications are those collected fiberwise in Theorem 25 and in the active signature-level results, giving (C3). The final exclusions, together with Remark 19, give (C4). This is therefore dimension-stratified input-and-defect completeness, not a universal property, a uniqueness assertion, or a cross-dimensional identification. □

9.3. Post-Tower Conditional Applications

The indexed controlled completion stops at the finite-dimensional (R/Q/C/H) signatures above. The positive-mass operator theorem Theorem A4 and its chiral-null analogue Corollary A7 are separate conditional comparisons. The former requires a supplied positive-mass representation of the full Poincaré group, whereas the latter requires a supplied nonzero right-moving representation of the longitudinal $1+1$ Poincaré group. Both require the appropriate translation and boost generators, operator-domain hypotheses, and an independent ℏ. Only the massive theorem’s Pauli principal-log comparison uses $n=2$. Two further external geometric groups use selected outputs of that tower only after additional data are supplied. The spacetime-color group of Proposition A3 has the three-stage bundle/connection, Euclidean Yang-Mills, and reduction-field inputs summarized in Section 6.7.8. The null-screen/local-horizon group Proposition A4, Theorem A2, and Corollary A5 separately requires a Lorentzian spacetime with the congruence/screen or shrinking-horizon data stated in those results; affine-null, boost, and Unruh normalizations; cutoff faithful KMS states; stress, area, and flux data; and an independent entropy-area coefficient ${\eta }_{\mathrm{hor}}>0$. The shear corollary further specializes to four-dimensional pure general relativity. Einstein closure assumes vanishing of the complete scaled defect for every null direction at every point and stress-energy conservation; the identification ${\eta }_{\mathrm{hor}}={(4\hslash G_{\mathrm{N}})}^{-1}$ is optional supplied calibration. These groups are conditional applications of selected tower outputs, not inputs to or outputs of Theorem 26.

10. Discussion, Limitations, and Outlook

The synthesis is an input-audited interface architecture, not a derivation of every structure from entropy (Theorems 25 and 26). Relative logarithms serve as Gibbs-coordinate differences, partition sources, and BKM forces, but away from a central reference they are not affine-invariant Riemannian logarithms. The shifted-LogDet and curvature bridges retain trace-normalization defects; endpoint actions are related only by the proved center/spectral and local-proxy formulas (Theorems 11, 12 and 19 and Proposition 27). Likewise, the output mixture-Born tuple is intrinsic to its Hessian quotient: a Hessian isomorphism intertwines it with an input tuple, but a general noninjective channel supplies neither that identification nor ambient descent (Remark 7).

Dimension-uniform backbone and dimension-specific fibers.

For each fixed finite $n\ge 2$, the general signatures give the common backbone under their declared auxiliary inputs. The activation rules in Equation (796) specify its enhancements: these are distinct dimension fibers, not nested constructions on one dimension-free carrier or a universal-property claim. Independently, Equations (797) and (798) distinguish the reduced Hilbert rank $r_{\mathsf{\Lambda }}$ from the accessible real dimension $q_{\mathsf{\Lambda }}$. The output Born carrier has real dimension $2q_{\mathsf{\Lambda }}$, including the legitimate point quotient at $q_{\mathsf{\Lambda }}=0$; output rank alone does not activate the full-state qutrit or AIII signatures.

At $n=2$, ${\mathcal{P}}_2^1$ is hyperbolic three-space with curvature $-1/\left(2\alpha \right)$, hence $-1$ at $\alpha =1/2$. The general backbone gives the Pauli log-cosh/Legendre dictionary and Thomas-Wigner compact residue; the central leading small-loop normalized-relative rotation is the inverse residue (Corollaries 30 and 34). These do not identify qubit BKM holonomy: its central cubic and curvature vanish. The separate massive and chiral-null operator results have no matrix-size hypothesis; they require supplied full or longitudinal Poincaré representations, generators, domains, and ℏ. Only the massive collinear Pauli-log comparison uses $n=2$ (Section 8.5).

The nonzero central cubic and full-BKM-holonomy enhancements start at $n=3$. The qutrit specialization has the determinant/d-symbol cubic, oriented metric-cubic stabilizer $\mathrm{PSU}\left(3\right)$, and central $\mathfrak{so}\left(8\right)={\mathfrak{h}}_8\oplus {\mathfrak{m}}_{20}$ resolution, where ${\mathfrak{h}}_8=ad\mathfrak{su}\left(3\right)$. Global reduction still requires the independently supplied affine-invariant metric and its resulting normalized connection, a parallel Cartan form, and central alignment; principal $\mathrm{SU}\left(3\right)$ also retains the Cartan lift. At $n=5$, an independent orthogonal complex-linear $3+2$ grading yields principal $K_{\mathrm{SM}}=\mathrm{S}(\mathrm{U}\left(3\right)\times \mathrm{U}\left(2\right))$, its $(\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right))/{\mathbb{Z}}_6$ global form, and the even-exterior associated module. The general central metric-cubic stabilizer $\mathrm{PSU}\left(5\right)⋊{\mathbb{Z}}_2$ does not select that grading. Nor is the qutrit principal $\mathrm{SU}\left(3\right)$/tangent $\mathrm{PSU}\left(3\right)$ canonically identified with the AIII color factor: no cross-dimensional carrier or connection intertwiner is supplied (Theorem 26).

Global-to-local-to-holonomy resolution.

On the supplied trace-one Hermitian carrier, equipped with its mixture-affine structure, the full smooth contrast determines the hierarchy

$$\mathfrak{D}(\rho ,\sigma ):=D_{\mathrm{U}}(\rho \parallel \sigma )⇝[log\rho -log\sigma ]⇝g_{\mathrm{BKM}}⇝{\mathcal{C}}^{\left(\mathrm{m}\right)}⇝R^D⇝{\mathfrak{hol}}_{{\rho }_{\ast }}\left(D\right)⇝{Hol}_{{\rho }_{\ast }}\left(D\right),$$

(800)

where $D={\nabla }_{g_{\mathrm{BKM}}}^{\mathrm{LC}}$ and ${\rho }_{\ast }=I/n$. The arrows are cumulative differential and transport operations, not injective maps or reconstruction from one scalar value. First variation gives the logarithmic cotangent class modulo $\mathbb{R}I$; diagonal second and pure first-argument third jets give $g_{\mathrm{BKM}}$ and ${\mathcal{C}}^{\left(\mathrm{m}\right)}={\nabla }^{\left(\mathrm{m}\right)}{g}_{\mathrm{BKM}}$ (Proposition 5). Holonomy generally requires the transported curvature field through Ambrose-Singer, not curvature in one fiber (Proposition 3).

For a fixed CPTP channel on its support-reduced output carrier, ${\mathcal{H}}_{\mathsf{\Lambda }}=\mathcal{F}-{\mathcal{F}}_{\mathsf{\Lambda }}\circ \mathsf{\Lambda }$ organizes the same response levels: its Bregman defect is finite information loss, its Hessian is BKM metric loss, and its third derivative is the signed cubic loss. The weighted-chord identity makes finite loss an exact accumulation of local metric loss (Corollary 12). Fixed affine channels on stagewise fixed supports give additive pullback cocycles for the potentials, endpoint defects, and all jets of order $r\ge 2$ (Corollary 13). Unlike the Bregman defect and Hessian, the potential itself and jets of order $r\ge 3$ have no general positivity. These are not curvature or holonomy subtraction laws: curvature is nonlinear, and a loss geometry requires a positive restriction or a separately justified smooth quotient.

Common recovery as zero defect for the intrinsic geometric stack.

Let $\mathcal{S}\subset {\mathcal{S}}_n^{+}$ and $\mathcal{N}=\mathsf{\Lambda }\left(\mathcal{S}\right)$ be smooth embedded faithful models, the latter on the fixed output support. One CPTP recovery map $\mathcal{R}$ on the reduced output algebra with ${\mathcal{R}\mathsf{\Lambda }|}_{\mathcal{S}}={\mathrm{id}}_{\mathcal{S}}$ makes ${f=\mathsf{\Lambda }|}_{\mathcal{S}}$ a contrast-preserving diffeomorphism. The restricted contrast-loss and its jets vanish; by Proposition 10, f preserves the intrinsic metric, cubic, dual connections and Levi-Civita midpoint, hence curvature and parallel transport, and conjugates both restricted and full holonomy. Exact model-wide recovery therefore preserves the intrinsic contrast geometry, not merely the states.

Pair-dependent recovery is insufficient, but ${\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,{\sigma }_0)=0$ for every $\rho \in \mathcal{S}$ at one fixed ${\sigma }_0\in \mathcal{S}$ supplies a common Petz map (Corollary 14). These are not ambient connection restrictions without autoparallel hypotheses, nor assertions of recovery of off-model directions or a channel kernel. For mixture-affine Hessian models and affine f, $Tf$ additionally intertwines their intrinsic mixture and exponential Sakamoto Born tuples (1). Supplied compact geometry, amplitude complex structures, and calibrated actions require further compatibility.

Why dual flatness and non-Abelian holonomy coexist.

Set $D_S:=D$ and $K=({\nabla }^{\left(\mathrm{m}\right)}-{\nabla }^{\left(\mathrm{e}\right)})/2$. The globally flat, torsion-free, $g_{\mathrm{BKM}}$-dual connections $D_S\pm K$ have a potentially curved midpoint: ${\mathrm{d}}^{D_S}K=0$ and $R^{D_S}(X,Y)=-[K_X,K_Y]$ (Equation (37) and Appendix C.1). Curvature is nonlinear under connection averaging. The global affine charts and central curvature span, together with contractibility, give

$${Hol}_{{\rho }_{\ast }}\left({\nabla }^{\left(\mathrm{m}\right)}\right)={Hol}_{{\rho }_{\ast }}\left({\nabla }^{\left(\mathrm{e}\right)}\right)=\left\{\mathrm{id}\right\},{Hol}_{{\rho }_{\ast }}^0\left(D_S\right)={Hol}_{{\rho }_{\ast }}\left(D_S\right)=\mathrm{SO}(n^2-1),n\ge 3.$$

(801)

No $n=2$ midpoint-holonomy identification is asserted (Corollary 27).

Independently, the canonical Cartan connection on ${\mathcal{P}}_n^1\simeq \mathrm{SL}(n,\mathbb{C})/\mathrm{SU}\left(n\right)$ has principal holonomy $\mathrm{SU}\left(n\right)$, while its associated affine-invariant tangent connection sees $Ad\left(\mathrm{SU}\right(n\left)\right)\simeq \mathrm{PSU}\left(n\right)$ ((Theorem 13 and Corollary 24). Positive normalization conjugates that pulled-back tangent connection to ${\widehat{D}}^S$ on the BKM metric bundle, not to $D_S$ (Corollary 25). Their interface is

$$\begin{array}{cc}\hfill D_S& ={\widehat{D}}^S+{\mathcal{B}}_S^{\mathrm{rel}},\hfill \\ \hfill {\mathcal{F}}_S^{\mathrm{rel}}& :=R^{D_S}-R^{{\widehat{D}}^S}={\mathrm{d}}^{{\widehat{D}}^S}{\mathcal{B}}_S^{\mathrm{rel}}+{\mathcal{B}}_S^{\mathrm{rel}}\wedge {\mathcal{B}}_S^{\mathrm{rel}}.\hfill \end{array}$$

(802)

Here ${\mathcal{B}}_S^{\mathrm{rel}}$ is a skew-adjoint difference tensor, not another connection, and ${\mathcal{F}}_S^{\mathrm{rel}}$ is the covariant curvature difference (Proposition 16). Relative transport obeys the twisted composition law (390). In a common oriented orthonormal frame, the normalized compact holonomy is a proper $\mathrm{PSU}\left(n\right)$ subgroup of the BKM $\mathrm{SO}(n^2-1)$ for $n\ge 3$: these connections coexist but are not conjugate.

The symmetric Hessian cubic as a typed cross-branch hinge.

With $\mathcal{C}={\mathcal{C}}^{\left(\mathrm{m}\right)}$, faithful $\sigma$, trace-zero Hermitian V, and sufficiently small $\epsilon$,

$$D_{\mathrm{U}}(\sigma +\epsilon V\parallel \sigma )-D_{\mathrm{U}}(\sigma \parallel \sigma +\epsilon V)=-\frac{{\epsilon }^3}{6}{\mathcal{C}}_{\sigma }(V,V,V)+O\left({\epsilon }^4\right).$$

(803)

This is the leading local orientation-sensitive correction, not the complete global asymmetry (Equation (78)). The same cubic determines the dual-horizontal Born shear through $g_{\mathrm{BKM}}(2K_{X}Y,Z)=-\mathcal{C}(X,Y,Z)$ (Equation (21)); its raised-endomorphism commutator, equivalently the associator in Equation (43), determines midpoint curvature. A nonzero cubic alone is insufficient: those endomorphisms must fail to commute. Third-order response also governs the channel cubic loss above and, under the mechanical and no-caustic hypotheses of Corollary 51, the leading directed action defect, canceled by the balanced $\theta =1/2$ generator (Equations (757) and (767)).

At the tracial qutrit point,

$${Stab}_{\mathrm{SO}(V_3,g_0)}\left({\mathcal{C}}_0^{\mathsf{\Psi }}\right)=Ad\left(\mathrm{SU}\left(3\right)\right)\simeq \mathrm{PSU}\left(3\right).$$

(804)

The metric, cubic, and orientation thus select the pointwise $8\oplus 20$ split (Equation (472)), not a global reduction of the BKM cubic field. The distinct parallel comparison cubic ${\mathcal{C}}^J$ requires independently supplied normalized affine-invariant holonomy and central alignment (Equations (507) and (508)); it is not identified with the BKM cubic away from the center, and neither cubic supplies the principal $\mathrm{SU}\left(3\right)$ lift. These roles relate third-order data without identifying the statistical, channel-loss, affine-invariant barrier, or parallel comparison cubics, or the alternating Cartan three-form. Beyond the metric means beyond the contrast’s pointwise second jet, not the full metric field and its derivatives, which already determine curvature.

Classical restriction and noncommutative curvature enrichment.

On a fixed maximal commutative ∗-algebra, Umegaki and BKM restrict to KL/Fisher geometry. The full faithful simplex is totally geodesic, with sectional curvature $1/4$ in dimensions at least two (Corollary 18). At the tracial qutrit point, commuting-plane bivectors are purely transverse and their Fisher curvature generators, over all fixed bases, span ${\mathfrak{m}}_{20}$ (Corollary 31). This is a curvature span, not a 20-dimensional classical state space; the normalization background is the full $\mathrm{id}/4=({\mathsf{\Pi }}_8+{\mathsf{\Pi }}_{20})/4$.

The Gram-normalized squared commutator $\chi$ (Equation (419)), invariant under simultaneous unitary conjugation and changes of plane generators, determines $r_8=2\chi /3$, $r_{20}=1-2\chi /3$, and ${sec}_0^{\mathsf{\Psi }}=1/4-3\chi /8$. Increasing $\chi$ changes the plane’s weights and lowers curvature, not the fixed channel eigenvalues: this is a static tracial resolution, not dynamics or a formula at arbitrary Gibbs states. It gives a noncommutative enrichment, not reconstruction from Fisher geometry alone. All Fisher-normalized monotone quantum metrics share the classical restriction [6]; the supplied Umegaki contrast selects BKM, while the Hermitian carrier and physical inputs remain supplied.

A grammar of typed composition.

Four recurrent composition types retain different carriers:

$$\begin{array}{cc}\hfill \mathrm{state}\mathrm{proximal}\text{:}& P_r^{\sigma }\circ P_s^{\sigma }=P_{r+s+rs}^{\sigma },r,s\ge 0,\hfill \\ \hfill \mathrm{moment}\mathrm{proximal}\text{:}& {\mathsf{Q}}_{\delta }^G{\mathsf{Q}}_{\epsilon }^{G}f={\mathsf{Q}}_{\epsilon +\delta }^{G}f,\epsilon ,\delta >0,\hfill \\ \hfill \mathrm{channel}\mathrm{defect}\text{:}& {\mathcal{H}}_{{\mathsf{\Lambda }}_2\circ {\mathsf{\Lambda }}_1}={\mathcal{H}}_{{\mathsf{\Lambda }}_1}+{\mathcal{H}}_{{\mathsf{\Lambda }}_2}\circ {\mathsf{\Lambda }}_1,\hfill \\ \hfill \mathrm{connection}\mathrm{transport}\text{:}& P_{{\gamma }_2\circ {\gamma }_1}=P_{{\gamma }_2}\circ P_{{\gamma }_1},\hfill \\ \hfill \mathrm{endpoint}\mathrm{relation}\text{:}& {\mathbb{S}}_{N,\mathsf{\Delta }t}^{\mathrm{on}}(q_0,q_N)=\underset{q_1,\dots ,q_{N-1}}{stat}{\mathbb{S}}_{N,\mathsf{\Delta }t}(q_0,\dots ,q_N).\hfill \end{array}$$

(805)

Even the proximal laws differ. For fixed faithful $\sigma$, ${\mathsf{\Phi }}_u^{\sigma }:=P_{{\mathrm{e}}^u-1}^{\sigma }$ is an additive state semigroup by relative-log contraction, a property of this Umegaki resolvent rather than general Bregman resolvents. Its barycentric values obey the discounted cocycle (620), not Hopf-Lax. Conversely, ${\mathsf{Q}}_{\epsilon }^G$ has the Hopf-Lax semigroup law on proper closed convex functions, not generally on iterates of its point resolvent ${(\mathrm{id}+\epsilon G\partial \mathsf{\Gamma })}^{-1}$ (Equation (192)). Neither scale is physical time. Identifying $u=t/\tau$ with BKM relaxation requires BKM mobility and $\tau$; the exact cumulative sampling times are Equations (626) and (627). The nonlinear state semigroup is not a CPTP channel semigroup.

For fixed affine CPTP maps on stagewise fixed support-reduced carriers, the channel law and its derivatives give the endpoint and $r\ge 2$ jet cocycles of Corollary 13. Contravariant denotes pullback along the first channel, not the variance of the response tensors; nonlinear or state-dependent maps add chain-rule terms. Ordinary transport instead maps composable paths to fiber isomorphisms, represented by group multiplication in compatible endpoint frames. BCH and Magnus compute local finite-product and time-ordered logarithms, respectively, with the convergence/formal-jet, parity, and exact-loop qualifications of Section 6.3 and Proposition 31; they are not additional products.

The defect laws nevertheless share a precise pattern: ${\mathcal{H}}_{\mathsf{\Lambda }}={\mathcal{F}}_{\mathrm{dom}}-{\mathsf{\Lambda }}^{\ast }{\mathcal{F}}_{\mathrm{cod}}$ is an Abelian difference of canonical negative-entropy potentials, whereas $U_{\gamma }^{\mathrm{rel}}={\left({\widehat{P}}_{\gamma }^{\mathrm{AI}}\right)}^{-1}{P}_{\gamma }^{\mathsf{\Psi }}$ is a quotient of transport functors. Composition therefore forces an additive pullback cocycle in the first case and a conjugation-twisted multiplicative cocycle in the second (Equation (390)). Different coefficient objects and actions prevent their identification.

For the regular generator of Theorem 23, with ${\mathbb{S}}_{N,\mathsf{\Delta }t}$ from Equation (720), choose a smooth critical branch with invertible internal endpoint Jacobian. Stationary elimination is then locally unique and

$$\mathrm{d}{\mathbb{S}}_{N,\mathsf{\Delta }t}^{\mathrm{on}}=-{\mathsf{A}}_0·\mathrm{d}{q}_0+{\mathsf{A}}_N·\mathrm{d}{q}_N,{\mathbb{S}}_{N,\mathsf{\Delta }t}^{\mathrm{on}}\mathrm{generates}{\mathsf{\Gamma }}_{L_d}^{\circ N}.$$

(806)

This is momentum-matched composition of exact Lagrangian relations; it does not establish global clean composition across caustics or multiple critical branches, nor does it make the separately reconstructed h-dependent endpoint maps an autonomous semigroup (Lemma 4). Thus the logarithmic, Legendre, metric-normalization, and cotangent interfaces relate associative operations within their respective domains, not one common product or clock: regularization scale, channel stage, path parameter, and mechanical time remain distinct.

Statistical directionality, operational loss, and dynamical dissipation.

Three notions often grouped under irreversibility remain distinct. The signed asymmetry ${\mathfrak{a}}_U(\rho ,\sigma ):=D_{\mathrm{U}}(\rho \parallel \sigma )-D_{\mathrm{U}}(\sigma \parallel \rho )$ is static, with leading cubic term (803); it specifies neither evolution nor entropy-production rate.

Channel loss is pair- and output-relative: it vanishes on recoverable pairs, and even a noninjective channel can preserve a recoverable model (Equation (A261) and Proposition 10). Subtracting the two orientations of Equation (227) gives

$${\mathfrak{a}}_U(\rho ,\sigma )-{\mathfrak{a}}_U(\mathsf{\Lambda }\rho ,\mathsf{\Lambda }\sigma )={\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma )-{\mathsf{\Delta }}_{\mathsf{\Lambda }}(\sigma ,\rho ).$$

(807)

Both losses are nonnegative, but their difference has no fixed sign. For faithful $\sigma$, trace-zero Hermitian V, and small $\epsilon$, setting $\rho =\sigma +\epsilon V$ gives the leading term $-{\epsilon }^3{\mathcal{C}}_{\mathsf{\Lambda },\sigma }^{\mathrm{loss}}(V,V,V)/6+O\left({\epsilon }^4\right)$ (Equation (238)). Identity and unitary channels retain asymmetry with zero loss; the symmetrized contrast also contracts (Equation (774)). Moreover, ${\mathsf{\Delta }}_{\mathsf{\Lambda }}=B_{{\mathcal{H}}_{\mathsf{\Lambda }}}\ge 0$ is not the raw entropy gain ${\mathcal{H}}_{\mathsf{\Lambda }}=S_{\mathrm{vN}}\circ \mathsf{\Lambda }-S_{\mathrm{vN}}$, which can be negative for nonunital channels.

Dissipation requires a selected law and clock. A positive-semidefinite Onsager mobility gives Equation (610), with possible zero decay away from equilibrium if degenerate. The full-state BKM choice gives strict decay away from $\sigma$ and exact dissipation (614); its proximal decrement samples that law only under the calibration in Equation (629). Decreasing a Lyapunov functional does not imply microscopic noninvertibility: the finite-time relative-log contraction (613) has a mathematical inverse on faithful states, and this nonlinear flow is not a CPTP semigroup.

For an independently supplied finite-dimensional CPTP semigroup ${\mathcal{T}}_t$ with faithful stationary reference $\tau$, Equation (A263) instead identifies ${\mathsf{\Delta }}_{{\mathcal{T}}_h}({\rho }_t,\tau )$ with accumulated nonnegative Spohn relative-entropy production on $[t,t+h]$ [69]. A fixed channel need not admit such an embedding, and static contrast selects neither the semigroup nor its mobility; detailed-balance gradient representations generally use generator-dependent transport metrics [97]. This is a reference-relative balance, not automatically total thermodynamic entropy production. For fixed Gibbs reference it is free-energy decrease, not necessarily system-entropy increase; the latter follows for uniform reference. Heat, work, and bath-entropy interpretations retain the physical inputs of Corollary 42; a moving reference adds the drift term in Equation (A225).

Neither proximal construction restores a channel kernel, crosses rank-changing strata, or resolves spacetime singularities. Channel contraction, exponential-tilt intertwining failure, and proximal deformation remain distinct (Theorems 6 and 7 and Corollary 16). The moment envelope’s Yosida and susceptibility responses use an inference/noise scale. For dimension-changing normalized-unital channels, data processing controls normalized entropy deficit; the covariant ray POVM provides reference tomography; it does not select an instrument (Remark 8).

Canonical Yang-Mills criticality is local and stationary on the statistical symmetric-space base, not spacetime propagation. The qutrit $8\oplus 20$ split describes curvature and connection channels, not particles; neither its adapted $\mathrm{PSU}\left(3\right)$ connection and retained principal $\mathrm{SU}\left(3\right)$ lift nor the separately graded AIII sector derives QCD or the Standard Model. A reduction-field action can give a conditional Euclidean quadratic coefficient, but physical mass needs spectral and dynamical input (Propositions 15 and A3 and Theorems 17 and 20). Transported frames can retain path data as reversible anholonomy, not by themselves dissipative hysteresis, Berry phase, or an observed Wilson loop.

A temporal Mori-Zwanzig kernel requires a generator; GENERIC retains its Jacobi, degeneracy, conservation, unit, smoothness, and quantum-admissibility hypotheses. The fixed conversion ${\kappa }_A$ differs from the mechanical ${\kappa }_h={\mu }_{\ast }/h$, which requires inertia and a clock; fixed-time error bounds give neither global stability nor long-time energy control. A faithful reference fixes only dimensionless modular ordering. Thermal, regulated Bisognano-Wichmann, and local-horizon interpretations retain their KMS/Unruh, cutoff, flux, area, conservation, focusing, and Bianchi inputs, with no Type-III theorem, microscopic area law, caustic continuation, or singularity resolution supplied by proximal control (Section 7.2, Remark 19, Corollary 52, and Theorem A2).

Open mathematical problems include restricted-family holonomy controllability, global output Type-II domains, intrinsic-torsion ${\mathsf{\Phi }}_{20}$ classification, metric channel-completion bundles, and rank-changing or thermodynamic limits. Physical development needs microscopic mobility and inertia, an apparatus realization of ray readout, operational tilt/curvature probes, and a reflection-positive spacetime qutrit reduction with controlled decoupling.

11. Conclusions

The controlled completion is the input- and defect-audited interface of Theorems 25 and 26, not a derivation of every structure from entropy. Its fixed-$n\ge 2$ family retains the activation rules of Equation (796): the qutrit and independently graded AIII packages require $n=3$ and $n=5$, respectively, with every auxiliary hypothesis retained. Channel outputs separately obey Equations (797) and (798).

On the supplied faithful-state affine carrier, the full contrast determines the cumulative relative-logarithm, metric, cubic, curvature, and holonomy hierarchy (Equation (800)). One channel-loss potential similarly organizes finite distinguishability loss and all mixture-affine response jets of order $r\ge 2$. Under the embedded-model hypotheses, a common recovery map preserves the entire intrinsic contrast-geometric hierarchy, including transport and holonomy (Equation (262)); Born preservation additionally requires mixture-affine Hessian models and an affine restricted channel. This does not imply preservation of ambient or independently supplied compact and mechanical geometry, nor does a degenerate loss Hessian itself define curvature or Born data.

Flatness and holonomy coexist because they concern different connections: the mixture/exponential pair is flat, its BKM midpoint has full $\mathrm{SO}(n^2-1)$ tangent holonomy for $n\ge 3$, and the independent affine-invariant connection has principal $\mathrm{SU}\left(n\right)$ and tangent $\mathrm{PSU}\left(n\right)$ holonomy. Their normalized defects compare, rather than conjugate, these structures. The cubic links directed response, dual-horizontal shear, midpoint curvature, channel-response loss, and the calibrated leading endpoint-action defect. With the metric and orientation fixed, its tracial qutrit stabilizer is $\mathrm{PSU}\left(3\right)$; a distinct parallel comparison cubic requires independently supplied normalized affine-invariant holonomy and central alignment.

Fixed commuting sectors recover KL/Fisher geometry. At the tracial qutrit point, their curvature generators span the transverse 20, while the normalized squared commutator determines a plane’s $8\oplus 20$ weights and signed curvature (Corollaries 28 and 31). This is a static noncommutative enrichment, not a 20-dimensional classical state space or a dynamical classical-to-quantum transition.

The synthesis also distinguishes four composition types (Equation (805)) and three notions often called irreversibility: static asymmetry, operational information loss, and dynamical dissipation. Their exact interfaces (Equations (807) and (A263)) establish neither a common product nor a common clock; dynamical and thermodynamic interpretations retain their additional hypotheses.

Thus supplied amplitude complex kinematics and separately constructed Born complex structures remain distinct. The construction relates non-Abelian compact transport and endpoint action without deriving spacetime dynamics, matter assignments, measurement implementation, or absolute action and time scales. The outstanding physical and global-limit requirements are those of Section 10.

Appendix A.1. State-Space Entropic Envelope

Proof of Theorem 6. 

Multiply Equation (135) by $1+1/r$ and put $t=t_r={(1+r)}^{-1}$. Its unique optimizer is $p_r={\gamma }_{t_r}$, and the rescaled minimum and residual give Equations (147)–(149) and (151); normalizing the optimizer gives (152). This is the standard left Bregman-Moreau formulation identified before the theorem. Analytic functional calculus proves the stated real analyticity. Taking $t_r\to 1$ and $t_r\to 0$, together with the first terms of the expansions below, proves Equation (150).

The envelope theorem and $D{log}_{\rho }={\mathsf{K}}_{\rho }^{-1}$ give

$$D{\mathcal{M}}_r^{\sigma }\left(\rho \right)\left[V\right]=-\frac{1}{r}Tr\left[p_{r}D{log}_{\rho }\left[V\right]\right].$$

(A1)

Self-adjointness of ${\mathsf{K}}_{\rho }$, together with ${\mathsf{K}}_{\rho }\left(I\right)=\rho$ and $TrV=0$, gives $Tr\left[\rho D{log}_{\rho }\left[V\right]\right]=0$ and proves the first two identities in Equation (154). Differentiating the optimized value only through its explicit $r^{-1}$ coefficient proves the last one. The same calculation at the reference endpoint gives

$$D_{\sigma }{\mathcal{M}}_r^{\sigma }\left(\rho \right)\left[W\right]=-Tr\left[p_{r}D{log}_{\sigma }\left[W\right]\right]=g_{\sigma }^{\mathrm{BKM}}(\sigma -p_r,W),$$

(A2)

which proves Equation (155). For the extended-real assertion following Equation (156), the BKM musical map makes $\alpha \mapsto q\left(\alpha \right):=\rho -r{\alpha }^{{♯}_{\rho }}$ an affine homeomorphism onto the trace-one Hermitian hyperplane. Pulling back the proper lower-semicontinuous convex function $q\mapsto D_{\mathrm{U}}(q\parallel \rho )+{\delta }_{{\mathcal{S}}_n}\left(q\right)$ therefore gives the claimed extended Hamiltonian by standard convex pullback theory [49]. On the relative interior, $q\mapsto Tr(qlogq)$ is analytic exactly for $q>0$; at a rank-deficient boundary point, the one-sided derivative of $xlogx$ in a newly opened eigenvalue is divergent. Thus the extended Hamiltonian is smooth exactly over faithful displaced states. Its explicit Legendre-Fenchel representation is supplied later by Corollary 9. Substitution of $p_r=\rho -r{grad}_{\mathrm{BKM}}{\mathcal{M}}_r^{\sigma }\left(\rho \right)$ proves Equation (157); the Hessian expansion in the first argument about $\omega =\rho$ gives $D_{\mathrm{U}}(\rho -r{V}_r\parallel \rho )=r^2{g}_{\rho }^{\mathrm{BKM}}(V_r,V_r)/2+O\left(r^3\right)$ and proves Equation (158). Finally, Taylor expansion of $D_t^{\mathrm{b}}={\phi }_{\rho \mid \sigma }\left(t\right)/(t-1)$ at $t=1$ and $t=0$, using $t_r={(1+r)}^{-1}$ and Equation (139), proves Equations (159) and (160). Integrating the scale derivative in Equation (154), using the two endpoint limits, proves Equations (161) and (162). □

Proof of Corollary 8. 

The common exponent in the two swapped bridges proves Equations (163) and (164). The chain rule, $r=(1-t)/t$, and ${\partial }_r{\mathcal{M}}_r^{\sigma }=-r^{-2}{D}_{\mathrm{U}}(p_r^{\sigma }\left(\rho \right)\parallel \rho )$ prove Equation (165); the endpoint limits follow from Equation (150). At the optimizer,

$${\mathcal{M}}_r^{\sigma }\left(\rho \right)=D_{\mathrm{U}}(p_r^{\sigma }\left(\rho \right)\parallel \sigma )+r^{-1}{D}_{\mathrm{U}}(p_r^{\sigma }\left(\rho \right)\parallel \rho ).$$

(A3)

Combining this with ${\partial }_r{\mathcal{M}}_r^{\sigma }=-r^{-2}{D}_{\mathrm{U}}(p_r^{\sigma }\left(\rho \right)\parallel \rho )$ proves Equation (167). Integration uses $r{\mathcal{M}}_r^{\sigma }\left(\rho \right)\to 0$ as $r\downarrow 0$ and $r{\mathcal{M}}_r^{\sigma }\left(\rho \right)\to D_{\mathrm{U}}(\sigma \parallel \rho )$ as $r\to \infty$. Adding the two scale integrals and applying Equation (140) proves the last identity. □

Proof of Corollary 9. 

The second expression in Equation (170) is the Fenchel certificate Equation (93); Taylor’s integral formula for ${\mathsf{\Lambda }}_{\rho }(-srP)$ gives the covariance representation. Substitute $H=-rP$ in the extended Legendre-Fenchel formula for $D_{\mathrm{U}}(q\parallel \rho )$. Since $q=\rho -r{\alpha }^{{♯}_{\rho }}$, division by $r^2$ gives Equation (171). If q is not positive semidefinite, finite-dimensional separation in the trace-one affine hyperplane makes the supremum $+\infty$. Faithful q gives the unique optimizer in Equation (172); for singular states the finite value is approached but not attained. For $q=p_r$, the bridge logarithm proves Equation (173). Equations Equations (149) and (165) and $r=(1-t)/t$ prove Equations (174) and (175). Finally, $\mathrm{d}r/\mathrm{d}u={\mathrm{e}}^u$ converts Equation (157) into Equation (177). □

Remark A1

(Facewise boundary extension and rank obstruction). The differential, BKM-gradient, and Hamilton-Jacobi statements above hold on the faithful manifold. The variational theorem has a facewise extension to singular endpoints, but no smooth extension across rank-changing strata. Let $P_{\rho },P_{\sigma }$ be the support projections and put $P:=P_{\rho }\wedge P_{\sigma }$. Let ${log}_{P_{\rho }}\rho$ and ${log}_{P_{\sigma }}\sigma$ denote the logarithms on the respective supports, write $\mathcal{S}\left(P\right):=\{\omega \in {\mathcal{S}}_n:supp\omega \le P\}$, and define on $P{\mathbb{C}}^n$

$$A_{\rho }^P:=P\left({log}_{P_{\rho }}\rho \right)P,A_{\sigma }^P:=P\left({log}_{P_{\sigma }}\sigma \right)P.$$

(A4)

If $P=0$, every competitor has infinite objective value. If $P\ne 0$, the unique finite-r optimizer is

$$\begin{array}{|c|}\hline p_{r,P}=\frac{{exp}_P[(1-t_r)A_{\sigma }^P+t_r{A}_{\rho }^P]}{{Tr}_P{exp}_P[(1-t_r)A_{\sigma }^P+t_r{A}_{\rho }^P]},\\ \hline\end{array}$$

(A5)

It is faithful relative to the face $\mathcal{S}\left(P\right)$ and satisfies the same residual identity for competitors supported in P. This is precisely the support-reduced log-Euclidean Rényi construction [22], Proposition V.29.

In particular, if σ is faithful and ρ is singular, then $P=P_{\rho }$ and $supp{p}_{r,P}=supp\rho$ for every finite $r>0$. The map preserves the input face exactly and cannot restore missing rank. Moreover, with

$$Z_P:={Tr}_P{exp}_P(Plog\sigma P),{\sigma }_P^{\mathrm{e}}:=Z_P^{-1}{exp}_P(Plog\sigma P),$$

(A6)

one has

$$p_{r,P}⟶{\sigma }_P^{\mathrm{e}},{\mathcal{M}}_r^{\sigma }\left(\rho \right)=-log{Z}_P+\frac{1}{r}{D}_{\mathrm{U}}({\sigma }_P^{\mathrm{e}}\parallel \rho )+O\left(r^{-2}\right).$$

(A7)

Thus the faithful limit $p_r\to \sigma$ and zero envelope limit must not be transferred to a rank-changing face: the endpoint is instead the reverse-information projection of σ onto that face. Indeed, finite objective value is equivalent to $\omega \in \mathcal{S}\left(P\right)$, so repeating the mirror Pythagorean calculation on $P{\mathbb{C}}^n$ proves Equation (A5) and its residual identity. For faithful σ and $P=P_{\rho }$, put

$${\phi }_P\left(t\right):=log{Tr}_P{exp}_P[(1-t)Plog\sigma P+t{log}_P\rho ].$$

(A8)

Then ${\phi }_P\left(0\right)=log{Z}_P$ and $-{\phi }_P^{\prime }\left(0\right)-{\phi }_P\left(0\right)=D_{\mathrm{U}}({\sigma }_P^{\mathrm{e}}\parallel \rho )$. Taylor expansion of $-{\phi }_P\left(t\right)/(1-t)$ at $t=0$, with $t=t_r$, proves Equation (A7).

Appendix B.1. Proof, Residuals, and Noisy-Record Extension

Proof of Theorem 7. 

Substitution of $log{\rho }_{\theta }=log\sigma +O\left(\theta \right)-\lambda \left(\theta \right)I$ and $D_{\mathrm{U}}({\rho }^m\parallel \sigma )=\mathsf{\Gamma }\left(m\right)$ proves Equation (181). Put $h_{\epsilon }\left(z\right)={\left(2\epsilon \right)}^{-1}{z}^{\mathsf{T}}{G}^{-1}z$. Standard infimal-convolution conjugacy and metric Moreau-Yosida theory give Equation (183), the unique proximal point, its Yosida relation, and the pointwise and epigraphical limit as $\epsilon \downarrow 0$ [16,49,50].

For the stronger formulas used here, set $f_{\epsilon }\left(\theta \right)=\lambda \left(\theta \right)+\epsilon {\theta }^{\mathsf{T}}G\theta /2$. It is real analytic and strongly convex, while $f_{\epsilon }\left(\theta \right)-m·\theta$ is coercive for every m. Its unique critical point satisfies Equation (184). Since

$$D^2{f}_{\epsilon }\left(\theta \right)=\chi \left(\theta \right)+\epsilon G\succ 0,$$

the analytic inverse-function theorem gives the global analytic inverse $m\mapsto {\theta }_{\epsilon }\left(m\right)$. Legendre differentiation proves Equations (185) and (186). The bound $0\prec \chi ⪯G/2$ from Equation (120) gives Equation (187). Its constants are uniform-optimal because, for $\sigma =I/2$ and $O={\sigma }_z$, $G=2$ and $\chi \left(\theta \right)={sech}^2\theta$.

At the optimum, $m-q_{\epsilon }=\epsilon G{\theta }_{\epsilon }$ and

$${\mathsf{\Gamma }}_{\epsilon }^G\left(m\right)=D_{\mathrm{U}}({\rho }^{q_{\epsilon }}\parallel \sigma )+\frac{\epsilon }{2}{\theta }_{\epsilon }^{\mathsf{T}}G{\theta }_{\epsilon }.$$

Expanding the quadratic penalty about $q_{\epsilon }$ and using Equation (181) gives the exact residual

$$\begin{array}{cc}\hfill & \mathsf{\Gamma }\left(q\right)+\frac{1}{2\epsilon }{(m-q)}^{\mathsf{T}}{G}^{-1}(m-q)-{\mathsf{\Gamma }}_{\epsilon }^G\left(m\right)\hfill \\ \hfill & =D_{\mathrm{U}}({\rho }^q\parallel {\rho }^{q_{\epsilon }})+\frac{1}{2\epsilon }{(q-q_{\epsilon })}^{\mathsf{T}}{G}^{-1}(q-q_{\epsilon }).\hfill \end{array}$$

(A9)

The proximal optimality condition is Equation (188). Applying the envelope theorem to the dual value proves 189, and differentiating Equation (184) at fixed m proves Equation (190). □

Proof of Proposition 7. 

The quadratic Moreau kernels form the Hopf-Lax semigroup and satisfy ${\left({\mathsf{Q}}_{\epsilon }^{G}f\right)}^{\ast }=f^{\ast }+\epsilon {∥·∥}_G^2/2$ [16,49,50]. With $f=\mathsf{\Gamma }$ and ${\mathsf{\Gamma }}^{\ast }=\lambda$, this proves Equations (183) and (192) and the stated uniqueness. Integrating Equation (189) over $[\delta ,R]$, with $0<\delta <R$, proves Equation (193). Since $\mathsf{\Gamma }\left(m_{\sigma }\right)=0$, the competitor $q=m_{\sigma }$ gives

$$0\le {\mathsf{\Gamma }}_R^G\left(m\right)\le \frac{1}{2R}{∥m-m_{\sigma }∥}_{G^{-1}}^2⟶0(R\to \infty ).$$

(A10)

Nonnegativity of the integrand then gives Equation (194) as $R\to \infty$. Finally, for $m\in {\mathcal{C}}_O$, monotonicity of the Moreau envelope and pointwise convergence of the closed convex regularization give ${\mathsf{\Gamma }}_{\delta }^G\left(m\right)\uparrow \mathsf{\Gamma }\left(m\right)$ as $\delta \downarrow 0$. Monotone convergence proves Equation (195), including $m\in \partial {\mathcal{C}}_O$.

For $q\in int{\mathcal{C}}_O$, the Legendre equality $\mathsf{\Gamma }\left(q\right)+\lambda \left(\theta \right)=\theta ·q$ and $m_{\epsilon }=q+\epsilon G\theta$ give the resolvent and value identities in Equation (196). Finally put $d:=m-m_{\sigma }$ and ${\chi }_{\sigma }:=\chi \left(0\right)$. Analyticity at the origin and the resolvent equation give

$$d=({\chi }_{\sigma }+\epsilon G){\theta }_{\epsilon }+O\left({∥{\theta }_{\epsilon }∥}^2\right),$$

(A11)

so ${\theta }_{\epsilon }={\epsilon }^{-1}{G}^{-1}d+O\left({\epsilon }^{-2}\right)$ and $q_{\epsilon }=m_{\sigma }+{\epsilon }^{-1}{\chi }_{\sigma }{G}^{-1}d+O\left({\epsilon }^{-2}\right)$. Expanding $\lambda \left(\theta \right)=m_{\sigma }·\theta +{\theta }^{\mathsf{T}}{\chi }_{\sigma }\theta /2+O\left({∥\theta ∥}^3\right)$ in the dual value formula proves Equation (198). □

For $m\in int{\mathcal{C}}_O$, $q_{\epsilon }\left(m\right)\to m$ and ${\theta }_{\epsilon }\left(m\right)\to \nabla \mathsf{\Gamma }\left(m\right)$. At a boundary moment, $q_{\epsilon }\left(m\right)\to m$ but its finite source can diverge; outside ${\mathcal{C}}_O$, ${\mathsf{\Gamma }}_{\epsilon }^G\left(m\right)\to +\infty$. Thus positive $\epsilon$ supplies an analytic interior continuation without removing the original moment-body boundary.

Corollary A1

(Interior bias and monotone equilibrium continuation). Fix $m\in int{\mathcal{C}}_O$ and set ${\theta }_0:=\nabla \mathsf{\Gamma }\left(m\right)$ and ${\chi }_0:=\chi \left({\theta }_0\right)$. Uniformly on compact subsets of the interior, as $\epsilon \downarrow 0$,

$$\begin{array}{cc}\hfill {\theta }_{\epsilon }\left(m\right)& ={\theta }_0-\epsilon {\chi }_0^{-1}G{\theta }_0+O\left({\epsilon }^2\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill q_{\epsilon }\left(m\right)& =m-\epsilon G{\theta }_0+{\epsilon }^{2}G{\chi }_0^{-1}G{\theta }_0+O\left({\epsilon }^3\right),\hfill \end{array}$$

(A12)

$$\begin{array}{cc}\hfill {\mathsf{\Gamma }}_{\epsilon }^G\left(m\right)& =\mathsf{\Gamma }\left(m\right)-\frac{\epsilon }{2}{\theta }_0^{\mathsf{T}}G{\theta }_0+\frac{{\epsilon }^2}{2}{\theta }_0^{\mathsf{T}}G{\chi }_0^{-1}G{\theta }_0+O\left({\epsilon }^3\right),\hfill \end{array}$$

(A13)

$$\begin{array}{cc}\hfill D_{\mathrm{U}}({\rho }^m\parallel {\rho }^{q_{\epsilon }})& =\frac{{\epsilon }^2}{2}{\theta }_0^{\mathsf{T}}G{\chi }_0^{-1}G{\theta }_0+O\left({\epsilon }^3\right).\hfill \end{array}$$

(A14)

For every fixed $m\in {\mathbb{R}}^d$ and $\epsilon >0$, the selected inference cost $I_{\epsilon }\left(m\right):=D_{\mathrm{U}}({\rho }^{q_{\epsilon }\left(m\right)}\parallel \sigma )$ obeys

$$\frac{\mathrm{d}{I}_{\epsilon }\left(m\right)}{\mathrm{d}\epsilon }=-{\theta }_{\epsilon }^{\mathsf{T}}\chi \left({\theta }_{\epsilon }\right){[\chi \left({\theta }_{\epsilon }\right)+\epsilon G]}^{-1}G{\theta }_{\epsilon }\le 0,I_{\epsilon }\left(m\right)⟶0.$$

(A15)

Thus increasing ε returns the selected state monotonically to the reference in relative entropy; reversing the scale is a well-conditioned inference homotopy, not automatically thermodynamic time evolution.

Proof. 

The analytic implicit-function theorem applied to $m=\nabla \lambda \left({\theta }_{\epsilon }\right)+\epsilon G{\theta }_{\epsilon }$ gives the source and proximal-moment expansions. The value expansion follows by integrating Equation (189); the information expansion is the quadratic Taylor expansion of the Fenchel certificate Equation (181) at ${\theta }_0$. At fixed m, Equation (190) and $q_{\epsilon }=\nabla \lambda \left({\theta }_{\epsilon }\right)$ give the derivative. Its quadratic form is nonnegative because

$$\chi {(\chi +\epsilon G)}^{-1}G=\frac{1}{\epsilon }{[{\chi }^{-1}+{\epsilon }^{-1}{G}^{-1}]}^{-1}⪰0.$$

(A16)

The limit is Equation (197). □

Corollary A2

(Moment resolvent iteration and exact information dissipation). Let $m_0\in {\mathcal{C}}_O$, $h_k>0$, and iterate the metric resolvent

$$m_{k+1}:={(\mathrm{id}+h_{k}G\partial \mathsf{\Gamma })}^{-1}{m}_k,{\theta }_{k+1}:=\nabla \mathsf{\Gamma }\left(m_{k+1}\right).$$

(A17)

Then $m_{k+1}\in int{\mathcal{C}}_O$ and

$$\begin{array}{cc}\hfill m_k-m_{k+1}& =h_{k}G{\theta }_{k+1},\hfill \end{array}$$

(A18)

$$\begin{array}{cc}\hfill \mathsf{\Gamma }\left(m_k\right)-\mathsf{\Gamma }\left(m_{k+1}\right)& =h_k{\theta }_{k+1}^{\mathsf{T}}G{\theta }_{k+1}+D_{\mathrm{U}}({\rho }^{m_k}\parallel {\rho }^{m_{k+1}}).\hfill \end{array}$$

(A19)

Consequently, for every N,

$$\mathsf{\Gamma }\left(m_0\right)-\mathsf{\Gamma }\left(m_N\right)={\displaystyle \sum _{k=0}^{N-1}}\left[h_k{∥{\theta }_{k+1}∥}_G^2+D_{\mathrm{U}}({\rho }^{m_k}\parallel {\rho }^{m_{k+1}})\right].$$

(A20)

Moreover, Γ is 2-strongly convex in the $G^{-1}$ metric, so

$${∥m_k-m_{\sigma }∥}_{G^{-1}}\le {\displaystyle \prod _{j=0}^{k-1}}{(1+2h_j)}^{-1}{∥m_0-m_{\sigma }∥}_{G^{-1}}.$$

(A21)

If the product tends to zero, the finite balance converges to the same formula with left side $\mathsf{\Gamma }\left(m_0\right)$. This is backward Euler for the separately chosen fixed-G flow $\dot{m}=-G\nabla \mathsf{\Gamma }\left(m\right)$; unlike the state-space entropic resolvent, it is not generally an exact flow map and its resolvents do not form a semigroup. For a Gibbs reference, $\mathsf{\Gamma }/\beta$ is the corresponding constrained Helmholtz excess.

Proof. 

The first line is the resolvent condition. Applying Equation (181) at source ${\theta }_{k+1}$ gives

$$\mathsf{\Gamma }\left(m_k\right)-\mathsf{\Gamma }\left(m_{k+1}\right)-{\theta }_{k+1}·(m_k-m_{k+1})=D_{\mathrm{U}}({\rho }^{m_k}\parallel {\rho }^{m_{k+1}}),$$

(A22)

which proves the exact balance. The bound ${\nabla }^2\mathsf{\Gamma }={\chi }^{-1}⪰2G^{-1}$ on the interior extends as strong convexity of the closed potential. The standard strongly monotone resolvent contraction then proves Equation (A21) [50]. □

Corollary A3

(Possibly noninjective noisy linear moment records). Under the hypotheses of 4, let $T:{\mathbb{R}}^d\to {\mathbb{R}}^k$ be linear, possibly with nontrivial kernel, let $R=R^{\mathsf{T}}\succ 0$, and let $\epsilon >0$. Define

$$\begin{array}{cc}\hfill {\mathsf{\Gamma }}_{\epsilon ,T,R}\left(y\right)& :={\displaystyle \underset{q\in {\mathcal{C}}_O}{min}}\left\{{\mathsf{\Gamma }}_{\sigma ,O}\left(q\right)+\frac{1}{2\epsilon }{(y-Tq)}^{\mathsf{T}}{R}^{-1}(y-Tq)\right\}\hfill \\ & ={\displaystyle \underset{\omega \in {\mathcal{S}}_n}{min}}\left\{D_{\mathrm{U}}(\omega \parallel \sigma )+\frac{1}{2\epsilon }{[y-TM\left(\omega \right)]}^{\mathsf{T}}{R}^{-1}[y-TM\left(\omega \right)]\right\}.\hfill \end{array}$$

(A23)

Then

$$\begin{array}{|c|}\hline {\mathsf{\Gamma }}_{\epsilon ,T,R}={\left[\eta ⟼{\lambda }_{\sigma ,O}\left(T^{\mathsf{T}}\eta \right)+\frac{\epsilon }{2}{\eta }^{\mathsf{T}}R\eta \right]}^{\ast }.\\ \hline\end{array}$$

(A24)

For each $y\in {\mathbb{R}}^k$, unique ${\eta }_{\epsilon }\left(y\right)\in {\mathbb{R}}^k$ and $q_{\epsilon }\left(y\right)\in int{\mathcal{C}}_O$ satisfy

$$\begin{array}{cc}\hfill y& =T\nabla {\lambda }_{\sigma ,O}\left(T^{\mathsf{T}}{\eta }_{\epsilon }\right)+\epsilon R{\eta }_{\epsilon },\hfill \end{array}$$

(A25)

$$\begin{array}{cc}\hfill q_{\epsilon }\left(y\right)& =\nabla {\lambda }_{\sigma ,O}\left(T^{\mathsf{T}}{\eta }_{\epsilon }\right),{\nabla }_y{\mathsf{\Gamma }}_{\epsilon ,T,R}\left(y\right)={\eta }_{\epsilon }\left(y\right).\hfill \end{array}$$

(A26)

Moreover,

$$\begin{array}{cc}\hfill {\nabla }_y^2{\mathsf{\Gamma }}_{\epsilon ,T,R}\left(y\right)& ={\left[T\chi \left(T^{\mathsf{T}}{\eta }_{\epsilon }\right)T^{\mathsf{T}}+\epsilon R\right]}^{-1},\hfill \end{array}$$

(A27)

$$\begin{array}{cc}\hfill {\left(\epsilon R+\frac{1}{2}TG{T}^{\mathsf{T}}\right)}^{-1}& ⪯{\nabla }_y^2{\mathsf{\Gamma }}_{\epsilon ,T,R}\left(y\right)⪯{\left(\epsilon R\right)}^{-1}.\hfill \end{array}$$

(A28)

For every $q\in {\mathcal{C}}_O$, the exact residual is

$$\begin{array}{cc}& {\mathsf{\Gamma }}_{\sigma ,O}\left(q\right)+\frac{1}{2\epsilon }{(y-Tq)}^{\mathsf{T}}{R}^{-1}(y-Tq)-{\mathsf{\Gamma }}_{\epsilon ,T,R}\left(y\right)\hfill \\ & =D_{\mathrm{U}}({\rho }^q\parallel {\rho }^{q_{\epsilon }})+\frac{1}{2\epsilon }{\left[T(q-q_{\epsilon })\right]}^{\mathsf{T}}{R}^{-1}T(q-q_{\epsilon }).\hfill \end{array}$$

(A29)

It also satisfies

$${\partial }_{\epsilon }{\mathsf{\Gamma }}_{\epsilon ,T,R}+\frac{1}{2}{\left({\nabla }_y{\mathsf{\Gamma }}_{\epsilon ,T,R}\right)}^{\mathsf{T}}R\left({\nabla }_y{\mathsf{\Gamma }}_{\epsilon ,T,R}\right)=0$$

(A30)

with extended-real initial datum

$${\mathsf{\Gamma }}_{0,T}\left(y\right)=\left\{\begin{array}{cc}{\displaystyle \underset{q:Tq=y}{min}}{\mathsf{\Gamma }}_{\sigma ,O}\left(q\right),\hfill & y\in T{\mathcal{C}}_O,\hfill \\ +\infty ,\hfill & y\notin T{\mathcal{C}}_O.\hfill \end{array}\right.$$

(A31)

Using Equation (191) on ${\mathbb{R}}^k$ with R in place of G,

$$\begin{array}{|c|}\hline {\mathsf{\Gamma }}_{\epsilon ,T,R}={\mathsf{Q}}_{\epsilon }^R{\mathsf{\Gamma }}_{0,T},{\mathsf{\Gamma }}_{\epsilon +\delta ,T,R}={\mathsf{Q}}_{\delta }^R{\mathsf{\Gamma }}_{\epsilon ,T,R}.\\ \hline\end{array}$$

(A32)

For $y\in T{\mathcal{C}}_O$, integration of the scale equation gives

$$\begin{array}{|c|}\hline {\mathsf{\Gamma }}_{0,T}\left(y\right)=\frac{1}{2}{\int }_0^{\infty }{\eta }_{\epsilon }{\left(y\right)}^{\mathsf{T}}R{\eta }_{\epsilon }\left(y\right)\mathrm{d}\epsilon .\\ \hline\end{array}$$

(A33)

Proof from linear infimal convolution. 

Write $h_{\epsilon ,R}\left(z\right)={\left(2\epsilon \right)}^{-1}{z}^{\mathsf{T}}{R}^{-1}z$. Writing $y=Tq+z$, direct conjugacy of the linear image together with the infimal-convolution rule [49], Theorem 16.4, gives

$${\mathsf{\Gamma }}_{\epsilon ,T,R}^{\ast }\left(\eta \right)={\mathsf{\Gamma }}_{\sigma ,O}^{\ast }\left(T^{\mathsf{T}}\eta \right)+h_{\epsilon ,R}^{\ast }\left(\eta \right)={\lambda }_{\sigma ,O}\left(T^{\mathsf{T}}\eta \right)+\frac{\epsilon }{2}{\eta }^{\mathsf{T}}R\eta ,$$

without requiring T to be injective. The positive quadratic term makes the dual objective strongly convex and coercive, proving the unique source and Equation (A25). Legendre differentiation gives Equation (A27); $0⪯\chi ⪯G/2$ gives both matrix bounds.

Expanding the penalty about $q_{\epsilon }$ and applying the exact Umegaki Fenchel certificate at source $T^{\mathsf{T}}{\eta }_{\epsilon }$ proves Equation (A29). Standard quadratic Moreau-Hopf-Lax theory [16,50] supplies the envelope semigroup. For the manuscript’s R-metric normalization, envelope differentiation and $y-T{q}_{\epsilon }=\epsilon R{\eta }_{\epsilon }$ retain the coefficient:

$${\partial }_{\epsilon }{\mathsf{\Gamma }}_{\epsilon ,T,R}=-\frac{1}{2{\epsilon }^2}{∥y-T{q}_{\epsilon }∥}_{R^{-1}}^2=-\frac{1}{2}{\eta }_{\epsilon }^{\mathsf{T}}R{\eta }_{\epsilon }.$$

This proves the displayed Hamilton-Jacobi equation; its zero-noise trace is the fiber minimum (A31), not an inverse pullback. This nonnegative potential vanishes at $TM\left(\sigma \right)$, so the competitor and monotone-limit argument proving Equation (195) also gives Equation (A33). Thus the standard convex core is cited while the noninjective carrier, quantum residual, sharp susceptibility bounds, and cumulative information action remain explicit. □

Corollary A4

(Gaussian noisy records and the smoothed rate function). Assume that $\sigma ,O_1,\dots ,O_d$ commute and let $X_1,X_2,\dots \in {\mathbb{R}}^d$ be their iid common-spectral records in the state σ, so that $log\mathbb{E}{\mathrm{e}}^{\theta ·X_1}={\lambda }_{\sigma ,O}\left(\theta \right)$. Put $M_N=N^{-1}{\sum }_{j=1}^N{X}_j$, take $Z\sim \mathcal{N}(0,R)$ independently, and define

$$Y_N:=T{M}_N+\sqrt{\frac{\epsilon }{N}}Z.$$

(A34)

Equivalently, if $Z_1,Z_2,\dots$ are iid $\mathcal{N}(0,R)$ and independent of the $X_j$, then

$$Y_N\stackrel{\mathrm{law}}{=}\frac{1}{N}{\displaystyle \sum _{j=1}^N}\left(T{X}_j+\sqrt{\epsilon }{Z}_j\right).$$

(A35)

Then, exactly for every N,

$$\frac{1}{N}log\mathbb{E}{\mathrm{e}}^{N\eta ·Y_N}={\lambda }_{\sigma ,O}\left(T^{\mathsf{T}}\eta \right)+\frac{\epsilon }{2}{\eta }^{\mathsf{T}}R\eta .$$

(A36)

Consequently $Y_N$ satisfies the large-deviation principle with good rate ${\mathsf{\Gamma }}_{\epsilon ,T,R}$. At $\epsilon =0$ the rate is the deterministically compressed potential ${\mathsf{\Gamma }}_{0,T}$; if T has a kernel, the discarded directions are minimized over rather than reconstructed. Independent Gaussian increments with covariances $\epsilon R/N$ and $\delta R/N$ add to covariance $(\epsilon +\delta )R/N$, providing the record-level realization of Equation (A32).

Proof from Cramér theory. 

Gaussian stability and independence give Equations (A35) and (A36). Cramér’s theorem identifies the convex conjugate (A24) as the good rate function [91]; the contraction principle gives the noninjective zero-noise limit, and Gaussian variance addition realizes the semigroup. □

For noncommuting observables, the augmented partition and all displayed convex-duality formulas remain exact, but a literal joint record requires a separately supplied measurement protocol. A kernel of T represents unresolved moment directions; the variational principle chooses their least-relative-entropy completion but cannot reconstruct them. This classical record map is a CPTP channel only when a physical output construction inducing it is supplied.

Appendix B.2. Observable Refinement and Exposed-Face Limits

Proposition A1

(Nested observable families and Schur-complement stiffness). Suppose $O_1,\dots ,O_d,B_1,\dots ,B_r$ are jointly minimal modulo I. Define

$$M_O\left(\omega \right):={(Tr\omega O_a)}_{a=1}^d,M_B\left(\omega \right):={(Tr\omega B_{\mu })}_{\mu =1}^r,{\mathcal{C}}_{O,B}:=\{(M_O\left(\omega \right),M_B\left(\omega \right)):\omega \in {\mathcal{S}}_n\}.$$

(A37)

For $(m,q)\in {\mathcal{C}}_{O,B}$, let

$${\rho }^{m,q}:=\underset{\begin{array}{c}\omega \in {\mathcal{S}}_n\\ M_O\left(\omega \right)=m,M_B\left(\omega \right)=q\end{array}}{\mathrm{argmin}}{D}_{\mathrm{U}}(\omega \parallel \sigma ),$$

(A38)

which is unique by strict convexity. Let ${\mathsf{\Gamma }}_{\sigma ,O,B}(m,q)$ be the effective potential for all their moments and ${\mathsf{\Gamma }}_{\sigma ,O}\left(m\right)$ the one retaining only the O moments. Then

$$\begin{array}{|c|}\hline {\mathsf{\Gamma }}_{\sigma ,O}\left(m\right)={\displaystyle \underset{q:(m,q)\in {\mathcal{C}}_{O,B}}{min}}{\mathsf{\Gamma }}_{\sigma ,O,B}(m,q).\\ \hline\end{array}$$

(A39)

For $m\in int{\mathcal{C}}_O$, the minimizer is uniquely $q_{\ast }\left(m\right):=M_B\left({\rho }^m\right)$, and every admissible q satisfies

$$\begin{array}{|c|}\hline {\mathsf{\Gamma }}_{\sigma ,O,B}(m,q)={\mathsf{\Gamma }}_{\sigma ,O}\left(m\right)+D_{\mathrm{U}}({\rho }^{m,q}\parallel {\rho }^m).\\ \hline\end{array}$$

(A40)

Thus resolving more observables adds exactly the information previously hidden by the coarse moment record.

At $(m,q_{\ast }\left(m\right))$, evaluate the fine susceptibility at the joint source $\left(\theta \right(m),0)$ corresponding to ${\rho }^m$, and block-decompose it and its inverse as

$$\chi =\left(\begin{array}{cc}{\chi }_{OO}& {\chi }_{OB}\\ {\chi }_{BO}& {\chi }_{BB}\end{array}\right),\mathcal{K}:={\chi }^{-1}=\left(\begin{array}{cc}{\mathcal{K}}_{OO}& {\mathcal{K}}_{OB}\\ {\mathcal{K}}_{BO}& {\mathcal{K}}_{BB}\end{array}\right).$$

(A41)

Then the coarse stiffness is the exact Schur complement

$$\begin{array}{|c|}\hline {\nabla }^2{\mathsf{\Gamma }}_{\sigma ,O}\left(m\right)={\chi }_{OO}^{-1}={\mathcal{K}}_{OO}-{\mathcal{K}}_{OB}{\mathcal{K}}_{BB}^{-1}{\mathcal{K}}_{BO}.\\ \hline\end{array}$$

(A42)

Proof. 

Minimizing first over states with fixed $(m,q)$ and then over q is identical to minimizing over states with only $M_O=m$, proving Equation (A39). Applying Equation (118) to ${\rho }^{m,q}$ proves Equation (A40) and uniqueness. At the minimum, the source dual to q is zero, so the coarse partition Hessian is the principal block ${\chi }_{OO}$. Legendre inversion and the block inverse identity prove Equation (A42). □

Proposition A2

(Exposed-face limit of a diverging source). For $u\ne 0$, let

$$h\left(u\right):={\lambda }_{max}\left(O\left(u\right)\right),P_u:={\mathbf{1}}_{\left\{h\right(u\left)\right\}}\left(O\left(u\right)\right).$$

(A43)

For fixed θ set $B_{\theta }:=log\sigma +O\left(\theta \right)$. Then

$$\begin{array}{cc}\hfill {\lambda }_{\sigma ,O}(\theta +su)-sh\left(u\right)& ⟶log{Tr}_{P_u}exp\left(P_u{B}_{\theta }{P}_u\right),\hfill \end{array}$$

(A44)

$$\begin{array}{cc}\hfill {\rho }_{\theta +su}& {\displaystyle \underset{s\to \infty }{\overset{{∥·∥}_1}{\to }}}\frac{P_{u}exp\left(P_u{B}_{\theta }{P}_u\right)P_u}{{Tr}_{P_u}exp\left(P_u{B}_{\theta }{P}_u\right)}.\hfill \end{array}$$

(A45)

Here ${Tr}_{P_u}$ denotes the trace on $ran{P}_u$. Every boundary moment has only singular states in its fiber and is therefore unreachable by a finite source. A fixed ray reaches the relative interior of the exposed face selected by u; nonexposed strata can require nested diverging sources.

Proof. 

The Gibbs variational formula writes the left side of Equation (A44) as

$$\underset{\omega \in {\mathcal{S}}_n}{max}\left\{Tr\left(\omega B_{\theta }\right)+S_{\mathrm{vN}}\left(\omega \right)+s[Tr\left(\omega O\left(u\right)\right)-h\left(u\right)]\right\}.$$

(A46)

The bracket is nonpositive and vanishes exactly on states supported in $P_u$. Compactness forces every cluster point of maximizers into that face; the remaining strictly concave problem has the unique compressed Gibbs maximizer displayed in Equation (A45). This proves both limits. Finally, a finite-dimensional linear map sends the relative interior of the state space-the faithful states-onto the relative interior of its convex support, proving the boundary assertion. □

The compressed reference in Equation (A45) is governed by $P_{u}log\sigma P_u$, not generally by $log\left(P_u\sigma P_u\right)$. Boundary maximum-entropy inference can be discontinuous for noncommuting observables [10]. Since the finite-dimensional partition is analytic and has positive Hessian at every finite source, a true finite-source thermodynamic singularity requires a thermodynamic limit; fixed finite n only has infinite-source boundary selection.

Example A1

(A genuinely noncommuting qubit effective potential). Take $\sigma =I/2$, $O_1={\sigma }_x$, and $O_2={\sigma }_z$. Although $[O_1,O_2]\ne 0$, direct exponentiation gives

$${\mathcal{C}}_O=\{m\in {\mathbb{R}}^2:∥m∥\le 1\},\lambda \left(\theta \right)=logcosh∥\theta ∥,m\left(\theta \right)=tanh∥\theta ∥\frac{\theta }{∥\theta ∥}.$$

(A47)

The moment formula is understood continuously as $m\left(0\right)=0$. With $0<r:=∥m∥<1$ and $\widehat{m}:=m/r$,

$$\begin{array}{cc}\hfill \mathsf{\Gamma }\left(m\right)& =\frac{1}{2}\left[(1+r)log(1+r)+(1-r)log(1-r)\right],\hfill \\ \hfill \nabla \mathsf{\Gamma }\left(m\right)& =artanh\left(r\right)\frac{m}{r},\hfill \end{array}$$

(A48)

$$\begin{array}{cc}\hfill {\nabla }^2\mathsf{\Gamma }\left(m\right)& =\frac{1}{1-r^2}\widehat{m}{\widehat{m}}^{\mathsf{T}}+\frac{artanhr}{r}(I-\widehat{m}{\widehat{m}}^{\mathsf{T}}).\hfill \end{array}$$

(A49)

At the removable origin these formulas extend as $\nabla \mathsf{\Gamma }\left(0\right)=0$ and ${\nabla }^2\mathsf{\Gamma }\left(0\right)=I$. The information cost tends to the finite value $log2$ at a pure-state boundary, while the required field and radial stiffness diverge. The moment body is a disk rather than a classical joint-spectrum polytope; the effective-potential theorem therefore has genuine noncommuting content even though it has no joint-record large-deviation interpretation without an added measurement protocol.

Appendix C.1. Curvature of Shifted Connections and the Dual-Flat Identities

For a torsion-free connection D and an endomorphism-valued one-form A, the standard shifted-curvature identity is $R^{D+A}=R^D+d^{D}A+[A,A]$ [1,3]. Apply it to $\nabla ={\nabla }^{\mathrm{LC}}+K$ and ${\nabla }^{\ast }={\nabla }^{\mathrm{LC}}-K$. If both are flat,

$$0=R^{\mathrm{LC}}\pm d^{\mathrm{LC}}K+[K,K].$$

(A50)

Adding and subtracting yields

$$d^{\mathrm{LC}}K=0,R^{\mathrm{LC}}=-[K,K].$$

(A51)

These are the Codazzi and Gauss-type identities used above.

Appendix C.2. Born Midpoint Mechanics and Cubic Transport

Proof of Propositions 2 and 3. 

For $A=X_u^{H,\nabla }+U_u^V$ and $B=Y_u^{H,\nabla }+V_u^V$, the corresponding ${\nabla }^{\ast }$-decompositions replace $U,V$ by $U-2K_{X}u,V-2K_{Y}u$. Therefore the change in the right-hand side of Equation (12) is

$$-2g(X,K_{Y}u)+2g(K_{X}u,Y)=0,$$

(A52)

by full symmetry of $C_K$. Thus ${\omega }_B^{\nabla }={\omega }_B^{{\nabla }^{\ast }}$. In ∇-affine coordinates, $b_g^{\ast }{\mathsf{\Omega }}_{\mathrm{can}}=\mathrm{d}\left(g_{ij}{u}^j\right)\wedge \mathrm{d}{x}^i=g_{ij}\mathrm{d}{u}^j\wedge \mathrm{d}{x}^i=-{\omega }_B$: symmetry of ${\partial }_k{g}_{ij}$ cancels the horizontal wedge term. Each horizontal bundle is Lagrangian directly from Equation (12), and flatness makes it integrable. Pulling ${\iota }_{X_H}{\mathsf{\Omega }}_{\mathrm{can}}=-\mathrm{d}H$ back through $b_g$ gives

$${\iota }_{{\left(b_g^{-1}\right)}_{\ast }{X}_H}{\omega }_B=\mathrm{d}(H\circ b_g)=-\mathrm{d}\widehat{H}.$$

(A53)

The two decompositions in Equation (27) then follow from Equation (20). Finally, $\mathrm{d}{\eta }_i=g_{ij}\mathrm{d}{\theta }^j$ and ${\pi }^i=g^{ij}{p}_j$ give ${\widehat{\ell }}^{\ast }\left({\pi }^i\mathrm{d}{\eta }_i\right)=p_i\mathrm{d}{\theta }^i$, proving Equation (28) and its Hamiltonian conjugacy statement.

Putting $X=u$ in that horizontal-lift identity proves Equation (30). In ∇-affine coordinates $(x^i,u^i)$,

$${\omega }_B=g_{ij}\mathrm{d}{x}^i\wedge \mathrm{d}{u}^j,\mathrm{d}E=g_{ij}{u}^i\mathrm{d}{u}^j+\frac{1}{2}{\mathcal{C}}_{kij}{u}^i{u}^j\mathrm{d}{x}^k.$$

(A54)

Since $C_K=-\mathcal{C}/2$, contraction with the three sprays gives Equation (31). Using D-normal extensions at a point,

$${\left(\mathrm{d}\beta \right)}_u(U^V,X^{H,D})=2C_K(U,u,X).$$

(A55)

If $K_x\ne 0$, polarization of its symmetric cubic gives $U,u,X\in T_{x}M$ for which this quantity is nonzero. Because ${\mathcal{L}}_S{\omega }_B=\mathrm{d}\left({\iota }_S{\omega }_B\right)$, neither affine spray is symplectic in a neighborhood of that u. Hence on $T\mathcal{O}$ either spray preserves ${\omega }_B$ if and only if ${K|}_{\mathcal{O}}=0$; in that case both equal $S^D$ and are Hamiltonian for $-E$.

Equation (21) gives $K_X=-{\mathsf{C}}_X^{♯}/2$. Substitution in Appendix C.1 proves Equation (37). The standard horizontal-lift bracket and infinitesimal rectangular-holonomy formulas then give Equations (38) and (40) [41]. Ambrose-Singer identifies the restricted holonomy algebra with that generated by the parallel translates of these curvature endomorphisms, and composition of parallel transports gives Equation (39) [45]. □

Appendix D.1. BCH Residues and Invariant Tensors

Define the group-log vertical residue

$${\mathcal{V}}_{\mathrm{grp}}(sx,ty):=logU(s,t)\in \mathfrak{su}\left(n\right),x,y\in \mathfrak{p}.$$

(A56)

The antisymmetric part of positive-log composition recovers the Lie bracket:

$$[x,y]={\left.\frac{{\partial }^2}{\partial s\partial t}\left[{\mathcal{V}}_{\mathrm{grp}}(sx,ty)-{\mathcal{V}}_{\mathrm{grp}}(ty,sx)\right]\right|}_{s=t=0}.$$

(A57)

Appendix D.1.1. Group, Quotient, and Embedded Normalizations

Let

$$q:\mathrm{SL}(n,\mathbb{C})\to \mathrm{SL}(n,\mathbb{C})/\mathrm{SU}\left(n\right),j\left(g\mathrm{SU}\left(n\right)\right)=g{g}^{†},\pi :=j\circ q.$$

(A58)

Because the base state $P={\mathrm{e}}^U$ is represented by $P^{1/2}={\mathrm{e}}^{U/2}$,

$${\mathcal{V}}_{\mathrm{base}}(sU,tV):={\mathcal{V}}_{\mathrm{grp}}(sU/2,tV/2)=\frac{st}{8}[U,V]+O\left(3\right).$$

(A59)

Therefore

$$4{\left.{\partial }_s{\partial }_t\left[{\mathcal{V}}_{\mathrm{base}}(sU,tV)-{\mathcal{V}}_{\mathrm{base}}(tV,sU)\right]\right|}_{s=t=0}=[U,V].$$

(A60)

Differentiating $\pi \left(g\right)=g{g}^{†}$ gives $\mathrm{d}{\pi }_e\left(x\right)=2x$, whereas the square-root section gives ${\left(\mathrm{d}{s}_0\right)}_I\left(U\right)=U/2$, proving Equation (344) and the factor-of-four conversion.

For Hermitian generators, write

$$x=x^b{T}_b,y=y^c{T}_c,[T_b,T_c]=\mathrm{i}{f^a}_{bc}{T}_a,$$

(A61)

and write the compact residue as

$${\mathcal{V}}_{\mathrm{grp}}(sx,ty)=\mathrm{i}{v}^a(sx,ty)T_a,$$

(A62)

Equation (343) gives

$$v^a(sx,ty)=\frac{st}{2}{f^a}_{bc}{x}^b{y}^c+O\left(3\right).$$

(A63)

Hence

$${f^a}_{bc}{x}^b{y}^c={\left.\frac{{\partial }^2}{\partial s\partial t}\left[v^a(sx,ty)-v^a(ty,sx)\right]\right|}_{s=t=0}.$$

(A64)

This is Equation (A57) in components. For embedded base logarithms $U,V$, the corresponding recovery formula is Equation (A60). The four-segment commutator path, closed in the base through order ${\epsilon }^2$, has endpoint

$${\mathrm{e}}^{\epsilon x}{\mathrm{e}}^{\epsilon y}{\mathrm{e}}^{-\epsilon x}{\mathrm{e}}^{-\epsilon y}=exp\left({\epsilon }^2[x,y]+O\left({\epsilon }^3\right)\right).$$

(A65)

The first-order base displacement cancels, leaving compact vertical transport at order ${\epsilon }^2$ and a possible base displacement only at order ${\epsilon }^3$. An $O\left({\epsilon }^3\right)$ closing segment makes the base loop exact without changing the displayed leading residue.

Appendix D.2. One matrix product contains δ_ab, d_abc, and f_abc

Let $T_a$, $a=1,\dots ,n^2-1$, be traceless Hermitian generators normalized by

$$Tr\left(T_a{T}_b\right)=\frac{1}{2}{\delta }_{ab}.$$

(A66)

Their product decomposes as [83]

$$T_a{T}_b=\frac{1}{2n}{\delta }_{ab}I+\frac{1}{2}{d_{ab}}^c{T}_c+\frac{\mathrm{i}}{2}{f_{ab}}^c{T}_c.$$

(A67)

Therefore

$$\begin{array}{cc}\hfill \left\{T_a,T_b\right\}& =\frac{1}{n}{\delta }_{ab}I+{d_{ab}}^c{T}_c,\hfill \end{array}$$

(A68)

$$\begin{array}{cc}\hfill [T_a,T_b]& =\mathrm{i}{f_{ab}}^c{T}_c.\hfill \end{array}$$

(A69)

The log-determinant metric and Hessian cubic become

$$\begin{array}{cc}\hfill g_{ab}& =\frac{\alpha }{2}{\delta }_{ab},\hfill \end{array}$$

(A70)

$$\begin{array}{cc}\hfill {\mathcal{C}}_{abc}& =-\alpha Tr\left(T_a\left\{T_b,T_c\right\}\right)=-\frac{\alpha }{2}{d}_{abc}.\hfill \end{array}$$

(A71)

The commutator supplies $f_{abc}$; one matrix product therefore contains the metric tensor ${\delta }_{ab}$, Hessian cubic $d_{abc}$, and bracket/holonomy tensor $f_{abc}$. In terms of the half-difference convention,

$${\left(C_K\right)}_{abc}=\frac{\alpha }{4}{d}_{abc}.$$

(A72)

For $n=2$, the symmetric invariant vanishes:

$$d_{abc}=0.$$

(A73)

For $n\ge 3$, it is generically nonzero.

Appendix D.3. Log-Determinant Derivatives and Curvature

For a differentiable matrix path $P\left(t\right)$,

$$\frac{\mathrm{d}}{\mathrm{d}t}{P}^{-1}=-P^{-1}\dot{P}{P}^{-1},\frac{\mathrm{d}}{\mathrm{d}t}logdetP=Tr\left(P^{-1}\dot{P}\right).$$

(A74)

For $F\left(P\right)=-\alpha logdetP$, repeated differentiation gives

$$\begin{array}{cc}\hfill D{F}_P\left[U\right]& =-\alpha Tr\left(P^{-1}U\right),\hfill \end{array}$$

(A75)

$$\begin{array}{cc}\hfill D^2{F}_P[U,V]& =\alpha Tr\left(P^{-1}U{P}^{-1}V\right),\hfill \end{array}$$

(A76)

$$\begin{array}{cc}\hfill D^3{F}_P[U,V,W]& =-\alpha Tr\left(P^{-1}U{P}^{-1}V{P}^{-1}W+P^{-1}U{P}^{-1}W{P}^{-1}V\right).\hfill \end{array}$$

(A77)

Proof of Theorem 11. 

Let ${\mathcal{F}}_{+}\left(X\right):=Tr(XlogX)$, put $\mathsf{\Delta }:=A-B$, and set $X_s:=B+s\mathsf{\Delta }$. The Fréchet derivative of the matrix logarithm has the resolvent representation

$$D{log}_X\left[V\right]={\int }_0^{\infty }{(X+tI)}^{-1}V{(X+tI)}^{-1}\mathrm{d}t,$$

(A78)

for $X>0$ [46]. With $a_{X,t}$ from Equation (314) on $Herm\left(n\right)$ and $f_t\left(X\right):=-\alpha logdet(X+tI)$, this gives

$$\begin{array}{cc}\hfill D^2{\mathcal{F}}_{+,X}[U,V]& =Tr\left[UD{log}_X\left[V\right]\right]={\int }_0^{\infty }{a}_{X,t}(U,V)\mathrm{d}t,\hfill \end{array}$$

(A79)

$$\begin{array}{cc}\hfill D^2{f}_t{|}_X[U,V]& =\alpha a_{X,t}(U,V).\hfill \end{array}$$

(A80)

The reversed orientation of Theorem A3, applied first to ${\mathcal{F}}_{+}$ and then to every $f_t$, together with Equations (A79) and (A80), reduces the identity to interchanging the s- and t-integrals. Writing $R={(X_s+tI)}^{-1}$,

$$Tr\left(\mathsf{\Delta }R\mathsf{\Delta }R\right)={∥R^{1/2}\mathsf{\Delta }{R}^{1/2}∥}_{\mathrm{HS}}^2\ge 0,$$

so Tonelli’s theorem permits interchange of the s- and t-integrals.

$$\begin{array}{cc}\hfill B_{{\mathcal{F}}_{+}}(A,B)& =\frac{1}{\alpha }{\int }_0^{\infty }{B}_{f_t}(A,B)\mathrm{d}t\hfill \\ & =\frac{1}{\alpha }{\int }_0^{\infty }{D}_{\mathrm{LD}}^{\left(\alpha \right)}(A+tI\parallel B+tI)\mathrm{d}t.\hfill \end{array}$$

(A81)

Direct differentiation gives $B_{{\mathcal{F}}_{+}}(A,B)={\mathcal{D}}_{+}(A\parallel B)$. Equation (313) follows from Equation (A79).

For direct convergence, let $E_t:={(B+tI)}^{-1/2}(A-B){(B+tI)}^{-1/2}=O\left(t^{-1}\right)$. Then

$$\frac{1}{\alpha }{D}_{\mathrm{LD}}^{\left(\alpha \right)}(A+tI\parallel B+tI)=Tr{E}_t-logdet(I+E_t).$$

(A82)

For large t, the scalar estimate $0\le x-log(1+x)\le C{x}^2$ on a fixed neighborhood of zero gives an $O\left(t^{-2}\right)$ bound after diagonalizing $E_t$. The integrand is continuous on bounded t-intervals because $A,B>0$, completing the proof. □

Appendix D.4. Resolvent Curvature Calculation

Proof of Proposition 12. 

Write $R_t={(Q+tI)}^{-1}$. Differentiating the inverse and using cyclicity of trace gives

$$c_{Q,t}(U,V,W)=-Tr\left(R_{t}U{R}_{t}V{R}_{t}W+R_{t}U{R}_{t}W{R}_{t}V\right).$$

(A83)

Consequently,

$$a_{Q,t}\left(\frac{1}{2}(U{R}_{t}V+V{R}_{t}U),W\right)=-\frac{1}{2}{c}_{Q,t}(U,V,W),$$

(A84)

which proves Equation (321). On every compact subset of ${\mathcal{P}}_n$, the integrands defining $a_{Q,t}$ and $c_{Q,t}$ are respectively $O\left({(1+t)}^{-2}\right)$ and $O\left({(1+t)}^{-3}\right)$. Differentiation under the integral in Equation (313) is therefore justified and gives

$$g_Q^{\mathrm{KM}}={\int }_0^{\infty }{a}_{Q,t}\mathrm{d}t,{\mathcal{C}}_Q^{\mathrm{KM}}={\int }_0^{\infty }{c}_{Q,t}\mathrm{d}t.$$

(A85)

For arbitrary $U,W$, the definition of ${\mathsf{T}}_{Q,t}$ and Equation (A85) imply

$$g_Q^{\mathrm{KM}}\left({\int }_0^{\infty }{\mathsf{T}}_{Q,t}U\mathrm{d}t,W\right)={\int }_0^{\infty }{a}_{Q,t}(U,W)\mathrm{d}t=g_Q^{\mathrm{KM}}(U,W).$$

(A86)

Nondegeneracy proves the first identity in Equation (323). Similarly,

$$\begin{array}{cc}\hfill g_Q^{\mathrm{KM}}\left({\int }_0^{\infty }{\mathsf{A}}_{Q,t}\left(U\right)V\mathrm{d}t,W\right)& ={\int }_0^{\infty }{a}_{Q,t}(J_U^{\left(t\right)}V,W)\mathrm{d}t\hfill \\ & =-\frac{1}{2}{\mathcal{C}}_Q^{\mathrm{KM}}(U,V,W),\hfill \end{array}$$

(A87)

which is the defining identity for $K_U^{\mathrm{KM}}V$.

The mixture and exponential connections of the full-cone entropy Hessian are flat, so the calculation in Appendix C.1 gives $R^{\mathrm{KM}}(U,V)=-[K_U^{\mathrm{KM}},K_V^{\mathrm{KM}}]$. Since ${\mathsf{A}}_{Q,t}\left(U\right)=O\left({(1+t)}^{-3}\right)$ in finite-dimensional operator norm, the two Bochner integrals are absolutely convergent; Fubini’s theorem then yields Equation (324). Applying the same dual-flat identity to the shifted LogDet Hessian, whose half-difference is $J^{\left(t\right)}$, gives Equation (325).

It remains to justify the normalized-state qualification. Suppose $Q\in {\mathcal{S}}_n^{+}$ and restrict all forms to $T_Q{\mathcal{S}}_n^{+}={Herm}_0\left(n\right)$. The $a_{Q,t}$-normal to this hyperplane is $N_t:={(Q+tI)}^2$, because, for every traceless W,

$$a_{Q,t}(N_t,W)=TrW=0,a_{Q,t}(N_t,N_t)=Tr\left[{(Q+tI)}^2\right].$$

(A88)

Thus the $a_{Q,t}$-orthogonal projection of an ambient vector is exactly Equation (326). The restricted shifted half-difference is therefore ${\mathsf{\Pi }}_{Q,t}{J}_U^{\left(t\right)}V$. Repeating the preceding metric-musical calculation with the restricted forms proves the state-slice statement. The projection is precisely the term that retains the trace-normalization contribution to curvature. □

Proof of Corollary 20. 

The matrices $P_0^{-1}{P}_1$ and $P_0^{-1/2}{P}_1{P}_0^{-1/2}={\mathrm{e}}^Z$ are similar. Hence

$$Tr\left(A_0^{-1}{A}_1\right)=qTr{\mathrm{e}}^Z,logdet\left(A_0^{-1}{A}_1\right)=nlogq,$$

(A89)

which gives Equation (330). The scalar inequality $q-1-logq\ge 0$ and $Tr{\mathrm{e}}^Z\ge n{(det{\mathrm{e}}^Z)}^{1/n}=n$ prove separate nonnegativity. Apply the identity pointwise to $A_i\left(t\right)={\rho }_i+tI$ and use Equation (312); because the two terms are nonnegative and their sum is integrable, each is integrable. □

At I, define the Jordan multiplication operator

$$L_{X}Y:=\frac{1}{2}\{X,Y\}.$$

(A90)

Then

$$\begin{array}{cc}\hfill [L_X,L_Y]Z& =\frac{1}{4}\left(\{X,\{Y,Z\left\}\right\}-\{Y,\{X,Z\left\}\right\}\right)\hfill \end{array}$$

(A91)

$$\begin{array}{cc}& =\frac{1}{4}\left(XYZ-YXZ-ZXY+ZYX\right)\hfill \end{array}$$

(A92)

$$\begin{array}{cc}& =\frac{1}{4}[[X,Y],Z].\hfill \end{array}$$

(A93)

Because $K_X=L_X$ in the primal-affine convention,

$$R^{\mathrm{LC}}(X,Y)Z=-[K_X,K_Y]Z=-\frac{1}{4}[[X,Y],Z].$$

(A94)

For Hermitian $X,Y$, the commutator $[X,Y]$ is anti-Hermitian, so the curvature is generated by the compact algebra.

Using ${\left(gP{g}^{†}\right)}^{-1}={\left(g^{†}\right)}^{-1}{P}^{-1}{g}^{-1}$, trace cyclicity proves congruence invariance of Equation (346).

Appendix D.5. Full-Tangent Normalization and Diagonal Metric Separation

Proof of Proposition 13 

Let ${\rho }_{\ast }=I/n$ and let $V,W\in T_{{\rho }_{\ast }}{\mathcal{S}}_n^{+}$ be arbitrary Hermitian trace-zero tangents. Differentiating Equation (48) and using Jacobi’s formula gives, at every faithful $\rho$,

$$D{\mathsf{\Phi }}_{\rho }\left[V\right]={(det\rho )}^{-1/n}\left(V-\frac{1}{n}Tr\left({\rho }^{-1}V\right)\rho \right).$$

(A95)

At ${\rho }_{\ast }$, one has $\mathsf{\Phi }\left({\rho }_{\ast }\right)=I$, ${(det{\rho }_{\ast })}^{-1/n}=n$, and $Tr\left({\rho }_{\ast }^{-1}V\right)=nTrV=0$. Hence

$$D{\mathsf{\Phi }}_{{\rho }_{\ast }}\left[V\right]=nV,{\left({\mathsf{\Phi }}^{\ast }{g}_{\mathrm{AI}}\right)}_{{\rho }_{\ast }}(V,W)=\alpha n^{2}Tr\left(VW\right).$$

(A96)

On the other hand, Equation (57) gives ${\mathsf{K}}_{{\rho }_{\ast }}\left(A\right)=A/n$ on the entire matrix space. Thus Equation (59) yields

$$g_{{\rho }_{\ast }}^{\mathrm{BKM}}(V,W)=nTr\left(VW\right),{\left({\mathsf{\Phi }}^{\ast }{g}_{\mathrm{AI}}\right)}_{{\rho }_{\ast }}=\alpha n{g}_{{\rho }_{\ast }}^{\mathrm{BKM}}.$$

(A97)

This proves Equation (348) on the full tangent space, including noncommuting directions.

It remains to rule out one state-independent proportionality constant. It is enough to restrict both metrics to commuting diagonal states:

$$\rho =diag(p_1,\dots ,p_n),V=diag(v_1,\dots ,v_n),\sum _i{v}_i=0.$$

(A98)

There BKM reduces to the Fisher metric,

$$g_{\rho }^{\mathrm{BKM}}(V,V)=\sum _i\frac{v_i^2}{p_i}.$$

(A99)

For $P=\mathsf{\Phi }\left(\rho \right)$,

$$\frac{\delta P_i}{P_i}=\frac{v_i}{p_i}-\frac{1}{n}\sum _j\frac{v_j}{p_j},$$

(A100)

and hence

$${\left({\mathsf{\Phi }}^{\ast }{g}_{\mathrm{AI}}\right)}_{\rho }(V,V)=\alpha \sum _i{\left(\frac{v_i}{p_i}-\frac{1}{n}\sum _j\frac{v_j}{p_j}\right)}^2.$$

(A101)

These diagonal formulas agree with the preceding full-tangent identity at $p_i=1/n$. For $n=2$, take $p=(q,1-q)$ and $v=(a,-a)$ with $a\ne 0$:

$$g_{\rho }^{\mathrm{BKM}}(V,V)=\frac{a^2}{q(1-q)},{\left({\mathsf{\Phi }}^{\ast }{g}_{\mathrm{AI}}\right)}_{\rho }(V,V)=\frac{\alpha a^2}{2q^2{(1-q)}^2}.$$

(A102)

The ratio $\alpha /\left[2q\right(1-q\left)\right]$ varies with q. For $n>2$, take

$$p_1=p_2=t,p_3=\dots =p_n=\frac{1-2t}{n-2},v=(a,-a,0,\dots ,0),$$

(A103)

with $0<t<1/2$. Then

$$g_{\rho }^{\mathrm{BKM}}(V,V)=\frac{2a^2}{t},{\left({\mathsf{\Phi }}^{\ast }{g}_{\mathrm{AI}}\right)}_{\rho }(V,V)=\frac{2\alpha a^2}{t^2},$$

(A104)

whose ratio $\alpha /t$ also varies. □

Appendix D.6. Legendre-Cartan Curvature Calculation

Proof of Theorem 12. 

The global Gibbs differential Equation (65), in the Hessian-operator notation of this theorem, is

$$D{\rho }_X\left[H\right]=-{\mathsf{K}}_{{\rho }_X}\left({\tilde{H}}_{{\rho }_X}\right)=-{\mathsf{G}}_{X}H.$$

(A105)

Factoring ${\rho }_X^{1/2}$ from the Kubo-Mori integral yields

$${\mathsf{K}}_{{\rho }_X}\left(H\right)={\rho }_X^{1/2}\left[{\int }_{-1/2}^{1/2}{\mathrm{e}}^{s{ad}_X}\mathrm{d}s\right]\left(H\right){\rho }_X^{1/2}={\rho }_X^{1/2}{\mathsf{S}}_X\left(H\right){\rho }_X^{1/2},$$

(A106)

which proves Equation (350). In an eigenbasis of X, ${\mathsf{S}}_X$ multiplies the $(i,j)$ matrix entry by $sinhc((x_i-x_j)/2)>0$. Together with the positive Hessian in Equation (64), this proves that ${\mathsf{G}}_X$ and ${\mathsf{S}}_X$ are positive isomorphisms on the indicated spaces.

The identity $\mathsf{\Phi }\left({\rho }_X\right)=P_X={\mathrm{e}}^{-X}$ and the chain rule prove the second relation in Equation (351). Moreover,

$$\begin{array}{cc}\hfill P_X^{-1/2}D{P}_X\left[H\right]P_X^{-1/2}& =-{\int }_0^1{\mathrm{e}}^{(s-1/2)X}H{\mathrm{e}}^{-(s-1/2)X}\mathrm{d}s\hfill \\ & =-{\mathsf{S}}_X\left(H\right),\hfill \end{array}$$

(A107)

which proves Equation (352).

For $V=-{\mathsf{G}}_{X}H$ and $W=-{\mathsf{G}}_{X}K$, the same pullback identity reads

$$g_{{\rho }_X}^{\mathrm{BKM}}(V,W)=Tr\left(H{\mathsf{G}}_{X}K\right)=g_X^{\mathsf{\Psi }}(H,K).$$

(A108)

Thus $\mathcal{X}$ is an isometry, and naturality of the Levi-Civita connection and curvature proves Equation (354).

In the flat X-coordinates, the Hessian cubic is

$$D{g}_X^{\mathsf{\Psi }}\left[H\right](K,L)=Tr\left(KD{\mathsf{G}}_X\left[H\right]L\right).$$

(A109)

Hence the half-difference tensor in the convention of Equation (340) is $K_H^{\mathsf{\Psi }}=-{\mathsf{Q}}_X\left(H\right)/2$. The flat-dual curvature identity proved in Appendix C.1 gives $R^{\mathsf{\Psi }}=-[K^{\mathsf{\Psi }},K^{\mathsf{\Psi }}]$ and therefore Equation (355). On the symmetric-space side, whitening the three tangent arguments at $P_X$ converts the canonical formula Equation (347) into

$${\mathsf{T}}_X{\overline{R}}_X^{\mathrm{AI}}(H,K)L=-\frac{1}{4}\left[[{\mathsf{T}}_{X}H,{\mathsf{T}}_{X}K],{\mathsf{T}}_{X}L\right],$$

(A110)

which is Equation (356). Since $\mathsf{\Phi }=P_{•}\circ \mathcal{X}$, the chain rule, the BKM intertwining relation Equation (354), and the definition of the pulled-back affine-invariant curvature give

$$\begin{array}{cc}\hfill D{\mathsf{\Phi }}_{\rho }\left(R_{\rho }^{\mathrm{BKM}}(V,W)Z\right)& =D{P}_X\left[R_X^{\mathsf{\Psi }}(H,K)L\right],\hfill \end{array}$$

(A111)

$$\begin{array}{cc}\hfill R_{P_X}^{\mathrm{AI}}(D{\mathsf{\Phi }}_{\rho }V,D{\mathsf{\Phi }}_{\rho }W)D{\mathsf{\Phi }}_{\rho }Z& =D{P}_X\left[{\overline{R}}_X^{\mathrm{AI}}(H,K)L\right].\hfill \end{array}$$

(A112)

Subtracting proves Equation (358).

It remains to compute the trace-state defect. At $X=0$,

$${\mathsf{G}}_0=\frac{1}{n}\mathrm{id},D^3{\mathsf{\Psi }}_0(H,K,L)=-\frac{1}{2n}Tr\left(H\{K,L\}\right).$$

(A113)

Let

$${\mathsf{\Pi }}_{0}A:=A-\frac{TrA}{n}I,{\mathsf{J}}_{H}K:={\mathsf{\Pi }}_0\{H,K\}.$$

(A114)

Equation (A113) gives ${\mathsf{Q}}_0\left(H\right)K=-{\mathsf{J}}_{H}K/2$, and direct expansion gives the Jordan-Lie identity

$$[{\mathsf{J}}_H,{\mathsf{J}}_K]L=[[H,K],L]-\frac{4}{n}\left[Tr\left(KL\right)H-Tr\left(HL\right)K\right].$$

(A115)

Therefore

$$R_0^{\mathsf{\Psi }}(H,K)L=-\frac{1}{16}[{\mathsf{J}}_H,{\mathsf{J}}_K]L.$$

(A116)

Combining Equation (A115) with ${\overline{R}}_0^{\mathrm{AI}}(H,K)L=-[[H,K],L]/4$ proves Equation (359). On the unrestricted positive cone at a scalar point $Q=cI$, the projection ${\mathsf{\Pi }}_0$ is absent and the corresponding calculation gives $R_Q^{\mathrm{KM}}=R_Q^{\mathrm{AI}}/4$; the additional space-form term is therefore exactly the trace-one normalization contribution.

For $\alpha =1/n$, both center metrics are $g_0^{\mathsf{\Psi }}$, so contraction on a two-plane gives Equation (360). For $n\ge 3$, two independent commuting traceless diagonal directions have zero affine-invariant curvature and nonzero BKM curvature. For $n=2$, Equation (A115) makes $R_0^{\mathsf{\Psi }}=0$, while ${\overline{R}}_0^{\mathrm{AI}}\ne 0$. These examples already rule out curvature intertwining by the displayed natural bridge.

Under $\alpha =1/n$, the adjoint Casimir contraction in the normalization $g_0^{\mathsf{\Psi }}(H,K)=Tr\left(HK\right)/n$ gives

$${Ric}_0^{\mathrm{AI}}=-\frac{n^2}{2}{g}_0^{\mathsf{\Psi }}.$$

(A117)

Contracting the two terms in Equation (359), in dimension $n^2-1$, now gives Equation (361), in agreement with the direct monotone-metric calculations after translating their metric and curvature normalization conventions [28,47].

Suppose more generally that an invertible real map $L:\mathfrak{p}\to \mathfrak{p}$ conjugated the two curvature operators. For fixed $K,Z$, the endomorphisms $H\mapsto R_0^{\mathsf{\Psi }}(H,K)Z$ and $H^{\prime }\mapsto {\overline{R}}_0^{\mathrm{AI}}(H^{\prime },LK)LZ$ would then be conjugate by L and have equal traces. Hence

$${Ric}_0^{\mathsf{\Psi }}(K,Z)={\overline{Ric}}_0^{\mathrm{AI}}(LK,LZ).$$

(A118)

For $n\ge 3$ the left side is positive definite and the right side negative definite; for $n=2$ the left side vanishes and the right side does not. This proves the unrestricted no-conjugacy statement in the theorem.

It remains to prove Corollary 23. From Equation (353) and Equation (A108),

$$g_X^{\mathsf{\Psi }}({\mathsf{Q}}_X\left(H\right)K,L)=D{g}_X^{\mathsf{\Psi }}\left[H\right](K,L)=D^3{\mathsf{\Psi }}_X(H,K,L),$$

(A119)

so $\mathsf{Q}$ is the raised entropy cubic. Applying the shifted curvature formula to $D^{\mathsf{\Psi }}={\overline{D}}^{\mathrm{AI}}+\mathcal{A}$ gives the second identity in Equation (363).

Because ${\mathsf{S}}_X=\mathrm{id}+O\left({∥X∥}^2\right)$, the pulled-back affine-invariant metric is in exponential normal coordinates at the origin: ${\left.\mathrm{d}{\overline{g}}^{\mathrm{AI}}\right|}_0=0$ and ${\overline{D}}_0^{\mathrm{AI}}=0$. The Levi-Civita connection of a Hessian metric in its flat coordinates is

$$D_H^{\mathsf{\Psi }}K=\frac{1}{2}{\mathsf{Q}}_X\left(H\right)K,$$

(A120)

at the origin. Together with ${\mathsf{Q}}_0\left(H\right)K=-{\mathsf{\Pi }}_0\{H,K\}/2$, this proves the connection formula in Equation (364).

In a common frame along $\gamma$, parallel transport satisfies $\dot{P}=-\mathsf{\Gamma }\left(\dot{\gamma }\right)P$. Differentiating ${\mathcal{Q}}_t={\left({\overline{P}}_t^{\mathrm{AI}}\right)}^{-1}{P}_t^{\mathsf{\Psi }}$ yields

$${\dot{\mathcal{Q}}}_t=-{\left({\overline{P}}_t^{\mathrm{AI}}\right)}^{-1}{\mathcal{A}}_{\gamma \left(t\right)}\left(\dot{\gamma }\left(t\right)\right){\overline{P}}_t^{\mathrm{AI}}{\mathcal{Q}}_t,{\mathcal{Q}}_0=\mathrm{id},$$

(A121)

whose solution is Equation (365). Applying $P_{{\gamma }_{\epsilon }}=\mathrm{id}-{\epsilon }^{2}R(H,K)+O\left({\epsilon }^3\right)$ to both connections and subtracting proves Equation (366). □

Proof by metric normalization and connection comparison. 

Trace self-adjointness of the even operator ${\mathsf{S}}_X$ gives

$${\overline{g}}_X(H,K)=Tr\left(H\alpha {\mathsf{S}}_X^{2}K\right)=g_X^{\mathsf{\Psi }}\left(H,{\mathsf{G}}_X^{-1}\left(\alpha {\mathsf{S}}_X^{2}K\right)\right),$$

which proves Equation (381). The positive-normalizer lemma gives $J={\mathsf{D}}^{1/2}$ and its isometry property. If $L_X=D{P}_X$, then $L_X^{†}{L}_X={\mathsf{D}}_X$, so ${\tilde{J}}_X=L_X{\mathsf{D}}_X^{-1/2}$ is precisely the positive polar isometry in Equation (383); this uses no commutation of ${\mathsf{G}}_X$ and ${\mathsf{S}}_X$.

Under $\alpha \mapsto c\alpha$, $\mathsf{D}\mapsto c\mathsf{D}$ and $J\mapsto \sqrt{c}J$. Constant metric rescaling leaves ${\overline{D}}^{\mathrm{AI}}$ unchanged and the scalar factors cancel from $J{\overline{D}}^{\mathrm{AI}}{J}^{-1}$, proving scale independence. Moreover, ${\overline{D}}^{\mathrm{AI}}\left(J^{-1}\right)=-J^{-1}\left({\overline{D}}^{\mathrm{AI}}J\right)J^{-1}$; hence

$${\widehat{D}}^{\mathrm{AI}}={\overline{D}}^{\mathrm{AI}}-\left({\overline{D}}^{\mathrm{AI}}J\right)J^{-1},$$

which is exactly Equation (387).

Transport of a connection through a bundle isometry, the skewness of the difference of two metric connections, the gauge laws for curvature and parallel transport, and the relative-transport differential equation are standard connection theory [41,82]. Applied to the positive J, these facts give the metricity of ${\widehat{D}}^{\mathrm{AI}}$, Equations (385) and (386), and

$${\widehat{R}}^{\mathrm{AI}}=J{\overline{R}}^{\mathrm{AI}}{J}^{-1},{\widehat{P}}_t^{\mathrm{AI}}=J_{\gamma \left(t\right)}{\overline{P}}_t^{\mathrm{AI}}{J}_x^{-1}.$$

Both transports preserve orientation and g, so their relative product is in $\mathrm{SO}(T_{x}M,g_x)$. The standard comparison ODE for $U_t={\left({\widehat{P}}_t^{\mathrm{AI}}\right)}^{-1}{P}_t^{\mathsf{\Psi }}$ gives Equation (389); factorizing the transports of two composable paths gives the manuscript-specific twisted cocycle (390).

Finally, the standard change-of-connection curvature identity, the two Bianchi identities, and the infinitesimal rectangular-holonomy formula [41,81] give, with the wedge and curvature signs fixed in the statement, Equations (391), (393) and (394). A simultaneous oriented frame change cancels at the terminal endpoint and conjugates $U_{\gamma }^{\mathrm{rel}}$ at x, proving the stated open-path conjugacy invariants. Thus standard gauge identities supply the comparison machinery, while the positive BKM-AI normalization, scale cancellation, raw-defect bridge, twisted composition, and signs remain explicit. □

Proof of Proposition 17. 

At the center, ${\mathsf{G}}_0=n^{-1}\mathrm{id}$, ${\mathsf{S}}_0=\mathrm{id}$, and ${\left(\mathrm{D}\mathsf{S}\right)}_0=0$. Differentiating ${\mathsf{D}}^{\left(\alpha \right)}={\mathsf{G}}^{-1}\alpha {\mathsf{S}}^2$, and using $\mathsf{Q}={\mathsf{G}}^{-1}\mathrm{D}\mathsf{G}$, gives ${\mathsf{D}}_0^{\left(\alpha \right)}=\alpha n\mathrm{id}$ and ${\left(\mathrm{D}{\mathsf{D}}^{\left(\alpha \right)}\right)}_0\left[H\right]=-\alpha n{\mathsf{Q}}_0\left(H\right)$. The Fréchet derivative of the square root at the scalar operator $\alpha n\mathrm{id}$ is multiplication by ${\left(2\sqrt{\alpha n}\right)}^{-1}$, proving Equation (406). In constant X-coordinate fields,

$${\left({\widehat{\mathsf{\Gamma }}}_0^{\mathrm{AI}}\right)}_{H}K=J_0^{\left(\alpha \right)}{\left(\mathrm{D}{\left(J^{\left(\alpha \right)}\right)}^{-1}\right)}_0\left[H\right]K=\frac{1}{2}{\mathsf{Q}}_0\left(H\right)K={\left({\mathsf{\Gamma }}_0^{\mathsf{\Psi }}\right)}_{H}K,$$

(A122)

because ${\overline{\mathsf{\Gamma }}}_0^{\mathrm{AI}}=0$. This proves Equation (407) for every $\alpha >0$. Substitution of Equation (359) proves Equation (408).

By 3, $b^{\ast }b=n^2{\mathsf{\Pi }}_{\mathrm{ad}}$. Substitution into the affine-invariant central curvature and Equation (359) gives Equation (409). Put $V=\mathfrak{p}$. For ${\widehat{\mathcal{F}}}_0^{\mathrm{rel}}$, small rectangles with side $\sqrt{t}$ give one-sided tangent curves at the identity in the directions $-{\mathcal{F}}_0^{\mathrm{rel}}(H,K)$; reversing the rectangles gives the opposite directions. Decomposable bivectors linearly span ${\mathsf{\Lambda }}^{2}V$, so the Lie algebra of the closed group in Equation (410) is all $\mathfrak{so}(n^2-1)$. This proves Equation (411). □

Proof of Corollary 27. 

For $n\ge 3$, ${\widehat{R}}_0^{\mathsf{\Psi }}$ is onto $\mathfrak{so}(n^2-1)$, so Ambrose-Singer [45] gives restricted holonomy $\mathrm{SO}(n^2-1)$. The global Gibbs chart identifies M with ${\mathbb{R}}^{n^2-1}$, so M is simply connected; full and restricted BKM holonomy therefore agree. □

Proof of Theorem 15. 

Specializing Lemma 3 and Equation (409) to $n=3$ proves .

Let $e_a=\sqrt{6}{T}_a$ and use the notation of Equation (431). The symmetric-cube and Weyl character formulas, using the standard tensor-product data and irreducible dimensions in [33], Tables 23-24, give Equation (430), with component dimensions $1,8,10,10,27,64$. Since $d_{abc}$ is a nonzero invariant symmetric cubic and the trivial summand has multiplicity one, it spans that summand.

Equation (A113) and Equation (A68) give

$$D^3{\mathsf{\Psi }}_0(e_a,e_b,e_c)=-\frac{1}{6}Tr\left(e_a\{e_b,e_c\}\right)=-\sqrt{\frac{3}{2}}{d}_{abc}.$$

(A123)

Because $\left\{e_a\right\}$ is $g_0$-orthonormal, raising an index gives ${\mathsf{Q}}_0\left(e_a\right)=-\sqrt{3/2}{\mathsf{d}}_a$ and hence $K_{0,e_a}^{\mathsf{\Psi }}=\sqrt{3/8}{\mathsf{d}}_a$.

Put ${\left(A_e\right)}_{cd}:=f_{ecd}$. In the normalization of Equation (A66), the contraction $f_{amn}{f}_{bmn}=3{\delta }_{ab}$ and the standard $SU\left(3\right)$f-d identity give directly

$${\left({\mathsf{\Pi }}_8{J}_{ab}\right)}_{cd}=\frac{2}{3}{f}_{abe}{f}_{ecd},[{\mathsf{d}}_a,{\mathsf{d}}_b]=\frac{5}{6}{\mathsf{\Pi }}_8{J}_{ab}-\frac{2}{3}{\mathsf{\Pi }}_{20}{J}_{ab}$$

[83]. Together with $R^{\mathsf{\Psi }}=-[K^{\mathsf{\Psi }},K^{\mathsf{\Psi }}]$, this proves the commutator and curvature identities in Equation (432). The two nonzero projector eigenvalues then prove Equation (434) and show that the 28 images of the basis bivectors are linearly independent. □

Proof of Theorem 16. 

The shared product formula Equation (461) gives Equation (464). Relative to $J_n=\mathbb{R}I\oplus V_n$, its multiplication operator is

$$L_X=\left(\begin{array}{cc}0& X^{♭}\\ X& M_X\end{array}\right).$$

(A124)

Its commutator has zero scalar row and column and lower block $W_{X,Y}+[M_X,M_Y]$. By Equation (A93), this lower block equals $\frac{1}{4}{ad}_{[X,Y]}$ on $V_n$, proving Equation (465). The omitted off-diagonal block acts by

$$P_0{L}_X{P}_{\mathbf{1}}{L}_Y{P}_0\left(Z\right)=g_0(Y,Z)X;$$

(A125)

antisymmetrization proves Equation (466). Notice that $\mathbb{R}I\oplus V_n$ is an orthogonal vector-space decomposition, not a decomposition into Jordan ideals.

Raising the index in Equation (460) and using $K^{\mathsf{\Psi }}=-\mathsf{Q}/2$ proves Equation (467). Dual flatness gives $R^{\mathsf{\Psi }}=-[K^{\mathsf{\Psi }},K^{\mathsf{\Psi }}]$, so Equation (465) yields Equation (468), giving the normalization interpretation of the BKM row in Equation (409).

For $n=3$, Newton’s identity for trace-free $3\times 3$ matrices is $Tr\left(X^3\right)=3detX$, proving the first part of Equation (469). If $X=x^a{T}_a$, then Equation (A67) and symmetry of $x^a{x}^b{x}^c$ give $Tr\left(X^3\right)=\frac{1}{4}{d}_{abc}{x}^a{x}^b{x}^c$; the antisymmetric f-term vanishes. Newton’s identity therefore also proves Equation (470). If $r^2=g_0(X,X)$, differentiation gives

$${grad}_{g_0}F=3\sqrt{2}{P}_0\left(X^2\right).$$

(A126)

Cayley-Hamilton gives $Tr\left(X^4\right)=\frac{1}{2}Tr{\left(X^2\right)}^2$, whence ${∥P_0\left(X^2\right)∥}_{g_0}^2=r^4/2$ and ${∥gradF∥}_{g_0}^2=9r^4$. The cubic is harmonic because the trace of its Hessian would be a $\mathrm{PSU}\left(3\right)$-invariant linear form on the nontrivial irreducible module $V_3$, and hence vanishes. This is Cartan’s eight-dimensional cubic normalization [119,120].

The stabilizer identity Equation (471) is the $n=3$ specialization of Proposition 22; here ${det}_{V_3}{\tau }_{\mathrm{J}}={(-1)}^3=-1$.

The reduced structure algebra of a simple Euclidean Jordan algebra is generated by its derivations and trace-free multiplication operators [77,121]. Here the explicit real representation $\rho \left(A\right)Z=AZ+Z{A}^{†}$ sends $A^{†}=-A$ to $[A,Z]$ and $A^{†}=A$, $TrA=0$, to $2L_{A}Z$. Dimension and faithfulness then prove Equation (473).

For qutrits, ${\mathsf{\Pi }}_{20}{ad}_{[X,Y]}=0$, so complementary projection of Equation (465) proves the first identity in Equation (474); projection of Equation (468) proves the second. The complex branching follows from Equation (482). This completes the pointwise and algebraic claims of the theorem. □

Proof of Corollary 30. 

Corollary 18 gives the diagonal simplex’s total geodesy and Fisher curvature $1/4$. Diagonal-unitary conjugations are also isometries of the pulled-back affine-invariant metric, so their common fixed set is totally geodesic for that metric as well. Its logarithmic-coordinate expression is constant and hence flat. With the manuscript curvature convention $R^{\mathrm{BKM}}(H,K)=-\mathsf{j}/4$, Abelian Stokes theory on the simply connected simplex gives Equation (490).

For $n=2$, the relative transporter is $\mathrm{SO}\left(3\right)$-valued. The Gibbs chart identifies the faithful qubit manifold with oriented, contractible ${\mathbb{R}}^3$, whose unique spin structure lifts metric connections to $Spin\left(3\right)\simeq \mathrm{SU}\left(2\right)$ [82,84]. The $n=2$ part of Equation (359) gives $R_0^{\mathsf{\Psi }}=0$. Substitution into Equation (394) proves Equation (491); the identification of the affine-invariant rank-one rotation with Thomas-Wigner rotation uses the quotient normalization in Equation (344) and [26,110].

For $n=3$, independent commuting diagonal $H,K$ span the diagonal Cartan plane. Its affine-invariant curvature is zero, while Equation (359) gives Equation (492). By Equation (428), $W_{H,K}={\mathsf{\Pi }}_{20}\left(W_{H,K}\right)\ne 0$ for these independent commuting directions. Thus Equation (394) excludes $\mathrm{PSU}\left(3\right)$-valued relative transport locally, without turning the relative comparisons into a connection holonomy.

For the stronger statement, the real split ${\mathsf{\Lambda }}^2\mathbf{8}=\mathbf{8}\oplus \mathbf{20}$ is the $\mathrm{PSU}\left(3\right)$ decomposition used by Puhle [31]; after complexification, ${\mathbf{20}}_{\mathbb{C}}=\mathbf{10}\oplus \overline{\mathbf{10}}$ follows from the standard adjoint branching [33], Table 24. The ${\mathbb{Z}}_3$-graded brackets in Equation (493) are Lie-algebraic [32], Remark 3.1; they require no global order-three symmetry of $\mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right)$. Complex conjugation exchanges $\mathbf{10}$ and $\overline{\mathbf{10}}$, so their sum is the complexification of a real irreducible 20-dimensional module.

For the global qualification, let $p:Spin\left(8\right)\to \mathrm{SO}\left(8\right)$. The adjoint subgroup $\mathrm{PSU}\left(3\right)\subset \mathrm{SO}\left(8\right)$ lifts injectively to $\tilde{H}\simeq \mathrm{PSU}\left(3\right)$ because ${\pi }_1\left(\mathrm{PSU}\left(3\right)\right)={\mathbb{Z}}_3\to {\pi }_1\left(\mathrm{SO}\left(8\right)\right)={\mathbb{Z}}_2$ is zero. Hence $Spin\left(8\right)/\tilde{H}$ is the simply connected double cover of $\mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right)$. The grading integrates to an order-three triality automorphism $\theta$ with fixed Lie algebra $\mathfrak{su}\left(3\right)$, giving this cover its global three-symmetry. Triality cycles the three nonidentity elements of $Z(Spin\left(8\right))\simeq {\mathbb{Z}}_2^2$ [122], Section 2, so $\theta$ does not preserve $kerp$. Since $\tilde{H}\cap Z(Spin\left(8\right))=\left\{1\right\}$, the induced coset symmetry conjugates the nontrivial deck translation by $z\in kerp$ to the distinct central translation by $\theta \left(z\right)$ outside the deck group. It therefore does not descend to $\mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right)$. The reduction orbit remains $\mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right)$, with the homotopy groups in Equation (A184).

If $H,K$ are independent, $W_{H,K}$ has rank two. Every nonzero $A\in \mathfrak{su}\left(3\right)$ is unitarily diagonalizable. Tracelessness excludes three equal eigenvalues, so their multiplicities are $1+1+1$ or $2+1$; the centralizer in $\mathfrak{su}\left(3\right)$ consequently has dimension two or four. Hence $rank\left({ad}_A\right)=8-dim{Z}_{\mathfrak{su}\left(3\right)}\left(A\right)\in \{6,4\}$. Thus $W_{H,K}$ cannot lie in $ad\mathfrak{su}\left(3\right)$. This proves Equation (494). The commuting case was established above. The separate principal/tangent distinction follows from Corollary 24. □

Oriented polarization in Remark 12. 

Use the convention $\left(d_{\beta }\xi \right)(X,Y)=-\xi \left(\beta (X,Y)\right)$ on one-forms and the induced exterior-power inner products. Compatibility ensures that $d_{\beta }^2=0$; the Cartan form is closed, and unimodularity gives $d_{\beta }{\mathsf{\Lambda }}^7{E}^{\ast }=0$ [30]. For two-forms $\eta ,\xi$, Hodge duality therefore gives

$$\begin{array}{cc}\hfill \langle \eta ,{\mathsf{A}}_{\phi }\xi \rangle \mathrm{vol}& =\phi \wedge d_{\beta }\eta \wedge \xi ,\hfill \\ \hfill \left(\langle \eta ,{\mathsf{A}}_{\phi }\xi \rangle +\langle \xi ,{\mathsf{A}}_{\phi }\eta \rangle \right)\mathrm{vol}& =-d_{\beta }(\phi \wedge \eta \wedge \xi )=0.\hfill \end{array}$$

Thus ${\mathsf{A}}_{\phi }$ is skew-adjoint. Under the metric identification, ${\mathfrak{h}}_8=im(d_{\beta }:{\mathsf{\Lambda }}^1{E}^{\ast }\to {\mathsf{\Lambda }}^2{E}^{\ast })$. Since $d_{\beta }^2=0$, the first identity gives $im{\mathsf{A}}_{\phi }\subseteq {\mathfrak{m}}_{20}$; skew-adjointness then gives ${\mathsf{A}}_{\phi }{|}_{{\mathfrak{h}}_8}=0$.

All defining operations are $\mathrm{PSU}\left(3\right)$-equivariant. On the real irreducible ${\mathfrak{m}}_{20}$, the self-adjoint equivariant operator ${\mathsf{A}}_{\phi }^2$ is scalar. To prove that it is strictly negative, it suffices, up to isometry and nonzero scaling, to use the orthonormal Cartan model, with $e^{ijk}:=e^i\wedge e^j\wedge e^k$ and orientation $e^{12345678}$,

$$\begin{array}{cc}\hfill {\phi }_f=& e^{123}+\frac{1}{2}(e^{147}-e^{156}+e^{246}+e^{257}+e^{345}-e^{367})\hfill \\ \hfill & +\frac{\sqrt{3}}{2}(e^{458}+e^{678}).\hfill \end{array}$$

Its Chevalley-Eilenberg differential $d_f$ gives directly

$$\ast ({\phi }_f\wedge e^{38})=d_f{e}^{12}=\frac{1}{2}(e^{146}+e^{157}-e^{247}+e^{256}).$$

Hence $\langle e^{12},{\mathsf{A}}_{{\phi }_f}{e}^{38}\rangle ={∥d_f{e}^{12}∥}^2=1$. This proves nonvanishing and therefore ${\mathsf{A}}_{\phi }^2=-a_{\phi }^2{\mathsf{\Pi }}_{20}$ with $a_{\phi }>0$. Taking traces gives the normalization in Equation (497); skew-adjointness then makes $J_{\perp }^{\mathrm{or}}$ orthogonal.

For $t\ne 0$, $d_{t\beta }=t{d}_{\beta }$ and ${\mathsf{A}}_{t\phi }=t^2{\mathsf{A}}_{\phi }$, so the normalized structure depends only on the Cartan line. Reversing the orientation changes the Hodge star’s sign and hence reverses $J_{\perp }^{\mathrm{or}}$. The construction is natural under oriented metric-Cartan isomorphisms; it therefore globalizes smoothly on an oriented Cartan-line reduction and is parallel for any metric connection preserving that line. Finally, the tracial metric-cubic stabilizer $H_0$ determines the line ${\left({\mathsf{\Lambda }}^3{E}^{\ast }\right)}^{H_0}$: it contains a nonzero Cartan form and is one-dimensional because the adjoint representation occurs once in ${\mathsf{\Lambda }}^2\mathbf{8}=\mathbf{8}\oplus \mathbf{20}$. This proves the central-fiber assertion without supplying a connection or globalizing the central BKM cubic itself. □

Proof of Proposition 24. 

Choose a local $P_G$-frame. Because ${\widehat{D}}^{\mathrm{AI}}$ preserves the reduction, its connection form is $\mathfrak{g}$-valued. The connection form of $D^{\mathsf{\Psi }}$ is its sum with ${\mathcal{B}}^{\mathrm{rel}}$, so the first two conditions are equivalent. If ${\mathcal{B}}^{\mathrm{rel}}$ is $\mathfrak{g}$-valued, conjugation by the reference G-transport keeps the integrand in Equation (389) inside $\mathfrak{g}$; hence every relative transporter lies in G. Conversely, along a short segment with initial velocity X in a reference-parallel G-frame,

$$U_{\epsilon }^{\mathrm{rel}}=\mathrm{id}-\epsilon {\mathcal{B}}^{\mathrm{rel}}\left(X\right)+O\left({\epsilon }^2\right).$$

(A127)

Membership in G for all such segments forces ${\mathcal{B}}^{\mathrm{rel}}\left(X\right)\in \mathfrak{g}$, proving equivalence. The curvature necessity follows from Equation (391); it is not sufficient because curvature membership alone neither supplies a preserved reduction nor controls global holonomy.

The frame-change law proved above implies conjugation invariance of Equation (498). From Equation (394), write $logU=-{\epsilon }^{2}F+O\left({\epsilon }^3\right)$, with every coefficient skew and hence traceless. Expansion of the matrix exponential gives

$$\frac{1}{d}TrU=1+\frac{{\epsilon }^4}{2d}Tr\left(F^2\right)+O\left({\epsilon }^5\right)=1-\frac{{\epsilon }^4}{2d}{∥F∥}_{\mathrm{HS}}^2+O\left({\epsilon }^5\right),$$

(A128)

which proves Equation (499). □

Proof of Theorem 17 and Corollary 32. 

The Cartan three-form is parallel for the affine-invariant connection by Proposition 25. Pullback preserves this statement. From the definition ${\widehat{D}}^{\mathrm{AI}}=J{\overline{D}}^{\mathrm{AI}}{J}^{-1}$, differentiating Equation (501) gives ${\widehat{D}}^{\mathrm{AI}}\phi =0$. At one oriented orthonormal frame its components are a nonzero multiple of the compact Cartan tensor $f_{abc}$. Define ${\beta }_0$ by $g_0({\beta }_0(U,V),W)={\phi }_0(U,V,W)$; it is a nonzero scalar multiple of $\mathrm{i}[U,V]$. If $R\in \mathrm{O}(V_3,g_0)$, then $R^{\ast }{\phi }_0={\phi }_0$ if and only if

$${\beta }_0(RU,RV)=R{\beta }_0(U,V).$$

(A129)

Consequently its full orthogonal stabilizer is $Aut\left(\mathfrak{su}\right(3\left)\right)$; the full linear-stabilizer equality follows from the general Cartan-three-form stabilizer theorem [29,30].

Now $[U^{\mathsf{T}},V^{\mathsf{T}}]=-{[U,V]}^{\mathsf{T}}$, so direct substitution gives ${\tau }_{\mathrm{J}}^{\ast }{\phi }_0=-{\phi }_0$. The nontrivial outer Lie automorphism is instead ${\tau }_{\mathfrak{g}}=-{\tau }_{\mathrm{J}}$, the Hermitian representative of complex conjugation on $\mathfrak{su}\left(3\right)$, and trilinearity gives ${\tau }_{\mathfrak{g}}^{\ast }{\phi }_0={\phi }_0$. Its $+1$ and $-1$ eigenspaces have dimensions three and five, respectively, so $det{\tau }_{\mathfrak{g}}=-1$. The determinant cubic is preserved by ${\tau }_{\mathrm{J}}$ and reversed by ${\tau }_{\mathfrak{g}}$, proving Equations (503)–(505) and the asserted oriented parallel reduction. The adjoint action is orthogonal, so $\mathfrak{so}\left(8\right)={\mathfrak{h}}_8\oplus {\mathfrak{m}}_{20}$ is reductive and the two associated subbundles and projections in Equation (509) are ${\widehat{D}}^{\mathrm{AI}}$-parallel.

At the center, Equation (406) gives $J_0^{\left(\alpha \right)}=\sqrt{3\alpha }\mathrm{id}$, so normalization does not conjugate the embedded adjoint subgroup. Hence Equations (378) and (401) and the two central stabilizer calculations prove Equation (506).

For two paths $\gamma ,\eta :0\to X$,

$$h_{\eta ,\gamma }:=P_{\eta }^{-1}{P}_{\gamma }\in {Hol}_0\left({\widehat{D}}^{\mathrm{AI}}\right)=H_0.$$

(A130)

Thus $h_{\eta ,\gamma }$ fixes ${\mathcal{C}}_0^{\mathsf{\Psi }}$, which proves that Equation (507) is path-independent. Standard parallel transport gives smoothness, and a parallel tensor with prescribed value at one point is unique on the connected Gibbs-coordinate manifold. If $u_0$ is the common central adjoint frame and $u_{\gamma }=P_{\gamma }{u}_0$, parallelness of both tensors gives ${\left(P_{{\mathcal{C}}^J}\right)}_X=u_{\gamma }{H}_0={\left(P_{\phi }\right)}_X$. This proves Equation (508) without using the later projected connection or its curvature, so the construction is not circular.

In a local $P_{\phi }$-frame the connection form of ${\widehat{D}}^{\mathrm{AI}}$ is ${\mathfrak{h}}_8$-valued. The BKM connection form is its sum with ${\mathcal{B}}^{\mathrm{rel}}$, so its orthogonal ${\mathfrak{h}}_8$ projection is precisely the connection form of $D^{\mathrm{col}}$. This proves metricity, preservation of the reduction, and the exact reconstruction Equation (510). Since ${\mathfrak{h}}_{\phi }$ is the infinitesimal stabilizer of both $\phi$ and ${\mathcal{C}}^J$, the same local connection form proves Equation (513); naturality of Equation (497) gives both stated polarization parallelness identities. Since

$$T^{\mathsf{\Psi }}(X,Y)=T^{\mathrm{col}}(X,Y)+{\mathsf{\Phi }}_{20}\left(X\right)Y-{\mathsf{\Phi }}_{20}\left(Y\right)X$$

(A131)

and $T^{\mathsf{\Psi }}=0$, Equation (514) follows.

The standard connection-shift identity quoted in Appendix C.1, applied to $D^{\mathsf{\Psi }}=D^{\mathrm{col}}+{\mathsf{\Phi }}_{20}$, proves Equation (516); reductivity then gives its two projections. Applied to the dual-flat pair $\nabla =D^{\mathsf{\Psi }}+K^{\mathsf{\Psi }}$, ${\nabla }^{\ast }=D^{\mathsf{\Psi }}-K^{\mathsf{\Psi }}$, the same identity gives $R^{\mathsf{\Psi }}=-[K^{\mathsf{\Psi }},K^{\mathsf{\Psi }}]$ and ${\mathrm{d}}^{D^{\mathsf{\Psi }}}{K}^{\mathsf{\Psi }}=0$ [1,3]. Since ${\mathrm{d}}^{D^{\mathsf{\Psi }}}{K}^{\mathsf{\Psi }}={\mathrm{d}}^{D^{\mathrm{col}}}{K}^{\mathsf{\Psi }}+[{\mathsf{\Phi }}_{20}\wedge K^{\mathsf{\Psi }}]$, this proves Equations (517)–(519). Finally, ${\widehat{R}}^{\mathrm{AI}}$ is ${\mathfrak{h}}_{\phi }$-valued, so complementary projection of ${\mathcal{F}}^{\mathrm{rel}}=R^{\mathsf{\Psi }}-{\widehat{R}}^{\mathrm{AI}}$ proves Equation (520). At the center, Equation (407) gives ${\mathcal{B}}_0^{\mathrm{rel}}=0$, hence both projected summands vanish there. Projecting Equation (429) and using ${\mathsf{\Phi }}_{20,0}\wedge {\mathsf{\Phi }}_{20,0}=0$ gives Equation (521). Substitution of Equation (428) then gives Equation (522).

The adjoint-frame bundle of the pulled-back Cartan bundle is $\tilde{P}/Z_3$. Transporting its frame embedding by J carries its parallel Cartan form to Equation (501), so its image is exactly $P_{\phi }$. The infinitesimal adjoint representation identifies $\mathfrak{su}\left(3\right)$ with ${\mathfrak{h}}_8$. Therefore the ${\mathfrak{h}}_8$-valued one-form ${\mathcal{B}}_8$ lifts uniquely on $\tilde{P}$; adding it to the normalized pullback of the canonical connection defines $A^{\mathrm{col}}$. Its associated adjoint curvature is $R^{\mathrm{col}}$, so Equation (522) implies Equation (523). Since $[{Herm}_0\left(3\right),{Herm}_0\left(3\right)]=\mathfrak{su}\left(3\right)$, these curvature values span the structure algebra. Ambrose-Singer gives restricted principal holonomy $\mathrm{SU}\left(3\right)$ and restricted tangent holonomy $\mathrm{PSU}\left(3\right)$. Each is already the entire connected structure group; therefore full and restricted holonomy agree, proving Equation (524).

Finally, applying the same connection-shift identity to $D^{\left(s\right)}=D^{\mathrm{col}}+s{\mathsf{\Phi }}_{20}$ proves Equation (526). Its center value follows from Equation (521); the torsion formula is the corresponding alternation of the connection difference. At $s=0$ the preceding holonomy result applies. For $s\ne 0$, both eigenvalues in Equation (527) are nonzero, so the curvature map at one point is onto $\mathfrak{so}\left(8\right)$. Ambrose-Singer then gives restricted, hence full, holonomy $\mathrm{SO}\left(8\right)$. This proves Corollary 32. □

Proof of Theorem 18. 

Put

$$L(H,K):={\left(D_H^{\mathrm{col}}{\mathsf{\Phi }}_{20}\right)}_0\left(K\right).$$

(A132)

The tracial state is fixed by conjugation and every ingredient of the construction is $\mathrm{PSU}\left(3\right)$-equivariant, so L is an equivariant bilinear map $V\otimes V\to {\mathfrak{m}}_{20}$. After complexification, the standard branchings ${Sym}^2\mathbf{8}=\mathbf{1}\oplus \mathbf{8}\oplus \mathbf{27}$ and ${\left({\mathfrak{m}}_{20}\right)}_{\mathbb{C}}=\mathbf{10}\oplus \overline{\mathbf{10}}$ have no common irreducible summand [33], Table 24, [31]. Hence the symmetric part of L vanishes, so L is alternating. For an endomorphism-valued one-form,

$$\left({\mathrm{d}}^{D^{\mathrm{col}}}{\mathsf{\Phi }}_{20}\right)(H,K)=\left(D_H^{\mathrm{col}}{\mathsf{\Phi }}_{20}\right)\left(K\right)-\left(D_K^{\mathrm{col}}{\mathsf{\Phi }}_{20}\right)\left(H\right)+{\mathsf{\Phi }}_{20}\left(T^{\mathrm{col}}(H,K)\right).$$

(A133)

The last term vanishes at the center because ${\mathsf{\Phi }}_{20,0}=0$. Consequently ${\left({\mathrm{d}}^{D^{\mathrm{col}}}{\mathsf{\Phi }}_{20}\right)}_0(H,K)=2L(H,K)$, and Equation (521) proves the final equality in Equation (532). The center specialization of Equation (518) gives

$$2L(H,K)=-{\pi }_{20}[K_{0,H}^{\mathsf{\Psi }},K_{0,K}^{\mathsf{\Psi }}],$$

(A134)

which proves its cubic-shear equality.

Let ${\left\{e_a\right\}}_{a=1}^8$ be $g_0$-orthonormal. The identification $H\wedge K\mapsto W_{H,K}$ is an isometry for Equation (530), whence

$${\displaystyle \sum _{a=1}^8}{∥W_{H,e_a}∥}_{\mathfrak{so}}^2={\displaystyle \sum _{a=1}^8}\left({∥H∥}_{g_0}^2-g_0{(H,e_a)}^2\right)=7{∥H∥}_{g_0}^2.$$

(A135)

From Equations (427) and (428), ${\mathsf{\Pi }}_8=\frac{1}{9}{b}^{\ast }b$ on ${\mathsf{\Lambda }}^{2}V$ and $b{b}^{\ast }=9\mathrm{id}$. Hence

$${∥{\mathsf{\Pi }}_8\xi ∥}^2=\frac{1}{9}{∥b\xi ∥}^2.$$

(A136)

The invariant quadratic form $C\left(H\right):={\sum }_a{∥[H,e_a]∥}^2$ equals $c{∥H∥}_{g_0}^2$. Tracing over another orthonormal index gives

$$8c=\sum _{i,a}{∥b(e_i\wedge e_a)∥}^2=2Tr\left(b{b}^{\ast }\right)=2·9·8,$$

(A137)

so $c=18$. Therefore

$$\sum _a{∥{\mathsf{\Pi }}_8\left(W_{H,e_a}\right)∥}^2=2{∥H∥}_{g_0}^2,\sum _a{∥{\mathsf{\Pi }}_{20}\left(W_{H,e_a}\right)∥}^2=5{∥H∥}_{g_0}^2.$$

(A138)

Taylor expansion of Equation (532) now gives

$${\mathsf{\Phi }}_{20,X}\left(H\right)=\frac{1}{8}{\mathsf{\Pi }}_{20}\left(W_{X,H}\right)+O\left({∥X∥}_{g_0}^2{∥H∥}_{g_0}\right).$$

(A139)

Because $g_X=g_0+O\left(∥X∥\right)$ and ${\mathsf{\Phi }}_{20,X}=O\left(∥X∥\right)$, variation of the tensor norm contributes only at cubic order. The second identity in Equation (A138) proves Equation (534).

Preservation of the reduction gives $D^{\mathrm{col}}\phi =0$. The connection-difference rule for covariant tensors and Equation (535) give $D^{\mathsf{\Psi }}\phi ={\mathsf{\Phi }}_{20}·\phi$. The equivariant map

$${\iota }_{\phi }:{\mathfrak{m}}_{20}⟶{\mathsf{\Lambda }}^3{V}^{\ast },A⟼A·\phi ,$$

(A140)

is injective: its kernel is the intersection of ${\mathfrak{m}}_{20}$ with the infinitesimal stabilizer ${\mathfrak{h}}_8$ of $\phi$. Irreducibility of the real ${\mathfrak{m}}_{20}$ module then makes the positive invariant form ${\iota }_{\phi }^{\ast }{\iota }_{\phi }$ a positive scalar multiple of the chosen metric. This proves Equation (536).

The same connection-difference rule and Equation (513) give $D^{\mathsf{\Psi }}{\mathcal{C}}^J={\mathsf{\Phi }}_{20}·{\mathcal{C}}^J$. At the center, Equation (432) gives the ordered-index components ${\left({\mathcal{C}}_0^J\right)}_{abc}=-\sqrt{3/2}{d}_{abc}$. The standard contraction $d_{abc}{d}_{abc}=40/3$ therefore gives ${∥{\mathcal{C}}_0^J∥}^2=20$. For an orthonormal basis ${\left\{A_r\right\}}_{r=1}^{28}$ of $\mathfrak{so}\left(8\right)$, the orthogonal Casimir on harmonic homogeneous cubics is $3(3+8-2)=27$, so

$${\displaystyle \sum _{r=1}^{28}}{∥A_r·{\mathcal{C}}_0^J∥}^2=27{∥{\mathcal{C}}_0^J∥}^2=540.$$

(A141)

The eight stabilizer directions contribute zero. On the irreducible real 20, Schur’s lemma gives ${\iota }_{J,0}^{\ast }{\iota }_{J,0}=\lambda {\mathsf{\Pi }}_{20}$; taking its trace in Equation (A141) yields $20\lambda =540$, hence $\lambda =27$. Parallel transport proves Equation (538) at every point, and contracting the one-form index gives Equation (539). The image of the infinitesimal action is the tangent space to the orthogonal orbit. Its complex branching is that of ${\left({\mathfrak{m}}_{20}\right)}_{\mathbb{C}}$ in Equation (482), proving the final orbit statement.

Finally, Equation (527) and orthogonality of the projectors give

$${∥{\widehat{R}}_0^{\left(s\right)}∥}_{\mathrm{HS}}^2=\frac{25}{256}Tr{\mathsf{\Pi }}_8+\frac{s^2}{16}Tr{\mathsf{\Pi }}_{20}=\frac{25}{32}+\frac{5}{4}{s}^2,$$

(A142)

because $rank{\mathsf{\Pi }}_8=8$ and $rank{\mathsf{\Pi }}_{20}=20$. This proves Equation (540). □

Proof of Proposition 26. 

Differentiate $g_X({\mathsf{Q}}_X\left(H\right)K,L)=C_X^{\left(3\right)}(H,K,L)$ in a constant affine direction M. Since ${\partial }_M{g}_X(U,L)=C_X^{\left(3\right)}(M,U,L)=g_X({\mathsf{Q}}_X\left(M\right)U,L)$, nondegeneracy gives Equation (547). The Hessian Levi-Civita connection is ${\mathsf{\Gamma }}_X\left(H\right)={\mathsf{Q}}_X\left(H\right)/2$. Inserting the preceding derivative in $R=\mathrm{d}\mathsf{\Gamma }+\mathsf{\Gamma }\wedge \mathsf{\Gamma }$ cancels the symmetric quartic terms and gives the curvature formula in Equation (355). Repeated covariant differentiation proves inductively that ${\left(D^{\mathsf{\Psi }}\right)}^r{R}^{\mathsf{\Psi }}$ is a universal contraction of $g^{-1},C^{\left(3\right)},\dots ,C^{(r+3)}$.

For a traceless $3\times 3$ matrix, its characteristic polynomial is exactly Equation (548). Define the qutrit spectral log-Laplace functions

$$Z_X\left(t\right):=\frac{1}{3}Tr{\mathrm{e}}^{-tX},K_X\left(t\right):=log{Z}_X\left(t\right)=\mathsf{\Psi }\left(tX\right)-log3.$$

(A143)

Multiplying Equation (548) by ${\mathrm{e}}^{-tX}$, taking the normalized trace, and using $Z_X^{\prime }=-(1/3)Tr\left(X{\mathrm{e}}^{-tX}\right)$ gives

$$Z_X^{\prime \prime \prime }=\frac{q}{2}{Z}_X^{\prime }-\frac{r}{3}{Z}_X.$$

(A144)

The identity $Z_X={\mathrm{e}}^{K_X}$ then gives

$$K_X^{\prime \prime \prime }+3K_X^{\prime }{K}_X^{\prime \prime }+{\left(K_X^{\prime }\right)}^3=\frac{q}{2}{K}_X^{\prime }-\frac{r}{3}.$$

(A145)

Writing $D^m{\mathsf{\Psi }}_0\left[X^m\right]:=D^m{\mathsf{\Psi }}_0[X,\dots ,X]$, successive differentiation at zero, with $Z_X\left(0\right)=1$ and $Z_X^{\prime }\left(0\right)=0$, gives

$$\begin{array}{cccccc}\hfill D^2{\mathsf{\Psi }}_0\left[X^2\right]& =\frac{q}{3},\hfill & \hfill D^3{\mathsf{\Psi }}_0\left[X^3\right]& =-\frac{r}{3},\hfill & \hfill D^4{\mathsf{\Psi }}_0\left[X^4\right]& =-\frac{q^2}{6},\hfill \\ \hfill D^5{\mathsf{\Psi }}_0\left[X^5\right]& =\frac{5qr}{6},\hfill & \hfill D^6{\mathsf{\Psi }}_0\left[X^6\right]& =\frac{13q^3}{36}-r^2.\hfill \end{array}$$

(A146)

Equivalently, these are the ordinary centered cumulant identities for the three eigenvalues of $-X$ with uniform weights.

Every conjugation-invariant polynomial on traceless Hermitian $3\times 3$ matrices is a symmetric polynomial in three real eigenvalues whose sum is zero. The elementary symmetric polynomials therefore reduce to q and r, proving $\mathbb{R}{\left[V\right]}^{\mathrm{SU}\left(3\right)}=\mathbb{R}[q,r]$ and the polarization statement. The qualifications away from the center and for larger n follow from the corresponding characteristic-polynomial invariant degrees.

For a fixed loop rescaled by $\epsilon$ about the center, ${\mathcal{B}}_0^{\mathrm{rel}}=0$ implies that its interaction-picture connection integrand is $O\left({\epsilon }^2\right)$. Hence the first Magnus term starts at $O\left({\epsilon }^2\right)$ and each additional nested integral adds at least two orders. Taylor expansion of the first term gives curvature at order two, its first covariant derivative at order three, and second derivatives at order four, where the second Magnus commutator also first appears. A nonzero order-two ${\mathfrak{m}}_{20}$ coefficient cannot be uniformly canceled by these higher orders, which proves the final claim. □

Appendix D.7. AIII Standard-Model Holonomy and Global Form

Proof from AIII symmetric-space theory. 

For the supplied $3+2$ grading, the standard AIII Cartan decomposition gives $\mathfrak{su}(3,2)={\mathfrak{k}}_{\mathrm{SM}}\oplus {\mathfrak{p}}_{\mathrm{SM}}$, $[{\mathfrak{p}}_{\mathrm{SM}},{\mathfrak{p}}_{\mathrm{SM}}]={\mathfrak{k}}_{\mathrm{SM}}$, the totally geodesic realization $G_{3,2}/K_{\mathrm{SM}}=exp\left({\mathfrak{p}}_{\mathrm{SM}}\right)$, its invariant integrable complex structure, and the restricted canonical connection [26]. In the manuscript’s normalization, direct block multiplication gives

$$I_{\mathrm{AIII}}\left(X_Z\right)=X_{\mathrm{i}Z}=-\frac{6}{5}[\mathrm{i}{Y}_H,X_Z],$$

and the $(1,0)$ upper-right block is Abelian. Thus this standard complex structure has exactly the stated sign and remains distinct from the supplied carrier and Born complex structures.

The Abelian factor is not inferred from notation: the explicit sum

$${\displaystyle \sum _{a=1}^3}{\displaystyle \sum _{\alpha =1}^2}[X_{E_{a\alpha }},X_{\mathrm{i}{E}_{a\alpha }}]=diag(-4\mathrm{i}{I}_3,6\mathrm{i}{I}_2)=12\mathrm{i}{Y}_H$$

(A147)

proves the full curvature-generating span

$$[{\mathfrak{p}}_{\mathrm{SM}},{\mathfrak{p}}_{\mathrm{SM}}]={\mathfrak{k}}_{\mathrm{SM}}=\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathbb{R}\mathrm{i}{Y}_H.$$

(A148)

At the embedded identity $E_I\left(U\right)=U/2$, the manuscript’s tangent normalization gives

$$F_{A_{\mathrm{SM}},I}(U,V)=-\frac{1}{4}[U,V].$$

(A149)

Ambrose-Singer [45] and this span give restricted holonomy $K_{\mathrm{SM}}$; connectedness of $K_{\mathrm{SM}}$ and ${Hol}^0\subseteq Hol\subseteq K_{\mathrm{SM}}$ give the full equality.

It remains to verify the statistical carrier. If $P{\mathsf{\Gamma }}_{3,2}P={\mathsf{\Gamma }}_{3,2}$, then ${\mathsf{\Gamma }}_{3,2}P{\mathsf{\Gamma }}_{3,2}=P^{-1}$, so principal functional calculus gives ${\mathsf{\Gamma }}_{3,2}(logP){\mathsf{\Gamma }}_{3,2}=-logP$ and $logP\in {\mathfrak{p}}_{\mathrm{SM}}$; the converse follows by exponentiation. Hence ${\mathcal{N}}_{\mathrm{SM}}=exp\left({\mathfrak{p}}_{\mathrm{SM}}\right)$. Finally, $\mathsf{\Phi }\left({\rho }_X\right)={\mathrm{e}}^{-X}$ and $\mathcal{X}\left({\rho }_X\right)=X$, so restriction of the global maps gives the two stated diffeomorphisms. Pullback by either diffeomorphism preserves the full principal holonomy. These are the normalization, curvature-span, and state-space steps not supplied by the abstract AIII classification. □

Proof of Corollary 40. 

Trace invariance gives the asserted $K_{\mathrm{SM}}$-invariance of the restricted log partition. The source-moment statement follows from Theorem 4, and pullback by a diffeomorphism preserves the holonomy group.

For the compact symmetric pair $\mathfrak{su}\left(5\right)={\mathfrak{k}}_{\mathrm{SM}}\oplus \mathrm{i}{\mathfrak{p}}_{\mathrm{SM}}$, Equation (A148) also gives $[\mathrm{i}{\mathfrak{p}}_{\mathrm{SM}},\mathrm{i}{\mathfrak{p}}_{\mathrm{SM}}]={\mathfrak{k}}_{\mathrm{SM}}$. Its canonical principal connection on $\mathrm{SU}\left(5\right)\to \mathrm{SU}\left(5\right)/K_{\mathrm{SM}}$ consequently has full $K_{\mathrm{SM}}$ holonomy by the same Ambrose-Singer argument. Direct determinant normalization of Equation (606) gives

$$-log\mathsf{\Phi }\left({\rho }_{3,2}\right)=\frac{6}{5}log\left(\frac{r_3}{r_2}\right)diag(-I_3/3,I_2/2),$$

(A150)

proving Equation (607). Conjugation equivariance of the principal logarithm maps the density orbit ${\mathcal{O}}_{\rho }$ onto the adjoint orbit ${\mathcal{O}}_{\lambda Y_H}$, where $\lambda =(6/5)log(r_3/r_2)$. Both are parametrized by the same coset coordinate $U{K}_{\mathrm{SM}}$; hence the displayed map intertwines the common homogeneous principal bundle and connection. If $r_3=r_2$, the logarithm is zero and the stabilizer is all of $\mathrm{SU}\left(5\right)$, proving the final qualification. □

Appendix D.8. Optional Compact-Dual Torsion Extension

On ${\mathcal{P}}_n^1=\mathrm{SL}(n,\mathbb{C})/\mathrm{SU}\left(n\right)$, the bracket of quotient tangents lies in $\mathfrak{k}$ and therefore represents curvature rather than base torsion. On the ambient complex Lie algebra induced by the supplied $(V,I_V)$, regarded as real, define

$$I_{\mathrm{amb}}:\mathfrak{sl}{\left(V\right)}_{\mathbb{R}}⟶\mathfrak{sl}{\left(V\right)}_{\mathbb{R}},I_{\mathrm{amb}}\left(Z\right)=\mathrm{i}Z,I_{\mathrm{amb}}^2=-\mathrm{id}.$$

(A151)

It exchanges the Cartan summands: $I_{\mathrm{amb}}\left(\mathfrak{k}\right)=\mathfrak{p}$ and $I_{\mathrm{amb}}\left(\mathfrak{p}\right)=\mathfrak{k}$. Its restriction gives the $\mathrm{SU}\left(n\right)$-equivariant compact-dual map

$$\mathcal{I}:=I_{\mathrm{amb}}{|}_{\mathfrak{k}}:\mathfrak{k}⟶\mathfrak{p},\mathcal{I}\left(A\right)=\mathrm{i}A.$$

(A152)

Thus $I_{\mathrm{amb}}\left(Z\right)=I_V\circ Z=Z\circ I_V$ is the induced ambient complex structure on the real direct sum $\mathfrak{k}\oplus \mathfrak{p}$, whereas $\mathcal{I}$ alone is not a complex structure on either summand: its domain and codomain differ, and it cannot be squared without the companion restriction $I_{\mathrm{amb}}{|}_{\mathfrak{p}}$. For homogeneous quotient coordinates $x,y\in \mathfrak{p}$, define

$${\beta }_q(x,y):=\mathcal{I}\left([x,y]\right)=\mathrm{i}[x,y].$$

(A153)

Since the embedded tangent is $U=2x$, transport through $\mathrm{d}{j}_e$ gives

$${\beta }_{\mathrm{emb}}(U,V):=\mathrm{d}{j}_e\left[{\beta }_q(U/2,V/2)\right]=\frac{\mathrm{i}}{2}[U,V].$$

(A154)

With the maps in Equation (368),

$$c_{\mathrm{emb}}\left({\beta }_{\mathrm{emb}}(U,V)\right)=-\frac{1}{2}[c_{\mathrm{emb}}\left(U\right),c_{\mathrm{emb}}\left(V\right)].$$

(A155)

Because $F_{A,I}(U,V)=-[U,V]/4$ in embedded normalization,

$${\beta }_{\mathrm{emb}}(U,V)=-2\mathrm{i}{F}_{A,I}(U,V).$$

(A156)

These formulas resolve the factor conversion between homogeneous and embedded tangents; an independently rescaled bracket tensor is possible but is not used here.

With $g_I(U,V)=\alpha Tr\left(UV\right)$, define

$$H(U,V,W):=g_I({\beta }_{\mathrm{emb}}(U,V),W)=\frac{\mathrm{i}\alpha }{2}Tr\left([U,V]W\right).$$

(A157)

For Hermitian $U,V,W$, the trace is purely imaginary, so H is real. For the normalized Hermitian generators of Equation (A66), the embedded tangent bracket and three-form have components

$$\begin{array}{cc}\hfill {\beta }_{\mathrm{emb}}(T_b,T_c)& =-\frac{1}{2}{f^a}_{bc}{T}_a,\hfill \end{array}$$

(A158)

$$\begin{array}{cc}\hfill H_{abc}:=H(T_a,T_b,T_c)& =-\frac{\alpha }{4}{f}_{abc}.\hfill \end{array}$$

(A159)

Thus $f_{abc}$ controls both vertical BCH curvature and, after the declared compact-dual identification, the optional common-torsion three-form. For $n=2$, $f_{abc}={ϵ}_{abc}$ gives the volume-form tensor of the parallelizing torsion on $S^3\simeq \mathrm{SU}\left(2\right)$ after the independent Cartan-Schouten rescaling specified below.

Since ${\mathcal{P}}_n^1$ is contractible and H is parallel, H is closed and exact on the noncompact base and supplies no quantized flux class there. Transport the identity metric and three-form through $c_{\mathrm{emb}}$ by

$$\tilde{g}(u,v):=g_I(c_{\mathrm{emb}}^{-1}u,c_{\mathrm{emb}}^{-1}v),\tilde{H}(u,v,w):=H(c_{\mathrm{emb}}^{-1}u,c_{\mathrm{emb}}^{-1}v,c_{\mathrm{emb}}^{-1}w).$$

Equation (A155) gives ${\tilde{H}}^{{♯}_{\tilde{g}}}(u,v)=-[u,v]/2$; hence the standard Cartan-Schouten convention is obtained by the independent rescaling $H_{\mathrm{CS}}=-2\tilde{H}$, up to chirality [123]. Treating this compact-dual form as a physical flux and choosing an integral level or action normalization are additional inputs.

Let $(M,g)$ now be any Riemannian manifold, $\mathcal{C}\in \mathsf{\Gamma }\left({Sym}^3{T}^{\ast }M\right)$, and $H\in {\mathsf{\Omega }}^3\left(M\right)$. Use g to raise both tensors:

$$g({\mathcal{C}}^{♯}(X,Y),Z)=\mathcal{C}(X,Y,Z),g(H^{♯}(X,Y),Z)=H(X,Y,Z).$$

(A160)

Related precedents include metric-dual torsion and teleparallel pairs in information geometry [124] and flat Cartan-Schouten metric connections with skew torsion [123].

Theorem A1

(Metric-dual pair with common skew torsion). The connections

$${\widehat{\nabla }}_X^{(\pm )}Y:={\nabla }_X^{\mathrm{LC}}Y\pm \frac{1}{2}{\mathcal{C}}^{♯}(X,Y)+\frac{1}{2}{H}^{♯}(X,Y)$$

(A161)

are dual with respect to g and have identical torsion

$$T^{{\widehat{\nabla }}^{(+)}}=T^{{\widehat{\nabla }}^{(-)}}=H^{♯}.$$

(A162)

Remark A2

(No canonical merger). Theorem A1 starts from independent $(g,\mathcal{C},H)$; metric duality selects none and implies no flatness. On the positive cone, quotient geometry selects $g_{\mathrm{AI}}$ up to scale and the construction above supplies $H^{♯}={\beta }_{\mathrm{emb}}$, but it does not select $\mathcal{C}$. The theorem therefore does not derive the BKM dual-flat pair from compact holonomy and is not used in the channel or discrete-generator results.

Under the standard realization $\left(\mathrm{SU}\left(n\right)\times \mathrm{SU}\left(n\right)\right)/\mathrm{SU}{\left(n\right)}_{\mathrm{diag}}\simeq \mathrm{SU}\left(n\right)$, the anti-diagonal tangent $(A,-A)$ maps by $(U,V)\mapsto U{V}^{-1}$ to $2A$. Symmetric-space dualization sends $\mathrm{i}A\mapsto (A,-A)$, proving the quotient-normalized map $c_q\left(\mathrm{i}A\right)=2A$ in Equation (368). Since an embedded tangent is twice its quotient representative, the corresponding embedded map is $c_{\mathrm{emb}}\left(U\right)=c_q(U/2)$.

Lemma A1

(Common skew-adjoint freedom). Let $(\nabla ,{\nabla }^{\ast })$ be g-dual, and let $S_X$ be g-skew-adjoint for every vector field X. Then

$$\tilde{\nabla }:=\nabla +S,{\tilde{\nabla }}^{\ast }:={\nabla }^{\ast }+S$$

(A163)

are g-dual, and

$$\begin{array}{cc}\hfill T^{\tilde{\nabla }}(X,Y)& =T^{\nabla }(X,Y)+S_{X}Y-S_{Y}X,\hfill \end{array}$$

(A164)

$$\begin{array}{cc}\hfill T^{{\tilde{\nabla }}^{\ast }}(X,Y)& =T^{{\nabla }^{\ast }}(X,Y)+S_{X}Y-S_{Y}X.\hfill \end{array}$$

(A165)

Thus metric duality alone does not select a common torsion.

Proof. 

Skew-adjointness gives $g(S_{X}Y,Z)+g(Y,S_{X}Z)=0$, so the added terms cancel in the duality identity. The torsion formulas follow directly from 3. □

Proof of Theorem A1. 

Metric compatibility supplies the Levi-Civita part of duality. The remaining terms cancel because $\mathcal{C}(X,Y,Z)=\mathcal{C}(X,Z,Y)$ and $H(X,Y,Z)=-H(X,Z,Y)$. Symmetry of ${\mathcal{C}}^{♯}$ gives no torsion, while skewness of $H^{♯}$ gives

$$\frac{1}{2}{H}^{♯}(X,Y)-\frac{1}{2}{H}^{♯}(Y,X)=H^{♯}(X,Y),$$

(A166)

which proves Equation (A162). □

With $\mathcal{C}=\nabla g$, $H=0$, and the convention of Section 6.2, the minus branch of Equation (A161) is ∇ and the plus branch is ${\nabla }^{\ast }$.

For the positive-cone specialization, take $M={\mathcal{P}}_n^1$, $g=g_{\mathrm{AI}}$ from Equation (346), and H from Equation (A157). At the identity, $H_I^{♯}(X,Y)={\beta }_{\mathrm{emb}}(X,Y)=\mathrm{i}[X,Y]/2$, extended globally by equivariance. Any symmetric three-tensor gives a $g_{\mathrm{AI}}$-dual pair, invariant when that tensor is invariant. This does not identify affine-invariant and BKM metrics: raising a BKM cubic with $g_{\mathrm{AI}}$ neither recovers its original dual-flat connections nor implies flatness, while raising H with BKM need not reproduce the affine-invariant ${\beta }_{\mathrm{emb}}$.

Under $G=\mathrm{SL}(n,\mathbb{C})$-invariance, the invariant symmetric cubic vanishes for $n=2$ and is proportional to $d_{abc}$ for $n\ge 3$; see Equation (A71). It is not the state-dependent BKM cubic away from the maximally mixed state. The torsion in Theorem A1 is ordinary affine torsion on ${\mathcal{P}}_n^1$. These connections neither live on $T\mathsf{\Theta }$ nor preserve its Born tensors and are not the generalized torsion-free Born connection of [42]. The quotient connection A remains torsion-free on the noncompact space. After the compact-dual rescaling following Equation (A155), the Cartan-Schouten connections ${\nabla }^{\mathrm{LC}}\pm \frac{1}{2}{H}_{\mathrm{CS}}^{♯}$ have opposite torsions; Equation (A161) instead uses the same sign of the chosen three-form in both dual branches.

Appendix D.9. Conditional Qutrit Spacetime Completion and Physical Qualifications

Proposition A3

(Conditional spacetime lift and transverse Yang-Mills Hessian). Let $(\mathcal{X},\gamma )$ be a compact oriented Riemannian four-manifold without boundary and let $Q\to \mathcal{X}$ be a principal $\mathrm{SO}\left(8\right)$ bundle supplied with a $\mathrm{PSU}\left(3\right)$ reduction $P\subset Q$. Set

$$E_Q:=Q{\times }_{\mathrm{SO}\left(8\right)}{\mathbb{R}}^8.$$

On $\mathfrak{so}\left(8\right)$ use ${\langle A,B\rangle }_8:=-\frac{1}{2}{Tr}_{{\mathbb{R}}^8}\left(AB\right)$; on ${\mathsf{\Lambda }}^3{E}_Q^{\ast }$ use the exterior-form norm ($1/3!$ times the ordered-index contraction), and on ${Sym}^3{E}_Q^{\ast }$ the induced ordered-index tensor norm, with no $1/3!$ factor. Spacetime form indices are contracted with γ. Write

$$\mathfrak{so}\left(8\right)={\mathfrak{h}}_8\oplus {\mathfrak{m}}_{20},{\mathfrak{h}}_8\simeq \mathfrak{su}\left(3\right),$$

(A167)

orthogonally. The restriction of an $\mathrm{SO}\left(8\right)$ connection $\mathcal{A}$ to P decomposes uniquely as

$$\mathcal{A}=A_c+\mathsf{\Phi },A_c\in {\mathsf{\Omega }}^1(P,{\mathfrak{h}}_8),\mathsf{\Phi }\in {\mathsf{\Omega }}^1(P,{\mathfrak{m}}_{20}),$$

(A168)

where $A_c$ is a $\mathrm{PSU}\left(3\right)$ connection and Φ is tensorial. With the wedge convention of Equation (515),

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill {\pi }_8{F}_{\mathcal{A}}& =F_{A_c}+{\pi }_8(\mathsf{\Phi }\wedge \mathsf{\Phi }),\hfill \\ \hfill {\pi }_{20}{F}_{\mathcal{A}}& ={\mathrm{d}}^{A_c}\mathsf{\Phi }+{\pi }_{20}(\mathsf{\Phi }\wedge \mathsf{\Phi }).\hfill \end{array}\\ \hline\end{array}$$

(A169)

Reductivity gives ${\mathrm{d}}^{A_c}\mathsf{\Phi }\in {\mathsf{\Omega }}^2(P,{\mathfrak{m}}_{20})$, so the first direct feedback into projected color curvature is quadratic: $F_{\mathrm{color}}^{\mathrm{eff}}:={\pi }_8{F}_{\mathcal{A}}=F_{A_c}+{\pi }_8(\mathsf{\Phi }\wedge \mathsf{\Phi })$. The convention $(\mathsf{\Phi }\wedge \mathsf{\Phi })(X,Y)=\left[\mathsf{\Phi }\right(X),\mathsf{\Phi }(Y\left)\right]$ already absorbs the customary factor $1/2$ in $\frac{1}{2}[\mathsf{\Phi },\mathsf{\Phi }]$. All objects in this paragraph are newly supplied spacetime fields; in particular, no identification of Φ with the derived state-space tensor ${\mathsf{\Phi }}_{20}$ is assumed. For the supplied Euclidean Yang-Mills functional

$$S_{\mathrm{YM}}\left[\mathcal{A}\right]:=\frac{1}{2g_8^2}{\int }_{\mathcal{X}}{∥F_{\mathcal{A}}∥}^2{vol}_{\gamma },$$

(A170)

the transverse second variation at ${\mathcal{A}}_0=A_c$ is

$$\begin{array}{|c|}\hline {\left.\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}{S}_{\mathrm{YM}}[A_c+t\varphi ]\right|}_{t=0}=\frac{1}{g_8^2}{\int }_{\mathcal{X}}\left\{{∥{\mathrm{d}}^{A_c}\varphi ∥}^2+2〈F_{A_c},{\pi }_8(\varphi \wedge \varphi )〉\right\}{vol}_{\gamma }.\\ \hline\end{array}$$

(A171)

Thus a flat color background yields only the covariant kinetic form $g_8^{-2}{∥{\mathrm{d}}^{A_c}\varphi ∥}_{L^2}^2$, with no positive zeroth-order term. At a curved background the second term is background dependent and has no universal sign. The reductive curvature feedback therefore does not by itself generate a transverse classical mass.

A gauge-covariant positive zeroth-order quadratic term requires additional data. Let ${\phi }_0$ be a normalized Cartan three-form with oriented stabilizer $\mathrm{PSU}\left(3\right)$ and ${\mathcal{O}}_{{\phi }_0}:=\mathrm{SO}\left(8\right)·{\phi }_0\simeq \mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right)$. If one supplies a dynamical reduction field

$$\sigma \in \mathsf{\Gamma }\left(Q{\times }_{\mathrm{SO}\left(8\right)}{\mathcal{O}}_{{\phi }_0}\right)$$

(A172)

then σ determines a reduction $P_{\sigma }\subset Q$. Write

$${\mathcal{A}|}_{P_{\sigma }}=A_{\sigma }+{\mathsf{\Phi }}_{\sigma }$$

(A173)

for its ${\mathfrak{h}}_8\oplus {\mathfrak{m}}_{20}$ decomposition. When $P_{\sigma }=P$, this is Equation (A168); otherwise it is a distinct, σ-adapted split. If one also supplies the nonlinear-sigma term

$$S_{\sigma }[\mathcal{A},\sigma ]:=\frac{f^2}{2}{\int }_{\mathcal{X}}{∥D_{\mathcal{A}}\sigma ∥}^2{vol}_{\gamma },$$

(A174)

then in an adapted gauge $\sigma ={\phi }_0$,

$$D_{\mathcal{A}}{\phi }_0={\mathsf{\Phi }}_{\sigma }·{\phi }_0,{∥{\mathsf{\Phi }}_{\sigma }·{\phi }_0∥}^2=c_{\phi }{∥{\mathsf{\Phi }}_{\sigma }∥}^2,c_{\phi }>0.$$

(A175)

After canonical normalization of the Yang-Mills kinetic term, the supplied sigma action contributes, in the adapted gauge $\sigma ={\phi }_0$, the same zeroth-order quadratic coefficient to all twenty complementary directions:

$${\mu }_{\perp ,\sigma }^2:=g_8^2{f}^2{c}_{\phi }.$$

(A176)

Here ${\mu }_{\perp ,\sigma }^2$ is a coefficient in the classical Euclidean Hessian. Its identification with a gauge-invariant pole mass requires a separately supplied Lorentzian or reflection-positive quantum completion and an analysis of the gauge-invariant spectrum. Choose a common model frame for the Cartan form and one of the two normalized signs of the symmetric cubic. The resulting equivariant identification of their $\mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right)$ orbits sends the same reduction field to a transported symmetric cubic ${\mathcal{C}}_{\sigma }^J\in \mathsf{\Gamma }\left({Sym}^3{E}_Q^{\ast }\right)$, normalized as in Equation (538). In an adapted gauge it obeys

$$\begin{array}{|c|}\hline D_{\mathcal{A}}{\mathcal{C}}_{\sigma }^J={\mathsf{\Phi }}_{\sigma }·{\mathcal{C}}_{\sigma }^J,{∥D_{\mathcal{A}}{\mathcal{C}}_{\sigma }^J∥}^2=27{∥{\mathsf{\Phi }}_{\sigma }∥}^2.\\ \hline\end{array}$$

(A177)

Consequently, if one instead supplies the independently normalized term

$$S_J[\mathcal{A},{\mathcal{C}}_{\sigma }^J]:=\frac{f_J^2}{2}{\int }_{\mathcal{X}}{∥D_{\mathcal{A}}{\mathcal{C}}_{\sigma }^J∥}^2{vol}_{\gamma },$$

(A178)

its common complementary quadratic coefficient is

$${\mu }_{\perp ,J}^2:=27g_8^2{f}_J^2.$$

(A179)

This is a coefficient-normalized alternative realization of the same reduction and 20-channel, not a second field or an additive quadratic contribution unless a second action term is independently supplied. Neither coefficient is, by itself, a gauge-invariant spectral mass. These are classical nonlinear-sigma realizations of the supplied reduction [125,126]; the field σ, the coefficient f or $f_J$, and its scale remain independent inputs, not consequences of the state-space split.

Proof of Proposition A3. 

Restriction to the supplied reductive reduction decomposes a connection into its ${\mathfrak{h}}_8$ connection part and horizontal equivariant ${\mathfrak{m}}_{20}$ part. The standard change-of-connection identity, followed by the two orthogonal projections, proves Equation (A169). It also gives

$${∥F_{\mathcal{A}}∥}^2={∥F_{A_c}+{\pi }_8(\mathsf{\Phi }\wedge \mathsf{\Phi })∥}^2+{∥{\mathrm{d}}^{A_c}\mathsf{\Phi }+{\pi }_{20}(\mathsf{\Phi }\wedge \mathsf{\Phi })∥}^2.$$

(A180)

Substitute $\mathsf{\Phi }=t\varphi$. Orthogonality eliminates the linear term, and extracting the coefficient of $t^2$ proves Equation (A171).

The fiberwise tensor-action argument establishing Equation (536), with $c_{\phi }$ fixed by the chosen normalization of ${\phi }_0$, gives Equation (A175). Comparing the supplied sigma and Yang-Mills kinetic coefficients gives Equation (A176). The common oriented stabilizer supplies the stated equivariant orbit identification. The connection-difference rule and Equation (538) give Equation (A177) in its specified ordered-index normalization. Comparing kinetic coefficients then proves Equation (A179). □

Remark A3

(Gauge-invariant bridge composites, screening, and closed excursions). In the conditional completion, the manifestly gauge-invariant local operator

$${\mathcal{O}}_{\sigma }:={∥D_{\mathcal{A}}\sigma ∥}^2$$

(A181)

reduces to $c_{\phi }{∥{\mathsf{\Phi }}_{\sigma }∥}^2$ in a σ-adapted gauge. This is the field-theoretic realization of the singlet in Equation (545); local gauge symmetry itself is not assigned a gauge-variant order parameter [127,128]. In a separate noncompact cylindrical completion, replacing the compact hypothesis of Proposition A3, suppose a reflection-positive quantum theory on $\mathcal{X}=\mathbb{R}\times \mathsf{\Sigma }$, with compact Σ, admits a Hilbert-space reconstruction with a self-adjoint Hamiltonian $H\ge E_{0}I$ and a normalized ground vector Ω satisfying $H\mathsf{\Omega }=E_0\mathsf{\Omega }$. Choose renormalized, spatially smeared gauge-invariant operators ${\overline{\mathcal{O}}}_i$ with $\mathsf{\Omega }\in Dom{\overline{\mathcal{O}}}_i$, and set $v_i=({\overline{\mathcal{O}}}_i-\langle \mathsf{\Omega },{\overline{\mathcal{O}}}_i\mathsf{\Omega }\rangle I)\mathsf{\Omega }$. For the inverse-energy semigroup parameter $t\ge 0$, the connected correlation matrix is

$$\begin{array}{cc}\hfill C_{ij}\left(t\right)& :=\langle v_i,{\mathrm{e}}^{-t(H-E_{0}I)}{v}_j\rangle ={\int }_{[0,\infty )}{\mathrm{e}}^{-tE}\mathrm{d}{\mu }_{ij}\left(E\right),\hfill \\ \hfill {\mu }_{ij}\left(B\right)& :=\langle v_i,{\mathbf{1}}_B(H-E_{0}I)v_j\rangle .\hfill \end{array}$$

(A182)

The spectral theorem makes μ a finite positive-semidefinite matrix measure on Borel subsets of $[0,\infty )$; no spatial translation symmetry is assumed. Centering removes the chosen vacuum vector, but a zero-energy atom can remain if the ground space is degenerate. If the relevant spectral measure is pure point, this becomes $C_{ij}\left(t\right)={\sum }_n\langle v_i,e_n\rangle \langle e_n,v_j\rangle {\mathrm{e}}^{-t(E_n-E_0)}$, with $H{e}_n=E_n{e}_n$ and an orthonormal eigenbasis on the generated spectral subspace, as in regulated glueball correlator calculations [129].

For a nonzero combination $v={\sum }_i{z}_i{v}_i$, the lower edge of its positive spectral measure controls the logarithmic decay rate of $z^{†}C\left(t\right)z$. An atom there gives a nonzero leading exponential coefficient, in particular at an isolated lowest overlapping eigenvalue; a continuous threshold need not represent a particle. The operator basis should include the ordinary scalar gluonic operator and bridge singlets, since operators with the same quantum numbers can mix. Neither the spectral threshold nor an eigenvalue is automatically ${\mu }_{\perp ,\sigma }$, ${\mu }_{\perp ,J}$, or a new particle.

This real module has no center-charge obstruction. For an irreducible $\mathrm{SU}\left(3\right)$ representation with Dynkin labels $(p,q)$, the center 3-ality is $t(p,q)=p+2q(mod3)$; hence

$$t\left(\mathbf{10}\right)=t(3,0)=0,t\left(\overline{\mathbf{10}}\right)=t(0,3)=0.$$

(A183)

Both conjugate components therefore descend to $\mathrm{PSU}\left(3\right)$. By the standard adjoint tensor product [33], Table 24 and Equation (482), $\mathbf{8}\otimes \mathbf{8}$ contains $\mathbf{10}\oplus \overline{\mathbf{10}}$, so, in a separately supplied Yang-Mills completion, a complementary decuplet component can be dressed by two adjoint gluonic components to form a singlet; generalized gluelump operators realize this representation-theoretic possibility [130]. Thus there is no center-charge obstruction to adjoint screening [131]. This does not prove confinement, a spectral gap, or the existence of a propagating bridge excitation. The conditional reduction has global form $\mathrm{PSU}\left(3\right)$; fundamental triplet Wilson lines require a separately supplied, topologically admissible principal $\mathrm{SU}\left(3\right)$ lift of the spacetime reduction. The lift in Theorem 17 is over the state manifold and supplies a spacetime lift only after an explicit pullback. If a separately supplied Lorentzian or reflection-positive quantum completion possesses a gauge-invariant heavy scale $M_{\perp }$ controlled by one of the preceding quadratic coefficients, and if a local Wilsonian decoupling expansion exists, integrating out those heavy states gives the color action plus symmetry-allowed operators suppressed by inverse powers of $M_{\perp }$. Neither the classical Hessian coefficient nor reductive geometry establishes these dynamical hypotheses or a pure-gauge spectral gap.

Finally, the Gibbs-coordinate qutrit state manifold is contractible, so unconstrained out-and-back state-space loops can be shrunk and have zero infimum of length and endpoint action. The separately introduced coset does have

$${\pi }_1(\mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right))\simeq {\mathbb{Z}}_2,{\pi }_2(\mathrm{SO}\left(8\right)/\mathrm{PSU}\left(3\right))\simeq {\mathbb{Z}}_3,$$

(A184)

as follows from the homotopy exact sequence and the necessarily zero map ${\mathbb{Z}}_3\to {\mathbb{Z}}_2$. These sectors become relevant only after a coset field such as σ is supplied and a topological class is fixed. A coset systole or minimum classical action is still not a Hamiltonian spectral gap in the sense of [132].

Remark A4

(Geometric anholonomy versus a physical gauge observable). For nonzero relative curvature on the loop’s tangent plane, the small-loop law in Equation (394) gives an area-order transport residue even when the endpoint state is unchanged. This reversible, many-to-one path dependence is not dissipative hysteresis. A physical phase, rotation, or mixing requires two realizable transport laws, a represented carrier, coherent preparation and comparison, and a readout. Berry-Simon, Wilczek-Zee, Hannay, Wilson, and Uhlmann effects require their respective adiabatic eigenbundles, action-angle family, gauge connection, or purification bundle [87,133,134,135,136,137]; none is supplied by ${\mathcal{B}}^{\mathrm{rel}}$ alone.

A metric polar factor of the raw transporter in Equation (365) need not obey a composition law or arise from a local connection. The smooth intrinsic normalizer J instead gives the metric-connection comparison, exact transgression, and subgroup test of Theorem 14 and Proposition 24. A physical Yang-Mills interpretation still requires the independent spacetime, action, coupling, and field-equation inputs described in Remark 16.

Remark A5

(Metric distortion is not a Hamiltonian identification). The positive metric distortion in Equation (381) is base dependent away from the center. Its square root canonically compares the two metric connections, but it is not generally a cotangent lift or a symplectomorphism. Hence the compact relative bridge does not identify the BKM and affine-invariant Hamiltonians, actions, or holonomy groups. More explicitly, for $p\in T_X^{\ast }\mathfrak{p}$ and $v={\left(g_X^{\mathsf{\Psi }}\right)}^{♯}p$ their natural kinetic Hamiltonians are

$$H_{\mathrm{BKM}}(X,p)=\frac{1}{2}{g}_X^{\mathsf{\Psi }}(v,v),{\overline{H}}_{\mathrm{AI}}(X,p)=\frac{1}{2}{g}_X^{\mathsf{\Psi }}(v,{\mathsf{D}}_X^{-1}v).$$

(A185)

They agree for every momentum exactly where ${\mathsf{D}}_X=\mathrm{id}$. The cotangent lift of any diffeomorphism, including Φ, is an exact symplectomorphism and conjugates a supplied Hamiltonian to its coordinate transform; it does not thereby identify these two different natural kinetic energies.

Appendix F.1. Measurement-Channel Proofs

Proof of Corollary 17. 

Writing $E_a={\sum }_j|e_{aj}\rangle \langle e_{aj}|$, the Kraus operators $K_{aj}:=|a\rangle \langle e_{aj}|$ obey ${\sum }_{a,j}{K}_{aj}^{†}{K}_{aj}=I$. Faithfulness gives $p_a^E\left(\rho \right)>0$. Apply Theorem 10 and evaluate Umegaki relative entropy and its BKM Hessian on diagonal matrices. The rank and informational-completeness assertions follow from $I={\sum }_a{E}_a$ and Hilbert-Schmidt duality.

If (294) failed for some faithful $\sigma$, the same Petz recovery map for $({\mathcal{M}}_E,\sigma )$ would recover every faithful $\rho$ [12,13]. By linearity its composition with ${\mathcal{M}}_E$ would then be the identity channel on ${\mathrm{M}}_n\left(\mathbb{C}\right)$. But a quantum-to-classical channel followed by a classical-to-quantum channel is measure-and-prepare and hence sends one half of a maximally entangled state to a separable state; it cannot be the identity for $n\ge 2$. If the two metrics agreed everywhere, then $Hess{\mathcal{H}}_E=0$ on the connected mixture domain, so ${\mathcal{H}}_E$ would be affine and $B_{{\mathcal{H}}_E}$ would vanish for all pairs, giving the same contradiction. The commuting-model clause follows from Corollary 18. □

Proof of Proposition 11. 

The connected group $\mathrm{U}\left(V\right)$ has trivial image under every continuous homomorphism to a finite permutation group. Hence each effect in the first clause commutes with every unitary and is scalar.

For the ray orbit, isotropy forces every covariant positive rank-one density to be $cP$, and normalization gives $c=n$. The standard projective Haar second moment then gives $\int p_{\rho }\left(P\right)P\mathrm{d}{\mu }_{\mathrm{FS}}=(\rho +I)/(n+1)$ and, for traceless $V,W$, $\int Tr\left(VP\right)Tr\left(WP\right)\mathrm{d}{\mu }_{\mathrm{FS}}=Tr\left(VW\right)/\left[n(n+1)\right]$ [140]. These identities prove the reconstruction formula and the central contraction.

For the data-processing claim, set ${\nu }_{\rho }\left(B\right):={\int }_B{p}_{\rho }\mathrm{d}{\mu }_{\mathrm{FS}}$. For every finite measurable partition $\mathcal{P}=\left\{B_j\right\}$ of $\mathbb{CP}\left(V\right)$, the effects $E_j^{\mathcal{P}}:=n{\int }_{B_j}P\mathrm{d}{\mu }_{\mathrm{FS}}\left(P\right)$ form a finite POVM with $Tr\left(\rho E_j^{\mathcal{P}}\right)={\nu }_{\rho }\left(B_j\right)$; ${\mu }_{\mathrm{FS}}$-null cells have zero effect and may be discarded. The finite-partition characterization of classical relative entropy [141], Theorem 10, $\alpha =1$, and finite-POVM data processing (Corollary 17) give

$$D_{\mathrm{ray}}(\rho \parallel \sigma )=\underset{\mathcal{P}}{sup}\sum _j{\nu }_{\rho }\left(B_j\right)log\frac{{\nu }_{\rho }\left(B_j\right)}{{\nu }_{\sigma }\left(B_j\right)}\le D_{\mathrm{U}}(\rho \parallel \sigma ),$$

(A222)

with the usual extended-real conventions. This proves Equation (303).

It remains to justify passage to the Fisher Hessian. At a faithful $\rho$, put $p_V\left(P\right):=nTr\left(VP\right)$. For sufficiently small $ϵ$,

$$p_{\rho +ϵV}\left(P\right)\ge \frac{n}{2}{\lambda }_{min}\left(\rho \right)\mathrm{uniformly}\mathrm{in}P.$$

(A223)

Since $p_V$ is uniformly bounded on the compact ray space, this lower bound supplies an integrable uniform dominator for two differentiations under the integral. Consequently,

$$\begin{array}{cc}\hfill {\left.\frac{{\mathrm{d}}^2}{\mathrm{d}{ϵ}^2}\right|}_{ϵ=0}{D}_{\mathrm{ray}}(\rho +ϵV\parallel \rho )& ={\int }_{\mathbb{CP}\left(V\right)}\frac{p_V{\left(P\right)}^2}{p_{\rho }\left(P\right)}\mathrm{d}{\mu }_{\mathrm{FS}}\left(P\right)\hfill \\ & =n{\int }_{\mathbb{CP}\left(V\right)}\frac{Tr{\left(VP\right)}^2}{Tr\left(\rho P\right)}\mathrm{d}{\mu }_{\mathrm{FS}}\left(P\right).\hfill \end{array}$$

(A224)

The corresponding diagonal Hessian of $D_{\mathrm{U}}(\rho +ϵV\parallel \rho )$ is $g_{\rho }^{\mathrm{BKM}}(V,V)$. Both divergences and their first derivatives vanish at the diagonal, so differentiating Equation (303) twice and polarizing gives the displayed bilinear formula and $g^{\mathrm{ray}}⪯g_{\mathrm{BKM}}$ in Equation (304). □

Appendix F.2. Moving References and Mobility Linearization

Let $H_{\mathrm{sys}}$ be time independent, $\beta \left(t\right)>0$ differentiable, and ${\tau }_{\beta \left(t\right)}={\mathrm{e}}^{-\beta \left(t\right)H_{\mathrm{sys}}}/Z_{\beta \left(t\right)}$. Along any differentiable faithful curve,

$$\begin{array}{cc}\hfill \frac{\mathrm{d}}{\mathrm{d}t}{D}_{\mathrm{U}}({\rho }_t\parallel {\tau }_{\beta \left(t\right)})& =Tr\left[{\dot{\rho }}_t(log{\rho }_t-log{\tau }_{\beta \left(t\right)})\right]+\dot{\beta }\left(t\right)Tr\left[({\rho }_t-{\tau }_{\beta \left(t\right)})H_{\mathrm{sys}}\right],\hfill \end{array}$$

(A225)

$$\begin{array}{cc}\hfill {\dot{\rho }}_t=-\frac{\mathrm{i}}{\hslash }[H_{\mathrm{sys}},{\rho }_t]& ⟹\frac{\mathrm{d}}{\mathrm{d}t}{D}_{\mathrm{U}}({\rho }_t\parallel {\tau }_{\beta \left(t\right)})=\dot{\beta }\left(t\right)Tr\left[({\rho }_t-{\tau }_{\beta \left(t\right)})H_{\mathrm{sys}}\right].\hfill \end{array}$$

(A226)

Thus reference calibration can account for the entire change without entropy production. If $D_{\mathrm{U}}(\tau +V\parallel \tau )=\frac{1}{2}〈{\mathbb{G}}_{\tau }V,V〉+O\left(V^3\right)$, then $v=-\mathbb{M}\mathrm{d}{D}_{\mathrm{U}}$ linearizes as

$$\dot{V}=-{\mathbb{M}}_{\tau }{\mathbb{G}}_{\tau }V,$$

(A227)

which separates the information Hessian from the constitutive mobility.

Appendix F.3. Magnus Parity and Soldered-Gradient Specialization

Proof of Proposition 31. 

Every homogeneous Magnus term is a sum of nested commutators with exactly r factors $X\left(t_j\right)$ [78,107]. Applying $[\mathfrak{p},\mathfrak{p}]\subset \mathfrak{k}$ and $[\mathfrak{k},\mathfrak{p}]\subset \mathfrak{p}$ inductively puts odd terms in $\mathfrak{p}$ and even terms in $\mathfrak{k}$. This standard grading argument does not replace the manuscript-specific exact-loop endpoint step that follows, which removes the possible order-${\epsilon }^2$ $\mathfrak{p}$ component. □

Let $X_{\epsilon }\left(t\right)=\epsilon X_1\left(t\right)+O\left({\epsilon }^2\right)$ be a perturbatively small horizontal family, assume that its base path is an exact loop, and suppose that its first-order noncompact displacement closes. First-order closure gives only ${\mathsf{\Omega }}_1=O\left({\epsilon }^2\right)$. Homogeneity of the Magnus terms and Equation (672) give

$${\mathsf{\Omega }}_1\in \mathfrak{p},{\mathsf{\Omega }}_2\in \mathfrak{k},{\mathsf{\Omega }}_2=O\left({\epsilon }^2\right),\sum _{r\ge 3}{\mathsf{\Omega }}_r=O\left({\epsilon }^3\right).$$

(A228)

Because the base path closes exactly, its relative horizontal endpoint

$$U_{\mathrm{hol},\epsilon }:={\tilde{\gamma }}_{\epsilon }{\left(0\right)}^{-1}{\tilde{\gamma }}_{\epsilon }\left(T\right)$$

(A229)

lies in $K=\mathrm{SU}\left(n\right)$. For sufficiently small $\epsilon$ it lies in the domain of the principal logarithm, so $log{U}_{\mathrm{hol},\epsilon }\in \mathfrak{k}$. On the Magnus convergence interval,

$$log{U}_{\mathrm{hol},\epsilon }={\mathsf{\Omega }}_1+{\mathsf{\Omega }}_2+O\left({\epsilon }^3\right).$$

(A230)

Taking the $\mathfrak{p}$-projection forces ${\mathsf{\Omega }}_1=O\left({\epsilon }^3\right)$ and therefore justifies

$$\begin{array}{|c|}\hline log{U}_{\mathrm{hol},\epsilon }={\mathsf{\Omega }}_2+O\left({\epsilon }^3\right).\\ \hline\end{array}$$

(A231)

Thus Equation (671) is the leading holonomy logarithm in this scaled loop setting. The exact-loop condition is essential: first-order closure alone leaves a possible $O\left({\epsilon }^2\right)$ noncompact component. For an unscaled finite loop, parity identifies compact and noncompact Magnus sectors but does not justify truncation after ${\mathsf{\Omega }}_2$.

Let E be the $\mathfrak{p}$-valued solder form along the horizontal lift. If the base velocity is $v_t=-{\mathbb{M}}_{\gamma \left(t\right)}\mathrm{d}\mathcal{D}$, then

$$X\left(t\right)=E_{\tilde{\gamma }\left(t\right)}\left(v_t\right)\in \mathfrak{p}.$$

(A232)

The moving Maurer-Cartan frame identifies cross-time tangents with one fixed Lie algebra, so their brackets are well defined. Substitution in Equation (671) gives

$${\mathsf{\Omega }}_{\mathfrak{k}}^{\left(2\right)}\left(T\right)=-\frac{1}{2}{\int }_{t_1>t_2}\left[E_{\tilde{\gamma }\left(t_1\right)}{\left(\mathbb{M}\mathrm{d}\mathcal{D}\right)}_{t_1},E_{\tilde{\gamma }\left(t_2\right)}{\left(\mathbb{M}\mathrm{d}\mathcal{D}\right)}_{t_2}\right]\mathrm{d}{t}_1\mathrm{d}{t}_2.$$

(A233)

The two descent signs cancel; the overall sign follows the convention in Equation (670). For $n=2$, identifying $X\left(t\right)={\dot{\eta }}^i\left(t\right)B_i$ turns Equation (671) into infinitesimal Thomas-Wigner rotation. For $n>2$, the parity theorem persists although the higher-rank base is not ordinary constant-curvature velocity space.

Appendix F.4. Finite Flagged-Extension Proof

Proof of Proposition 32. 

The Kraus operators $M_r=\sqrt{p_r}|r\rangle \otimes U_r$ satisfy ${\sum }_r{M}_r^{†}{M}_r=I$, so ${\widehat{\mathcal{E}}}_{\mathrm{flag}}$ is CPTP; tracing out $\mathsf{\Gamma }$ gives $\mathcal{E}$. For faithful $\rho$, the cq relative-entropy chain rule-equivalently, Donald’s finite-ensemble identity [14,108]-gives

$$\sum _r{p}_r{D}_{\mathrm{U}}({\rho }_r\parallel {\tau }_{\mathrm{out}})=\sum _r{p}_r{D}_{\mathrm{U}}({\rho }_r\parallel \mathcal{E}\left(\rho \right))+D_{\mathrm{U}}(\mathcal{E}\left(\rho \right)\parallel {\tau }_{\mathrm{out}}).$$

(A234)

The first term on the right is $I{(\mathsf{\Gamma }:S)}_{\mathsf{\Omega }}$. Direct expansion and unitary invariance give

$$D_{\mathrm{U}}(\rho \parallel {\tau }_{\mathrm{in}})-\sum _r{p}_r{D}_{\mathrm{U}}({\rho }_r\parallel {\tau }_{\mathrm{out}})=Tr\left(\rho {\mathfrak{M}}_{{\tau }_{\mathrm{in}},{\tau }_{\mathrm{out}}}\right),$$

(A235)

which proves Equation (677). The fixed-input criterion follows immediately. If the uncorrected equality holds for every faithful input, then $Tr\left(\rho \mathfrak{M}\right)=0$ on an open subset of the trace-one affine hyperplane, hence $\mathfrak{M}=0$. Conversely, $\mathfrak{M}=0$ removes the correction. Moreover,

$$\sum _r{p}_r{D}_{\mathrm{U}}\left({\tau }_{\mathrm{in}}\parallel U_r^{†}{\tau }_{\mathrm{out}}{U}_r\right)=-Tr\left({\tau }_{\mathrm{in}}{\mathfrak{M}}_{{\tau }_{\mathrm{in}},{\tau }_{\mathrm{out}}}\right).$$

(A236)

Thus $\mathfrak{M}=0$, or merely the uncorrected equality at $\rho ={\tau }_{\mathrm{in}}$, makes the nonnegative left-hand side vanish. Since every $p_r>0$, each term vanishes, which is equivalent to Equation (679); its converse is immediate. Under that condition,

$$D_{\mathrm{U}}({\mathsf{\Omega }}_{\mathsf{\Gamma }S}\parallel {\mathsf{\Omega }}_{\mathsf{\Gamma }}\otimes {\tau }_{\mathrm{out}})=\sum _r{p}_r{D}_{\mathrm{U}}({\rho }_r\parallel {\tau }_{\mathrm{out}})=D_{\mathrm{U}}(\rho \parallel {\tau }_{\mathrm{in}}),$$

(A237)

Substitution into the same cq chain rule yields Equation (680). Arbitrary (possibly singular) $\rho$ follow by approximation, since the references are faithful and all spaces are finite-dimensional. □

Appendix F.5. Proof and Consequences of BKM Contraction

Proof from monotone-metric contraction. 

For faithful $\rho$, choose $0<c_{\rho }\le C_{\rho }$ with $c_{\rho }I\le \rho \le C_{\rho }I$. Complete positivity gives $c_{\rho }\mathsf{\Lambda }\left(I\right)\le \mathsf{\Lambda }\left(\rho \right)\le C_{\rho }\mathsf{\Lambda }\left(I\right)$, so every output is faithful on the fixed carrier $P_{\mathsf{\Lambda }}{\mathbb{C}}^m$. A finite-dimensional linear map is open onto its image; consequently ${\mathcal{Q}}_{\mathsf{\Lambda }}$ is relatively open and convex, its fibers are the stated equivalence classes, and its tangent is $imL$.

For $q,r\in {\mathcal{Q}}_{\mathsf{\Lambda }}$ and trace-zero $H,K\in imL$, differentiation of the reduced negative entropy gives

$$D^2{\mathcal{F}}_{\mathsf{\Lambda },q}[H,K]=Tr\left[H{\mathsf{K}}_q^{-1}\left(K\right)\right]=g_q^{\mathrm{BKM}}(H,K),B_{{\mathcal{F}}_{\mathsf{\Lambda }}}(q,r)=D_{\mathrm{U}}(q\parallel r).$$

The restricted Hessian is positive definite, so the quotient is dually flat and Sakamoto’s theorem gives the intrinsic mixture Born lift [15, Theorem 3.1; see also Facts 3.2-3.3].

The BKM metric is a Petz monotone metric; CPTP monotonicity therefore gives the displayed semimetric contraction [6,47]. Equivalently, it is the second diagonal variation of Lindblad data processing. Since the output BKM form is positive definite on $imL$, the pullback radical is exactly $kerL$. This proves the quotient metric and Born packaging while retaining the fixed-support and relative-openness steps that the general monotonicity theorem does not supply. □

Proof of Corollary 12. 

The input Umegaki divergence is generated by $\rho \mapsto Tr(\rho log\rho )$, while its output counterpart is generated by the pullback of $q\mapsto Tr(qlogq)$. Linearity in the generating potential gives the Bregman identity; differentiating and applying BKM contraction gives the Hessian inequality.

For faithful $\rho ,\sigma$, put $Z:=\rho -\sigma$ and ${\rho }_t:=\sigma +tZ$. Convexity keeps ${\rho }_t$ faithful, while Theorem 8 keeps every $\mathsf{\Lambda }\left({\rho }_t\right)$ faithful on the fixed support $P_{\mathsf{\Lambda }}$. Applying the reversed $C^2$ remainder identity in Theorem A3 to ${\mathcal{H}}_{\mathsf{\Lambda }}$, whose possibly degenerate Hessian is $g_{\mathsf{\Lambda }}^{\mathrm{loss}}$, proves Equation (239). Differentiating $B_{{\mathcal{H}}_{\mathsf{\Lambda }}}(x,\sigma )$ twice at $x=\sigma$ proves Equation (240). Because $\mathsf{\Lambda }$ is affine on the trace-one hyperplane, a third derivative introduces no mixed chain-rule terms and gives ${\left({\nabla }^{\left(\mathrm{m}\right)}\right)}^3{\mathcal{H}}_{\mathsf{\Lambda }}={\left({\nabla }^{\left(\mathrm{m}\right)}\right)}^3\mathcal{F}-{\mathsf{\Lambda }}^{\ast }{\left({\overline{\nabla }}^{\left(\mathrm{m}\right)}\right)}^3{\mathcal{F}}_{\mathsf{\Lambda }}$; this proves Equation (238). Taylor expansion through cubic order gives Equation (241). On compact subsets of the faithful domain, the fourth derivatives of ${\mathcal{H}}_{\mathsf{\Lambda }}$ are bounded, which makes the displayed remainder locally uniform for bounded tangents. Finally, positive semidefiniteness and continuity of $g_{\mathsf{\Lambda },{\rho }_t}^{\mathrm{loss}}$ show that its weighted integral vanishes if and only if its value on $(Z,Z)$ vanishes for every $t\in [0,1]$. □

Set $r_{\mathsf{\Lambda }}:=rank{P}_{\mathsf{\Lambda }}=dim\left(P_{\mathsf{\Lambda }}{\mathbb{C}}^m\right)$. For $q\in {\mathcal{Q}}_{\mathsf{\Lambda }}$ and $H\in imL$, differentiating the reduced-carrier log chart gives

$${\left.D{\mathcal{X}}_{P_{\mathsf{\Lambda }}}\right|}_q\left[H\right]=-{\mathsf{K}}_q^{-1}\left(H\right)+\frac{1}{r_{\mathsf{\Lambda }}}{Tr}_{P_{\mathsf{\Lambda }}}\left[{\mathsf{K}}_q^{-1}\left(H\right)\right]I_{P_{\mathsf{\Lambda }}}\equiv -{\mathsf{K}}_q^{-1}\left(H\right)(mod\mathbb{R}{I}_{P_{\mathsf{\Lambda }}}).$$

(A238)

For an input tangent generated by the centered Hermitian score $\delta \tilde{X}$ in Equation (71), the output score is generally not $\mathsf{\Lambda }\left(\delta \tilde{X}\right)$: channels are linear on states, not logarithms. Define instead

$$\delta {\tilde{X}}^{\prime }:=-{\mathsf{K}}_{\mathsf{\Lambda }\left(\rho \right)}^{-1}\left(\mathsf{\Lambda }\left(V\right)\right).$$

(A239)

Then the metric inequality in Equation (235) becomes

$${∥\delta {\tilde{X}}^{\prime }∥}_{\mathsf{\Lambda }\left(\rho \right),\mathrm{BKM}}^2\le {∥\delta \tilde{X}∥}_{\rho ,\mathrm{BKM}}^2.$$

(A240)

Thus physical coarse-graining contracts the local BKM norm of centered Lie-log displacements.

Appendix F.5.1. Complete Qubit Dephasing: A Genuinely Noninjective Quotient

For $\parallel \mathit{r}\parallel <1$, let

$${\rho }_{\mathit{r}}:=\frac{1}{2}(I+\mathit{r}·\sigma ),P_{\pm }:=\frac{1}{2}(I\pm {\sigma }_z),{\mathsf{\Pi }}_z\left(A\right):=P_{+}A{P}_{+}+P_{-}A{P}_{-}=\frac{1}{2}(A+{\sigma }_{z}A{\sigma }_z).$$

(A241)

This channel is CPTP and ${\mathsf{\Pi }}_z\left({\rho }_{\mathit{r}}\right)={\rho }_{r_z{\mathit{e}}_z}$. For $V_{\mathit{v}}:=\mathit{v}·\sigma /2$,

$$L_{{\mathsf{\Pi }}_z}\left(V_{\mathit{v}}\right)=\frac{v_z}{2}{\sigma }_z,ker{L}_{{\mathsf{\Pi }}_z}={span}_{\mathbb{R}}\{{\sigma }_x,{\sigma }_y\},rank{L}_{{\mathsf{\Pi }}_z}=1,{\rho }_{\mathit{r}}{\sim }_{{\mathsf{\Pi }}_z}{\rho }_{\mathit{s}}\iff r_z=s_z.$$

(A242)

Thus the fibers are genuinely positive-dimensional. The quotient and its mixture potential are

$$\begin{array}{cc}\hfill {\mathcal{Q}}_{{\mathsf{\Pi }}_z}& =\left\{\frac{1}{2}(I+z{\sigma }_z):-1<z<1\right\}\cong (-1,1),\hfill \end{array}$$

(A243)

$$\begin{array}{cc}\hfill {\mathcal{F}}_{{\mathsf{\Pi }}_z}\left(z\right)=f\left(z\right)& :=\frac{1+z}{2}log\frac{1+z}{2}+\frac{1-z}{2}log\frac{1-z}{2}.\hfill \end{array}$$

(A244)

Consequently

$${\overline{D}}_{{\mathsf{\Pi }}_z}(\left[{\rho }_{\mathit{r}}\right]\parallel \left[{\rho }_{\mathit{s}}\right])=B_f(r_z,s_z),{\overline{g}}_{{\mathsf{\Pi }}_z}=\frac{\mathrm{d}{z}^2}{1-z^2},g_{{\mathsf{\Pi }}_z,{\rho }_{\mathit{r}}}^{\mathrm{obs}}(V_{\mathit{v}},V_{\mathit{w}})=\frac{v_z{w}_z}{1-r_z^2}.$$

(A245)

The pullback radical is exactly $ker{L}_{{\mathsf{\Pi }}_z}$, while

$${\mathcal{B}}_{{\mathsf{\Pi }}_z,\mathrm{m}}:={\mathcal{B}}_{{\overline{\nabla }}^{\left(\mathrm{m}\right)}}({\mathcal{Q}}_{{\mathsf{\Pi }}_z},{\overline{g}}_{{\mathsf{\Pi }}_z})$$

(A246)

is the intrinsic strongly integrable Sakamoto structure on $T{\mathcal{Q}}_{{\mathsf{\Pi }}_z}$. The defect potential and its two-point divergence are explicitly

$${\mathcal{H}}_{{\mathsf{\Pi }}_z}\left(\mathit{r}\right)=f\left(\parallel \mathit{r}\parallel \right)-f\left(r_z\right)=D_{\mathrm{U}}({\rho }_{\mathit{r}}\parallel {\mathsf{\Pi }}_z\left({\rho }_{\mathit{r}}\right)),{\mathsf{\Delta }}_{{\mathsf{\Pi }}_z}({\rho }_{\mathit{r}},{\rho }_{\mathit{s}})=B_{{\mathcal{H}}_{{\mathsf{\Pi }}_z}}(\mathit{r},\mathit{s}).$$

(A247)

For $0<a<1$, the distinct faithful states ${\rho }_{a{\mathit{e}}_x}$ and $I/2$ have the same output, but

$${\mathsf{\Delta }}_{{\mathsf{\Pi }}_z}({\rho }_{a{\mathit{e}}_x},I/2)=\frac{1+a}{2}log(1+a)+\frac{1-a}{2}log(1-a)>0.$$

(A248)

This is complete loss of distinguishability within one quotient fiber, not merely strict contraction by an injective parametrization. Since ${\mathsf{\Pi }}_z=\frac{1}{2}(\mathrm{id}+{Ad}_{{\sigma }_z})$, the common reference $I/2$ satisfies the branchwise condition for the particular two-outcome flagged extension with branches $I,{\sigma }_z$ and equal weights. Hence Proposition 32 gives $I{(\mathsf{\Gamma }:S)}_{\mathsf{\Omega }}={\mathcal{H}}_{{\mathsf{\Pi }}_z}$. The reduced dephasing map alone does not select this flag. Moreover, $f^{\prime }\left(z\right)=artanhz$ maps $(-1,1)$ onto $\mathbb{R}$, so the pure output Type-II generator is global; strict depolarization below instead has a proper Legendre domain.

Appendix F.5.2. Strict Qubit Depolarization: The Injective Proper-Image Complement

For the Bloch states in Equation (A241), let

$${\mathsf{\Lambda }}_{\lambda }\left(\rho \right):=\lambda \rho +(1-\lambda )\frac{I}{2},0<\lambda <1.$$

(A249)

Then ${\mathsf{\Lambda }}_{\lambda }\left({\rho }_{\mathit{r}}\right)={\rho }_{\lambda \mathit{r}}$ and the faithful output image is the open Bloch ball ${\mathcal{Q}}_{\lambda }=\{{\rho }_{\mathit{s}}:\parallel \mathit{s}\parallel <\lambda \}$. Its tangent map is $L_{\lambda }=\lambda \mathrm{id}$, so $ker{L}_{\lambda }=0$: this example complements Equation (A242) rather than supplying another quotient with positive-dimensional fibers. The mixture potential is $\mathcal{F}\left({\rho }_{\mathit{r}}\right)=f\left(r\right)$, where $r=\parallel \mathit{r}\parallel$. For $V=\mathit{v}·\sigma /2$ and $W=\mathit{w}·\sigma /2$, its BKM Hessian is

$$g_{\mathit{r}}^{\mathrm{BKM}}(\mathit{v},\mathit{w})=\frac{artanhr}{r}{\mathit{v}}_{\perp }·{\mathit{w}}_{\perp }+\frac{(\mathit{v}·\widehat{\mathit{r}})(\mathit{w}·\widehat{\mathit{r}})}{1-r^2},$$

(A250)

with the continuous value $g_{\mathbf{0}}^{\mathrm{BKM}}(\mathit{v},\mathit{w})=\mathit{v}·\mathit{w}$. Thus ${\mathcal{Q}}_{\lambda }$ is an explicit nontrivial mixture-Hessian, and hence strongly integrable Born, output manifold.

The channel-defect potential becomes

$${\mathcal{H}}_{\lambda }\left(\mathit{r}\right)=f\left(r\right)-f\left(\lambda r\right).$$

(A251)

Its Hessian has tangential and radial eigenvalues

$$\begin{array}{cc}\hfill A_{\perp }\left(r\right)& =\frac{artanhr-\lambda artanh\left(\lambda r\right)}{r},\hfill \end{array}$$

(A252)

$$\begin{array}{cc}\hfill A_{\Vert }\left(r\right)& =\frac{1-{\lambda }^2}{(1-r^2)(1-{\lambda }^2{r}^2)}.\hfill \end{array}$$

(A253)

Both tend to $1-{\lambda }^2$ at $r=0$ and are positive because $q_{\lambda }\left(r\right):=artanhr-\lambda artanh\left(\lambda r\right)$ has $q_{\lambda }\left(0\right)=0$ and $q_{\lambda }^{\prime }=A_{\Vert }>0$. Thus contraction is strict. For the noncommuting pair $\mathit{a}=r{\mathit{e}}_x$, $\mathit{b}=r{\mathit{e}}_z$, $[{\rho }_{\mathit{a}},{\rho }_{\mathit{b}}]\ne 0$ and

$${\mathsf{\Delta }}_{{\mathsf{\Lambda }}_{\lambda }}({\rho }_{\mathit{a}},{\rho }_{\mathit{b}})=r\left[artanhr-\lambda artanh\left(\lambda r\right)\right]>0.$$

(A254)

Hence ${\kappa }_A{B}_{{\mathcal{H}}_{\lambda }}$ is a global regular Type-I generator with mixed Hessian $-{\kappa }_{A}Hess{\mathcal{H}}_{\lambda }$.

For the output Umegaki generator, define

$$\mathit{y}\left(\mathit{s}\right):=\frac{artanhs}{s}\mathit{s},\chi \left(\mathit{y}\right):=log\left(2cosh\parallel \mathit{y}\parallel \right),{\alpha }_{\lambda }:=artanh\lambda .$$

(A255)

where $s=\parallel \mathit{s}\parallel$ and $\mathit{y}\left(\mathbf{0}\right)=0$. Here $\mathit{y}·\sigma$ represents $log{\rho }_{\mathit{s}}$ modulo $\mathbb{R}I$. If $P_1={\mathit{p}}_1·\sigma ={\kappa }_A[\mathit{y}\left({\mathit{s}}_1\right)-\mathit{y}\left({\mathit{s}}_0\right)]·\sigma$, set $\mathit{z}:={\mathit{y}}_0+{\mathit{p}}_1/{\kappa }_A$. The generator and reconstructed endpoint are

$$\begin{array}{cccc}\hfill H_{d,\lambda }^{+}({\mathit{y}}_0,{\mathit{p}}_1)& ={\kappa }_A[\chi \left(\mathit{z}\right)-\chi \left({\mathit{y}}_0\right)],\hfill & \hfill {\mathit{s}}_1& =\frac{tanh\parallel \mathit{z}\parallel }{\parallel \mathit{z}\parallel }\mathit{z},\hfill \end{array}$$

(A256)

with continuous value ${\mathit{s}}_1=\mathbf{0}$ at $\mathit{z}=\mathbf{0}$. Both endpoints lie in the strictly depolarized image, and the regular Type-II generator is attained, precisely on

$${\mathcal{D}}_{\lambda }^{+}=\left\{({\mathit{y}}_0,{\mathit{p}}_1):\parallel {\mathit{y}}_0\parallel <{\alpha }_{\lambda },∥{\mathit{y}}_0+{\mathit{p}}_1/{\kappa }_A∥<{\alpha }_{\lambda }\right\}.$$

(A257)

Outside ${\mathcal{D}}_{\lambda }^{+}$, the analytic expression is an ambient continuation, not an attained output Type-II generator. Thus Type-I is defined on every output endpoint pair, but regular Type-II attainment requires the natural Legendre image.

Finally, choose the Pauli random-unitary decomposition

$${\mathsf{\Lambda }}_{\lambda }\left(\rho \right)={\displaystyle \sum _{\mu =0}^3}{p}_{\mu }{\sigma }_{\mu }\rho {\sigma }_{\mu },p_0=\frac{1+3\lambda }{4},p_1=p_2=p_3=\frac{1-\lambda }{4},$$

(A258)

where ${\sigma }_0=I$, and equip this decomposition with a chosen corresponding orthogonally flagged CPTP extension. For the common invariant reference $\tau =I/2$, Proposition 32 gives

$$I{(\mathsf{\Gamma }:S)}_{\mathsf{\Omega }}=S_{\mathrm{vN}}\left({\mathsf{\Lambda }}_{\lambda }\left(\rho \right)\right)-S_{\mathrm{vN}}\left(\rho \right)={\mathcal{H}}_{\lambda }\left(\mathit{r}\right).$$

(A259)

This equality belongs to the specified Pauli flagged extension; the reduced map alone neither selects a unique flagged extension nor establishes that its flag is a physical record.

Appendix F.5.3. Petz Recovery and Stationary Decay

For faithful $\sigma$, the Petz map is

$${\mathcal{R}}_{\sigma ,\mathsf{\Lambda }}\left(X\right):={\sigma }^{1/2}{\mathsf{\Lambda }}^{\ast }\left[\mathsf{\Lambda }{\left(\sigma \right)}^{-1/2}X\mathsf{\Lambda }{\left(\sigma \right)}^{-1/2}\right]{\sigma }^{1/2},$$

(A260)

where the inverse is taken on $P_{\mathsf{\Lambda }}$. Petz sufficiency gives [12,13]

$${\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,\sigma )=0⟺{\mathcal{R}}_{\sigma ,\mathsf{\Lambda }}\left(\mathsf{\Lambda }\left(\rho \right)\right)=\rho ,$$

(A261)

and the same map recovers $\sigma$.

Now independently supply a finite-dimensional norm-continuous CPTP semigroup ${\mathcal{T}}_t=exp\left(t{\mathcal{L}}_{\ast }\right)$, $t\ge 0$, with faithful stationary state $\tau$. Its generator includes the chosen time scale. For faithful ${\rho }_0$, the curve ${\rho }_t:={\mathcal{T}}_t\left({\rho }_0\right)$ remains faithful: ${\rho }_0⪰c\tau$ for some $c>0$ implies ${\rho }_t⪰c\tau$. DPI and stationarity make $D_{\mathrm{U}}({\rho }_t\parallel \tau )$ nonincreasing. For faithful $\rho$, the trace-preserving chain rule gives Spohn’s relative-entropy production functional [69]:

$$\mathsf{\Sigma }\left(\rho \right):=-Tr\left[{\mathcal{L}}_{\ast }\left(\rho \right)(log\rho -log\tau )\right]=-{\left.\frac{\mathrm{d}}{\mathrm{d}u}{D}_{\mathrm{U}}({\mathcal{T}}_u\rho \parallel \tau )\right|}_{u=0^{+}}\ge 0.$$

(A262)

For all $t,h\ge 0$, the exact global channel-dissipation bridge is

$$\begin{array}{cc}\hfill {\mathsf{\Delta }}_{{\mathcal{T}}_h}({\rho }_t,\tau )& =D_{\mathrm{U}}({\rho }_t\parallel \tau )-D_{\mathrm{U}}({\rho }_{t+h}\parallel \tau )\hfill \\ & ={\int }_t^{t+h}\mathsf{\Sigma }\left({\rho }_s\right)\mathrm{d}s.\hfill \end{array}$$

(A263)

The first equality is stationarity and the semigroup law; the second is integration of Equation (A258). In this supplied stationary realization, the channel cocycle becomes additivity of accumulated relative-information dissipation. No detailed-balance assumption is needed for the identity, but strict positivity and decay rates do not follow from stationarity alone.

Differentiating BKM monotonicity at $t=0$ gives, for trace-zero Hermitian V,

$${\mathcal{Q}}_{\tau }^{\mathrm{BKM}}\left(V\right):=-g_{\tau }^{\mathrm{BKM}}(V,{\mathcal{L}}_{\ast }V)\ge 0.$$

(A264)

Taylor expansion of the matrix logarithm at faithful $\tau$, using ${\mathcal{L}}_{\ast }\left(\tau \right)=0$, gives in the present conventions

$$\mathsf{\Sigma }(\tau +ϵV)={ϵ}^2{\mathcal{Q}}_{\tau }^{\mathrm{BKM}}\left(V\right)+O\left({ϵ}^3\right).$$

(A265)

Monotonicity alone gives no rate. If $\mathsf{\Sigma }\left(\rho \right)\ge 2\alpha D_{\mathrm{U}}(\rho \parallel \tau )$, then Grönwall yields

$$D_{\mathrm{U}}({\rho }_t\parallel \tau )\le {\mathrm{e}}^{-2\alpha t}{D}_{\mathrm{U}}({\rho }_0\parallel \tau ).$$

(A266)

Existence of $\alpha >0$ is an additional semigroup property [70].

Appendix F.6. Based-Loop Convolution Semigroups

For based loops, assume that the represented holonomy measures ${\mu }_t$ are weakly continuous with ${\mu }_0={\delta }_e$ and stationary independent increments. Then ${\mu }_{t+s}={\mu }_t\ast {\mu }_s$ and the averaged channels form a semigroup [142]. BCH motivates compact increments but fixes neither their scaling nor a Brownian limit; microscopic rate, covariance, and a small-jump condition remain dynamical input.

• Modeling assumption: isotropic Brownian holonomy. Let $T_a$, $a=1,\dots ,n^2-1$, be traceless Hermitian generators in the defining representation, normalized by $Tr\left(T_a{T}_b\right)=\frac{1}{2}{\delta }_{ab}$. We postulate that the unresolved compact transport is the $\mathrm{SU}\left(n\right)$-valued Stratonovich diffusion

  $$\mathrm{d}{U}_t=-\mathrm{i}\sqrt{\gamma }{\displaystyle \sum _{a=1}^{n^2-1}}{T}_a{U}_t\circ \mathrm{d}{W}_t^a,U_0=I,$$

  (A267)

  where the $W_t^a$ are independent standard Brownian motions and $\gamma \ge 0$ has units of inverse time. This conditional model does not follow from BCH alone [143].

Proposition A5

(Channel generated by isotropic Brownian holonomy). Define ${\mathcal{E}}_t^{\mathrm{hol}}\left(\rho \right):=\mathbb{E}\left[U_t\rho U_t^{†}\right]$. Then ${\left\{{\mathcal{E}}_t^{\mathrm{hol}}\right\}}_{t\ge 0}$ is a random-unitary CPTP semigroup with generator

$${\mathcal{L}}_{\mathrm{hol}}\left(\rho \right)=-\frac{\gamma }{2}\sum _a[T_a,[T_a,\rho ]]=\gamma \sum _a\left(T_a\rho T_a-\frac{1}{2}\{T_a^2,\rho \}\right).$$

(A268)

In the defining representation,

$${\mathcal{L}}_{\mathrm{hol}}\left(\rho \right)=\frac{\gamma n}{2}\left(\frac{I}{n}-\rho \right),$$

(A269)

and hence

$${\mathcal{E}}_t^{\mathrm{hol}}\left(\rho \right)=\frac{I}{n}+{\mathrm{e}}^{-\gamma nt/2}\left(\rho -\frac{I}{n}\right).$$

(A270)

Proof. 

Stratonovich-Itô conversion and the product rule for $U_t\rho U_t^{†}$ give the double-commutator generator [143], Chapters 3-4; independent right increments give the semigroup law. In the defining representation, ${\sum }_a{T}_{a}A{T}_a=(Tr\left(A\right)I-A/n)/2$ and ${\sum }_a{T}_a^2=(n^2-1)I/\left(2n\right)$ reduce it to Equation (A269); solving that linear equation gives Equation (A270). □

For a unitary representation R, set

$$T_a^{\left(R\right)}:=\mathrm{i}\mathrm{d}R(-\mathrm{i}{T}_a).$$

(A271)

Equation (A268) then holds with $T_a^{\left(R\right)}$. The Fierz identity and single-rate formula are specific to the defining representation; general R requires decomposing its conjugation representation.

Appendix G.1. Complete Endpoint Characteristics

Proof of Lemma 4. 

The regular Type-I generating-function theorem gives the local endpoint Legendre diffeomorphisms and symplectic map [36], Sections 1.3 and 1.6. The selected inverse and Equation (695) give Equation (696); pulling back the extended canonical one-form gives Equation (697). Differentiate the Hamilton-Jacobi identity in $q_0$ and compare with the derivative of $p_0=-D_0{S}_h(q_0,q_h)$ at fixed incoming data:

$$D_0{D}_1{S}_h\left({\dot{q}}_h-D_p{\mathcal{H}}_h\right)=0.$$

Invertibility gives the first Hamilton equation; differentiating the Hamilton-Jacobi identity in q and using $p_h=D_1{S}_h(q_0,q_h)$ gives the second. Together they prove Equation (698) with convention Equation (686). Integration along the characteristic gives Equation (700), with the zero lower limit under the stated extension and vanishing-boundary hypotheses. Nondegeneracy of ${\omega }_A$ gives uniqueness up to $c\left(h\right)$; the scalar-gauge and clock formulas follow by the chain rule. □

Appendix G.2. Regularity, Legendre Images, and the Exact Canonical Relation

Affine differentiation gives the momenta and mixed Hessian in Equations (712)–(714); the endpoint-separated potential contributes no mixed derivative. The two natural Legendre images are

$$\begin{array}{cc}\hfill im{\mathbb{F}}^{-}{L}_d=\{(q_0,{\mathsf{A}}_0):& \eta \left(q_0\right)+{\kappa }_{\mathsf{\Delta }t}^{-1}{g}_{q_0}^{-1}\left({\mathsf{A}}_0-\frac{\mathsf{\Delta }t}{2}\mathrm{d}{V}_{q_0}\right)\in \eta \left(M\right)\},\hfill \end{array}$$

(A272)

$$\begin{array}{cc}\hfill im{\mathbb{F}}^{+}{L}_d=\{(q_1,{\mathsf{A}}_1):& \mathrm{d}{\psi }_{q_1}-{\kappa }_{\mathsf{\Delta }t}^{-1}\left({\mathsf{A}}_1+\frac{\mathsf{\Delta }t}{2}\mathrm{d}{V}_{q_1}\right)\in \mathrm{d}\psi \left(M\right)\}.\hfill \end{array}$$

(A273)

The first inversion is affine in $q_1$; the second uses the injective gradient map of the strictly convex potential. Thus both maps are diffeomorphisms onto these open images, but need not cover all of $T^{\ast }M$.

Since Equation (712) is nondegenerate, the standard discrete Poincaré-form and variational theorems imply that (718) is exact Lagrangian, is the graph of (717) on the two natural images, and that stationarity, endpoint variation, and composition are exactly Equations (719) and (721) and ${\mathsf{\Gamma }}_{L_d}^{\circ N}$ ([36], Sections 1.3 and 1.5) ([37], Sections 5.1-5.2). The sign follows from $j_d^{\ast }(-{\lambda }_{A,0}+{\lambda }_{A,1})=\mathrm{d}{L}_d$ under ${\omega }_A=\mathrm{d}{\lambda }_A$.

Appendix G.3. The Umegaki Type-II Transform

For $V=0$ and ${\kappa }_{\mathsf{\Delta }t}\equiv {\kappa }_A$, apply the Gibbs variational identity Equation (87) with reference ${\rho }_0$ and source $P_1/{\kappa }_A$, where $P_1$ represents the trace-one cotangent class. Its value and unique faithful optimizer give Equations (739) and (740). The affine lift Equation (744) accounts for scalar shifts without changing that state. The canonical relation has the natural Legendre images above; it is not a quantum channel, a global automorphism of $T^{\ast }M$, or a short-time mechanical approximation.

Appendix G.4. Directed Free-Endpoint Reconstruction

Proof of Theorem 24. 

The outgoing relation $p=D_1{S}_h^{\to }=({\mu }_{\ast }/h)({\vartheta }_q-{\vartheta }_{q_0})$ is equivalent to $q_0=Q_h(q,p)$, and $D_p{Q}_h\left[\alpha \right]=-(h/{\mu }_{\ast }){\left(g_{Q_h}^{♭}\right)}^{-1}\alpha$. Since $D_Q{B}_{\psi }(q,Q)\left[Y\right]=-g_Q(q-Q,Y)$, differentiation proves Equation (725). At $Q=q-hv$, its inverse fiber Legendre relation is $p=({\mu }_{\ast }/h)({\vartheta }_q-{\vartheta }_Q)$. The identity

$$({\vartheta }_q-{\vartheta }_Q)[q-Q]=B_{\psi }(q,Q)+B_{\psi }(Q,q)$$

(A274)

then gives $p\left[v\right]-{\mathcal{H}}_h^{\to }(q,p)=({\mu }_{\ast }/h^2)B_{\psi }(Q,q)$, proving Equation (726). Taylor expansion on compact subsets gives Equation (727).

At fixed endpoints, $-{\partial }_h{S}_h^{\to }=({\mu }_{\ast }/h^2)B_{\psi }(q,q_0)$, which evaluates to ${\mathcal{H}}_h^{\to }$ at $q_0=Q_h$. The mixed Hessian $D_0{D}_1{S}_h^{\to }=-({\mu }_{\ast }/h)g_{q_0}$ is nondegenerate, and the explicit smooth inverse restricts locally to the common right-Legendre product chart required by Lemma 4.

For Equation (728), $p_0=-D_0{S}_t^{\to }={\mu }_{\ast }{g}_{q_0}(v,·)$ is independent of t, so it is the selected characteristic. Taylor expansion gives $q_t\to q_0$, $p_t\to {\mu }_{\ast }{g}_{q_0}(v,·)=p_0$, and $S_t^{\to }(q_0,q_t)=O\left(t\right)\to 0$, verifying both ${\mathsf{\Phi }}_0^{\to }=\mathrm{id}$ and the zero-boundary normalization. Thus Equation (700) and the exact fiber Legendre relation give Equation (729). Hyperregular Legendre duality makes the Hamilton and Euler-Lagrange characteristics equivalent. If $\psi$ is quadratic, g is constant and direct substitution into Equation (724) gives ${\mathcal{H}}_h^{\to }=g^{-1}(p,p)/\left(2{\mu }_{\ast }\right)$ for every h, proving the stated exact special case. □

Proposition A6

(Endpoint-induced Umegaki Hamilton-Jacobi representation). For $h>0$, define the mechanically scaled two-endpoint function

$$S_h(\sigma ,\rho ):=\frac{{\mu }_{\ast }}{h}{D}_{\mathrm{U}}(\rho \parallel \sigma ).$$

(A275)

For a Hermitian representative P of a final cotangent class, put

$$\begin{array}{cc}\hfill {\sigma }_h(\rho ,P)& :=\frac{exp(log\rho -(h/{\mu }_{\ast })P)}{Trexp(log\rho -(h/{\mu }_{\ast })P)},\hfill \end{array}$$

(A276)

$$\begin{array}{cc}\hfill {\mathcal{H}}_h(\rho ,\left[P\right])& :=\frac{Tr\left(\rho P\right)}{h}+\frac{{\mu }_{\ast }}{h^2}logTrexp\left(log\rho -\frac{h}{{\mu }_{\ast }}P\right).\hfill \end{array}$$

(A277)

The second expression is invariant under $P\mapsto P+cI$ and equals

$${\mathcal{H}}_h(\rho ,\left[P\right])=\frac{{\mu }_{\ast }}{h^2}{D}_{\mathrm{U}}\left(\rho \parallel {\sigma }_h(\rho ,P)\right).$$

(A278)

With the negative-source mirror segment ${\rho }_s^{-}:=E_{\rho }(-shP/{\mu }_{\ast })$, so that ${\rho }_1^{-}={\sigma }_h(\rho ,P)$, it also has the exact response form

$$\begin{array}{|c|}\hline {\mathcal{H}}_h(\rho ,\left[P\right])=\frac{1}{{\mu }_{\ast }}{\int }_0^1(1-s){Cov}_{{\rho }_s^{-}}^{\mathrm{KM}}(P,P)\mathrm{d}s.\\ \hline\end{array}$$

(A279)

Its fiber derivative is hyperregular between its natural open momentum and velocity domains. More precisely,

$$\left[P\right]⟼W:=\frac{\rho -{\sigma }_h(\rho ,P)}{h}$$

(A280)

is a diffeomorphism from the cotangent quotient onto

$${\mathcal{V}}_{h,\rho }:=\{W=W^{†}:TrW=0,\rho -hW\in {\mathcal{S}}_n^{+}\},\left[P\right]=\frac{{\mu }_{\ast }}{h}[log\rho -log(\rho -hW)].$$

(A281)

For Hermitian representatives $A,B$,

$$\begin{array}{cc}\hfill D_P{\mathcal{H}}_h(\rho ,\left[P\right])\left[A\right]& =\frac{1}{h}Tr\left[(\rho -{\sigma }_h)A\right],\hfill \\ \hfill D_P^2{\mathcal{H}}_h(\rho ,\left[P\right])[A,B]& =\frac{1}{{\mu }_{\ast }}{Cov}_{{\sigma }_h}^{\mathrm{KM}}(A,B),\hfill \end{array}$$

(A282)

and its exact fiber Legendre transform is

$${\mathcal{L}}_h^{\mathrm{U}}(\rho ,W)=\frac{{\mu }_{\ast }}{h^2}{D}_{\mathrm{U}}(\rho -hW\parallel \rho ),\rho -hW\in {\mathcal{S}}_n^{+}.$$

(A283)

This specialization of Theorem 24 satisfies the common identities of Lemma 4, and

$${\mathcal{H}}_h(\rho ,\left[P\right])=H_{\mathrm{BKM}}(\rho ,\left[P\right])+O\left(h\right)$$

(A284)

uniformly for ρ in a compact subset of faithful states and $\left[P\right]$ in a bounded cotangent set. Its explicit affine characteristic and on-shell action specialize Equation (729):

$${\rho }_t=\sigma +\frac{t}{h}(\rho -\sigma ),{\int }_0^h{\mathcal{L}}_t^{\mathrm{U}}({\rho }_t,{\dot{\rho }}_t)\mathrm{d}t=\frac{{\mu }_{\ast }}{h}{D}_{\mathrm{U}}(\rho \parallel \sigma ).$$

(A285)

This h-dependent classical cotangent generator has the reconstruction scope of Section 8.2. The complete integral fixes it on the natural final-Legendre domain, not an extension beyond that domain. Here “quantum log-partition” refers to Kubo-Mori derivatives of the log normalizer; no two-point-measurement or stochastic work distribution is implied.

Proof. 

Work in the mixture-affine chart and apply Theorem 24 with $\psi =\mathcal{F}$, $q_0=\sigma$, and $q=\rho$. By Equation (74), its outgoing endpoint function is $S_h=({\mu }_{\ast }/h)D_{\mathrm{U}}(\rho \parallel \sigma )$. On trace-zero variations,

$$\left[D_{\rho }{S}_h\right]=\frac{{\mu }_{\ast }}{h}[log\rho -log\sigma ].$$

Consequently the normalized exponential in Equation (A276) is precisely the global right-Legendre inverse $Q_h$. Substitution of its logarithm into Equation (724) gives both Equations (A277) and (A278).

The fiber diffeomorphism, hyperregularity, exact Legendre transform, on-shell action, and compact-uniform small-h limit follow from Theorem 24. Under the cotangent-score identification, the inverse BKM metric is the Kubo-Mori covariance by Equation (61), giving the displayed momentum derivatives. Finally, the negative-source case of Equation (96), with source $-hP/{\mu }_{\ast }$, proves Equation (A275). Normalization and the cotangent quotient make every formula invariant under $P\mapsto P+cI$. □

Proposition A7

(Endpoint-induced Rényi-Moreau Hamilton-Jacobi family and midpoint improvement). Fix $0<\theta <1$, $h>0$, and define the mechanically scaled endpoint function

$$S_{h,\theta }(\sigma ,\rho ):=\frac{{\mu }_{\ast }}{h}{\mathfrak{B}}_{\theta }^{\mathrm{RM}}(\rho ,\sigma )=\frac{{\mu }_{\ast }}{h\theta }{D}_{\theta }^{\mathrm{b}}(\rho \parallel \sigma ).$$

(A286)

For a final cotangent class $\left[P\right]$, put $w:=P^{{♯}_{\rho }}$ and

$${\gamma }_{h,\theta }(\rho ,\left[P\right]):=\rho -\frac{h(1-\theta )}{{\mu }_{\ast }}w.$$

(A287)

On the natural right-Legendre domain

$${\mathcal{D}}_{h,\theta }^{\mathrm{RM}}:=\{(\rho ,\left[P\right])\in T^{\ast }{\mathcal{S}}_n^{+}:{\gamma }_{h,\theta }(\rho ,\left[P\right])\in {\mathcal{S}}_n^{+}\},$$

(A288)

define the inverse endpoint and Hamiltonian by

$$\begin{array}{cc}\hfill {\sigma }_{h,\theta }(\rho ,\left[P\right])& :=\frac{exp\left[\left(log{\gamma }_{h,\theta }-\theta log\rho \right)/(1-\theta )\right]}{Trexp\left[\left(log{\gamma }_{h,\theta }-\theta log\rho \right)/(1-\theta )\right]},\hfill \end{array}$$

(A289)

$$\begin{array}{cc}\hfill {\mathcal{H}}_{h,\theta }(\rho ,\left[P\right])& :=\frac{{\mu }_{\ast }}{h^2}{\mathfrak{B}}_{\theta }^{\mathrm{RM}}\left(\rho ,{\sigma }_{h,\theta }(\rho ,\left[P\right])\right).\hfill \end{array}$$

(A290)

This nonnegative Hamiltonian is independent of the scalar representative of $\left[P\right]$. The common identities of Lemma 4 apply on every selected common right-Legendre chart in these natural domains. Its on-shell action is Equation (765) with $V=0$ and $\psi =\mathcal{F}$. Uniformly for ρ in a compact faithful set and $\left[P\right]$ in a bounded cotangent set that remains in the natural domain,

$$\begin{array}{|c|}\hline {\mathcal{H}}_{h,\theta }(\rho ,\left[P\right])=H_{\mathrm{BKM}}(\rho ,\left[P\right])+\frac{h(2\theta -1)}{6{\mu }_{\ast }^2}{\mathcal{C}}_{\rho }(w,w,w)+O\left(h^2\right).\\ \hline\end{array}$$

(A291)

Consequently,

$${\mathcal{H}}_{h,1/2}(\rho ,\left[P\right])=H_{\mathrm{BKM}}(\rho ,\left[P\right])+O\left(h^2\right),$$

(A292)

whereas a directed endpoint has a generically nonzero $O\left(h\right)$ cubic correction. As $\theta \uparrow 1$, the reconstruction and Hamiltonian converge to Equations (A276) and (A278). This h-dependent classical endpoint Hamiltonian has the same reconstruction scope (Section 8.2); it is distinct from the regularization-scale Hamiltonian ${\mathcal{H}}_r^{\mathrm{U}}$.

Proof. 

Equation (706) gives

$${\left(D_{\rho }{S}_{h,\theta }\right)}^{{♯}_{\rho }}=\frac{{\mu }_{\ast }}{h(1-\theta )}(\rho -{\gamma }_{h,\theta }).$$

Substitution therefore reconstructs the internal state and then the initial endpoint through $log{\gamma }_{h,\theta }=(1-\theta )log\sigma +\theta log\rho$ modulo scalars. At fixed endpoints, $-{\partial }_h{S}_{h,\theta }=({\mu }_{\ast }/h^2){\mathfrak{B}}_{\theta }^{\mathrm{RM}}$, identifying the generator in Equation (695). The nondegenerate mixed Hessian Equation (707) and the explicit smooth inverse verify the lemma’s hypotheses after restriction to a common product chart. Put $ϵ=h/{\mu }_{\ast }$. In mixture-affine coordinates centered at the final endpoint, the dual-barycenter equation gives

$${\sigma }_{h,\theta }=\rho -ϵw-\frac{\theta {ϵ}^2}{2}{\left(g_{\rho }^{♭}\right)}^{-1}{\mathcal{C}}_{\rho }(w,w,·)+O\left({ϵ}^3\right).$$

(A293)

The corresponding final-endpoint form of Equation (766) is

$${\mathfrak{B}}_{\theta }^{\psi }(\rho ,\rho +\delta )=\frac{1}{2}{g}_{\rho }(\delta ,\delta )+\frac{1+\theta }{6}{\mathcal{C}}_{\rho }(\delta ,\delta ,\delta )+{O(\parallel \delta \parallel }^4).$$

(A294)

Substitution of the preceding $\delta ={\sigma }_{h,\theta }-\rho$ gives

$${\mathfrak{B}}_{\theta }^{\mathrm{RM}}\left(\rho ,{\sigma }_{h,\theta }\right)=\frac{{ϵ}^2}{2}{g}_{\rho }(w,w)+\frac{(2\theta -1){ϵ}^3}{6}{\mathcal{C}}_{\rho }(w,w,w)+O\left({ϵ}^4\right),$$

(A295)

and hence Equation (A291). Analyticity on compact faithful sets makes the remainder uniform. Finally, $D{log}_{\rho }\left[w\right]=P-Tr\left(\rho P\right)I$ for $w=P^{{♯}_{\rho }}$, so the quotient in Equation (A289) converges modulo scalars to $log\rho -(h/{\mu }_{\ast })P$ as $\theta \uparrow 1$. Hence ${\sigma }_{h,\theta }\to {\sigma }_h$. In addition, Equation (708) and smooth convergence on compact faithful sets give

$${\mathfrak{B}}_{\theta }^{\mathrm{RM}}\left(\rho ,{\sigma }_{h,\theta }\right)⟶D_{\mathrm{U}}(\rho \parallel {\sigma }_h),$$

(A296)

which proves convergence of the Hamiltonians as well as of the reconstructed endpoints. □

Appendix G.5. Thermal Calibration Details

For ${\tau }_{\beta }={\mathrm{e}}^{-\beta H_{\mathrm{sys}}}/Z_{\beta }$, direct calculation gives

$$Tr\left[(\rho -{\tau }_{\beta })H_{\mathrm{sys}}\right]=V_{\beta }\left(\rho \right)+{\beta }^{-1}\left[S_{\mathrm{vN}}\left(\rho \right)-S_{\mathrm{vN}}\left({\tau }_{\beta }\right)\right].$$

(A297)

Only $V_{\beta }$ is necessarily nonnegative. If an output Hamiltonian is independently supplied with $\mathsf{\Lambda }\left({\tau }_{\beta }\right)={\mathrm{e}}^{-\beta H_{\mathrm{out}}}/Z_{\beta }^{\mathrm{out}}$, then

$$V_{\beta }\left(\rho \right)-V_{\beta }^{\mathrm{out}}\left(\mathsf{\Lambda }\left(\rho \right)\right)={\beta }^{-1}{\mathsf{\Delta }}_{\mathsf{\Lambda }}(\rho ,{\tau }_{\beta })\ge 0,V_{\beta }^{\mathrm{out}}\left(q\right):={\beta }^{-1}{D}_{\mathrm{U}}(q\parallel \mathsf{\Lambda }\left({\tau }_{\beta }\right)).$$

(A298)

Under a supplied finite cq flagged extension satisfying the branchwise condition in Equation (679), Equation (680) rewrites the right-hand side as ${\beta }^{-1}I{(\mathsf{\Gamma }:S)}_{\mathsf{\Omega }}$. This is loss of reference-relative free-energy availability, not total internal-energy loss or, without a work protocol, universally extractable work.

Appendix G.5.1. Bidirectional Gibbs Quenches

Corollary A6

(Gibbs perturbations and bidirectional sudden-quench dissipation). Fix $\beta >0$, let $H_1=H_0+V$ and $Z_i:=Tr{\mathrm{e}}^{-\beta H_i}$, ${\tau }_i:={\mathrm{e}}^{-\beta H_i}/Z_i$, $F_{\mathrm{eq}}\left(H_i\right):=-{\beta }^{-1}log{Z}_i$, and set $\mathsf{\Delta }F:=F_{\mathrm{eq}}\left(H_1\right)-F_{\mathrm{eq}}\left(H_0\right)$. Then

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_{{\tau }_0}(-\beta V)& =-\beta \mathsf{\Delta }F,\hfill \\ \hfill \mathsf{\Delta }F& =\underset{\omega \in {\mathcal{S}}_n}{min}\left\{Tr\left(\omega V\right)+{\beta }^{-1}{D}_{\mathrm{U}}(\omega \parallel {\tau }_0)\right\},\hfill \end{array}$$

(A299)

$$\begin{array}{cc}\hfill Tr\left(\omega V\right)+{\beta }^{-1}{D}_{\mathrm{U}}(\omega \parallel {\tau }_0)-\mathsf{\Delta }F& ={\beta }^{-1}{D}_{\mathrm{U}}(\omega \parallel {\tau }_1).\hfill \end{array}$$

(A300)

For separately prepared forward and reverse instantaneous quenches, use inclusive two-point-measurement (TPM) mean work. Because each initial Gibbs state commutes with its initial Hamiltonian, its mean is equivalently the corresponding mean energy jump. The dissipated works are

$$\begin{array}{cc}\hfill W_{\mathrm{diss}}^{\to }& :=Tr\left({\tau }_{0}V\right)-\mathsf{\Delta }F={\beta }^{-1}{D}_{\mathrm{U}}({\tau }_0\parallel {\tau }_1),\hfill \\ \hfill W_{\mathrm{diss}}^{\leftarrow }& :=-Tr\left({\tau }_{1}V\right)+\mathsf{\Delta }F={\beta }^{-1}{D}_{\mathrm{U}}({\tau }_1\parallel {\tau }_0).\hfill \end{array}$$

(A301)

Writing the auxiliary equilibrium interpolation ${\tau }_t={\mathrm{e}}^{-\beta (H_0+tV)}/Z_t$, which is not either quench’s nonequilibrium trajectory, one consequently has

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill \beta (W_{\mathrm{diss}}^{\to }+W_{\mathrm{diss}}^{\leftarrow })& =D_{\mathrm{U}}({\tau }_0\parallel {\tau }_1)+D_{\mathrm{U}}({\tau }_1\parallel {\tau }_0)\hfill \\ & ={\beta }^2{\int }_0^1{Cov}_{{\tau }_t}^{\mathrm{KM}}(V,V)\mathrm{d}t\hfill \\ & \ge L_{\mathrm{BKM}}{\left({\tau }_{•}\right)}^2\ge d_{\mathrm{BKM}}{({\tau }_0,{\tau }_1)}^2.\hfill \end{array}\\ \hline\end{array}$$

(A302)

For $\alpha >0$, using the full-cone distance of Equation (332), the bidirectional cost also obeys the independent shifted-cone bound

$$\beta (W_{\mathrm{diss}}^{\to }+W_{\mathrm{diss}}^{\leftarrow })\ge \frac{1}{\alpha }{\int }_0^{\infty }{d}_{\mathrm{AI}}^{\left(\alpha \right)}{({\tau }_0+sI,{\tau }_1+sI)}^2\mathrm{d}s,$$

(A303)

and, for every non-scalar observable A,

$$W_{\mathrm{diss}}^{\to }+W_{\mathrm{diss}}^{\leftarrow }\ge \frac{4{\left|Tr\left[({\tau }_1-{\tau }_0)A\right]\right|}^2}{\beta osc{\left(A\right)}^2}.$$

(A304)

Conversely,

$$W_{\mathrm{diss}}^{\to },W_{\mathrm{diss}}^{\leftarrow }\le \frac{\beta }{8}osc{\left(V\right)}^2,W_{\mathrm{diss}}^{\to }+W_{\mathrm{diss}}^{\leftarrow }\le \frac{\beta }{4}osc{\left(V\right)}^2.$$

(A305)

These work statements use the standard inclusive TPM convention [144], specialize the relative-entropy irreversible-work framework of [145] to the stated sudden quenches, and assume separate canonical preparations at the same temperature. Their sum is a bidirectional aggregate, not by itself the work of a closed hysteresis cycle. The BKM length is unconditional geometry of the auxiliary equilibrium bridge; a finite-time open-system dissipation metric generally also depends on its relaxation generator [146].

Proof. 

The first three identities are the Gibbs variational formula and its Fenchel gap at source $-\beta V$. Substituting each initial Gibbs state in the sudden-quench work definition gives Equation (A301). Equation (140) applied to the Gibbs bridge gives the first hierarchy; Equation (333) gives Equation (A303). Equations (144), (100), and (101) give the remaining bounds after using $osc(log{\tau }_1-log{\tau }_0)=\beta osc\left(V\right)$. □

Appendix G.6. Weighted Affine-Chord Remainder

Theorem A3

(Bregman remainder as a weighted affine-chord Hessian). Let U be an open convex subset of a finite-dimensional affine space, let $F\in C^2\left(U\right)$, and let $x,y\in U$. Put

$$\sigma \left(t\right):=x+t(y-x),\overline{\sigma }\left(t\right):=y+t(x-y).$$

Then

$$\begin{array}{cc}\hfill B_F(x,y)& ={\int }_0^{1}t{D}^2{F}_{\sigma \left(t\right)}[\dot{\sigma },\dot{\sigma }]\mathrm{d}t\hfill \\ & ={\int }_0^1(1-t)D^2{F}_{\overline{\sigma }\left(t\right)}[\dot{\overline{\sigma }},\dot{\overline{\sigma }}]\mathrm{d}t.\hfill \end{array}$$

(A306)

The identities require no convexity. For convex F, the Hessian is positive semidefinite and may be degenerate, even under strict convexity (e.g. $F\left(s\right)=s^4$ has $F^{\prime \prime }\left(0\right)=0$). The squared-speed interpretation requires the additional hypothesis $g:=D^{2}F\succ 0$ on U; for smooth F this is a Riemannian Hessian metric.

Under this positive-definiteness hypothesis, use affine coordinates and assume the gradient image $U^{\ast }:=DF\left(U\right)$ is convex. Then $DF:U\to U^{\ast }$ is a $C^1$ diffeomorphism onto an open dual domain, the Legendre conjugate $F^{\ast }$ is $C^2$ there, and, for $z\in U$,

$$D^2{F}_{D{F}_z}^{\ast }={\left(D^2{F}_z\right)}^{-1},B_F(x,y)=B_{F^{\ast }}(D{F}_y,D{F}_x).$$

Thus the same remainder identity, applied on $U^{\ast }$ with reversed dual endpoints, gives the dual-affine form.

Proof. 

For $f=F\circ \sigma$, integration by parts gives

$${\int }_0^{1}t{f}^{\prime \prime }\left(t\right)\mathrm{d}t=f^{\prime }\left(1\right)-f\left(1\right)+f\left(0\right)=B_F(x,y).$$

Since $\overline{\sigma }\left(t\right)=\sigma (1-t)$, the second weighted identity follows by $t\mapsto 1-t$. Convexity gives semidefinite second derivatives along every line. Under $D^{2}F\succ 0$, integration along chords makes $DF$ injective, and the inverse-function theorem makes it locally invertible. Fenchel equality then gives $D{F}^{\ast }={\left(DF\right)}^{-1}$ on $U^{\ast }$. Differentiating this inverse relation proves the Hessian formula; substitution in the Bregman definition proves the endpoint reversal. □

Appendix G.7. Continuous Hamilton-Jacobi Obstruction

The weighted identity is not by itself a variational principle. For example, the singular proposal $\mathcal{L}=(t/2)g_{ij}{\dot{\xi }}^i{\dot{\xi }}^j$ [147] already fails in flat one dimension: ${\left(t\dot{\xi }\right)}^{\prime }=0$ gives $alogt+b$, and finiteness at $t=0$ forces a constant curve. Its value on the straight segment is $\frac{1}{4}{(q-p)}^2$, whereas the divergence is $\frac{1}{2}{(q-p)}^2$.

Provided ${\tau }_{\beta }\in M$, Equation (780) does not make the fixed-reference action a stationary Hamilton characteristic. Indeed, for $W={\mathcal{A}}_{\tau }^{\left({\kappa }_A\right)}=t_A{V}_{\beta }$, both W and $\mathrm{d}W$ vanish at ${\tau }_{\beta }$. The stationary Hamilton-Jacobi equation

$$\frac{1}{2}{\mathsf{m}}^{-1}(\mathrm{d}W,\mathrm{d}W)+V_{\beta }=E$$

(A307)

therefore gives $E=0$ at ${\tau }_{\beta }$, while positivity forbids the same equality away from equilibrium. Thus the discrete-generator construction is substantive but cannot be promoted to an exact continuous principal-function identity by this natural positive Hamiltonian.

Appendix G.8. Short-Time Defect and Phase-Map Order

Every estimate in this subsection uses the mechanical specialization ${\kappa }_h={\mu }_{\ast }/h$; the fixed endpoint conversion ${\kappa }_A$ does not enter the comparison with the exact discrete Lagrangian.

Assume first that $\psi \in C^4$, $V\in C^3$, and work in the no-caustic neighborhood of Corollary 51. Put $\eta \left(q_1\right)=\eta \left(q_0\right)+hv$. Taylor expansion gives

$$\begin{array}{cc}\hfill \frac{{\mu }_{\ast }}{h}{B}_{\psi }(q_1,q_0)& =\frac{{\mu }_{\ast }h}{2}g(v,v)+\frac{{\mu }_{\ast }{h}^2}{6}\mathcal{C}(v,v,v)+O\left(h^3\right),\hfill \end{array}$$

(A308)

$$\begin{array}{cc}\hfill -\frac{h}{2}(V\left(q_0\right)+V\left(q_1\right))& =-hV\left(q_0\right)-\frac{h^2}{2}\mathrm{d}{V}_{q_0}\left[v\right]+O\left(h^3\right).\hfill \end{array}$$

(A309)

Writing the exact extremal in affine coordinates as $\eta \left(q\left(s\right)\right)=\eta \left(q_0\right)+shv+h^{2}w\left(s\right)+O\left(h^3\right)$, $0\le s\le 1$, with $w\left(0\right)=w\left(1\right)=0$, one has, uniformly on the stated compact sets,

$$\begin{array}{cc}\hfill g_{q\left(s\right)}& =g_{q_0}+sh{\mathcal{C}}_{q_0}(v,·,·)+O\left(h^2\right),\hfill \end{array}$$

(A310)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}\eta \left(q\right(s\left)\right)}{h\mathrm{d}s}& =v+h\dot{w}\left(s\right)+O\left(h^2\right).\hfill \end{array}$$

(A311)

Therefore the order-$h^2$ kinetic correction is

$${\mu }_{\ast }{h}^2{\int }_0^1{g}_{q_0}(v,\dot{w})\mathrm{d}s+\frac{{\mu }_{\ast }{h}^2}{2}{\int }_0^{1}s{\mathcal{C}}_{q_0}(v,v,v)\mathrm{d}s=\frac{{\mu }_{\ast }{h}^2}{4}{\mathcal{C}}_{q_0}(v,v,v),$$

(A312)

because ${\int }_0^1{g}_{q_0}(v,\dot{w})\mathrm{d}s=g_{q_0}(v,w\left(1\right)-w\left(0\right))=0$ and ${\int }_0^{1}s\mathrm{d}s=1/2$. Taylor expansion of the potential along the same curve then gives

$$L_{d,h}^E=h\left[\frac{{\mu }_{\ast }}{2}g(v,v)-V\left(q_0\right)\right]+h^2\left[\frac{{\mu }_{\ast }}{4}\mathcal{C}(v,v,v)-\frac{1}{2}\mathrm{d}{V}_{q_0}\left[v\right]\right]+O\left(h^3\right).$$

(A313)

Subtraction proves Equation (757).

The scaled reverse divergence has expansion

$$\frac{{\mu }_{\ast }}{h}{B}_{\psi }(q_0,q_1)=\frac{{\mu }_{\ast }h}{2}g(v,v)+\frac{{\mu }_{\ast }{h}^2}{3}\mathcal{C}(v,v,v)+O\left(h^3\right).$$

(A314)

For the normalized dual-Jensen family, set ${\vartheta }_i=\mathrm{d}{\psi }_{q_i}$. Taylor expansion of Equation (705), using ${\vartheta }_1=\mathrm{d}{\psi }_{q_0+hv}$, $D^2{\psi }^{\ast }=g^{-1}$, and differentiation of this inverse, gives

$$z_{\theta }=q_0+\theta hv+\frac{\theta (1-\theta )h^2}{2}{\left(g_{q_0}^{♭}\right)}^{-1}{\mathcal{C}}_{q_0}(v,v,·)+O\left(h^3\right)$$

(A315)

and, on substitution,

$${\mathfrak{B}}_{\theta }^{\psi }(q_1,q_0)=\frac{h^2}{2}g(v,v)+\frac{(2-\theta )h^3}{6}\mathcal{C}(v,v,v)+O\left(h^4\right),$$

(A316)

which proves Equation (766). Combining this with Equations (A309) and (A313) proves Equation (767). Endpoint exchange in Equation (708) proves the adjoint relation; at $\theta =1/2$ it proves exact self-adjointness.

Averaging Equations (A308) and (A314) gives the coefficient $\mathcal{C}/4$ in Equation (A313), proving the scalar estimate in Equation (760). Moreover, differentiating Equation (A274) gives the mixed Hessian there. For the signed-step continuation, the adjoint convention in Equation (758) gives ${\left(L_{d,h}^{\mathrm{sym}}\right)}^{\ast }=L_{d,h}^{\mathrm{sym}}$. These scalar estimates are uniform for $q_0$ in a fixed compact subset and bounded v; fixed separated endpoints and caustics admit no such single-valued expansion.

Under the stronger hypotheses of Corollary 52, Taylor’s theorem may be differentiated twice on compact scaled-endpoint sets. With ${\mathsf{\Sigma }}_h$ from Equation (769), this upgrades the two action remainders to

$$\begin{array}{cc}\hfill R_h^{\to }& :=\left(L_{d,h}^{\to }-L_{d,h}^E\right)\circ {\mathsf{\Sigma }}_h=-\frac{{\mu }_{\ast }{h}^2}{12}{\mathcal{C}}_q(v,v,v)+O_{C^2\left(K\right)}\left(h^3\right),\hfill \end{array}$$

(A317)

$$\begin{array}{cc}\hfill R_h^{\mathrm{sym}}& :=\left(L_{d,h}^{\mathrm{sym}}-L_{d,h}^E\right)\circ {\mathsf{\Sigma }}_h=O_{C^2\left(K\right)}\left(h^3\right).\hfill \end{array}$$

(A318)

On the faithful-state specialization, analyticity of ${\mathfrak{B}}_{1/2}^{\mathrm{RM}}$ on compact subsets and the same Taylor argument give

$$R_h^{\mathrm{RM},\mathrm{mid}}:=\left(L_{d,h}^{\mathrm{RM},\mathrm{mid}}-L_{d,h}^E\right)\circ {\mathsf{\Sigma }}_h=O_{C^2\left(K\right)}\left(h^3\right).$$

(A319)

The derivative qualification is essential: a scalar action estimate alone does not control the endpoint derivatives defining the phase map.

Proposition A8

(Derivative-controlled variational-error transfer). Let $L:TQ\to \mathbb{R}$ be a regular autonomous Lagrangian and $L_{d,h}^E$ its exact discrete Lagrangian on a fixed no-caustic branch. In a ∇-affine chart $U\subset Q$, set

$${\ell }_h:=L_{d,h}\circ {\mathsf{\Sigma }}_h,{\ell }_h^E:=L_{d,h}^E\circ {\mathsf{\Sigma }}_h,R_h:={\ell }_h-{\ell }_h^E.$$

(A320)

Let $0<h\le h_0$ and let $K⋐TU$ be a compact scaled-endpoint set in the affine chart on which

$${\ell }_h^E=hL+O_{C^3\left(K\right)}\left(h^2\right),{\parallel R_h\parallel }_{C^2\left(K\right)}\le C_K{h}^{r+1},r\ge 1.$$

(A321)

For every compact phase set $\mathcal{K}$ whose exact incoming Legendre inverse lies uniformly in the interior of K for all sufficiently small h, the numerical branch exists and

$$\parallel {\mathsf{\Phi }}_h-{\phi }_h{\parallel }_{C^1\left(\mathcal{K}\right)}\le C_{\mathcal{K}}{h}^{r+1}.$$

(A322)

Fix $T>0$ and a compact regular phase tube $\mathcal{T}$ in the common local domain. Assume the exact incoming Legendre inverses on a fixed neighborhood of $\mathcal{T}$ lie uniformly in $intK$ for all sufficiently small h. Then, uniformly for $z\in \mathcal{K}$ whose exact and numerical iterates remain in $\mathcal{T}$ for $0\le kh\le T$,

$$\underset{0\le kh\le T}{max}\parallel {\mathsf{\Phi }}_h^k\left(z\right)-{\phi }_{kh}\left(z\right)\parallel \le C_{\mathcal{K},\mathcal{T},T}{h}^r.$$

(A323)

Proof. 

In the affine trivialization, the chain rule gives

$$P_{0,h}=h^{-1}{D}_v{\ell }_h-D_q{\ell }_h,P_{1,h}=h^{-1}{D}_v{\ell }_h,P_{1,h}-P_{0,h}=D_q{\ell }_h.$$

(A324)

At the same scaled endpoint pair,

$$P_{0,h}-P_{0,h}^E=h^{-1}{D}_v{R}_h-D_q{R}_h=O_{C^1\left(K\right)}\left(h^r\right).$$

(A325)

Equation (A321) implies $D_v{P}_{0,h}^E=L_{vv}+O_K\left(h\right)$, which is uniformly invertible after shrinking the branch. The implicit-function theorem therefore gives a numerical solution $v_h$ to the same incoming momentum equation as the exact solution $v_h^E$, with

$$v_h-v_h^E=O_{C^1\left(\mathcal{K}\right)}\left(h^r\right).$$

(A326)

The outgoing base error is $h(v_h-v_h^E)=O\left(h^{r+1}\right)$. Because the incoming momenta agree, the last identity in Equation (A324) gives

$$\begin{array}{cc}\hfill P_{1,h}-P_{1,h}^E& =D_q{\ell }_h^E(q,v_h)-D_q{\ell }_h^E(q,v_h^E)+D_q{R}_h(q,v_h)\hfill \\ & =O_{C^1\left(K\right)}\left(h\right)O_{C^1\left(\mathcal{K}\right)}\left(h^r\right)+O_{C^1\left(K\right)}\left(h^{r+1}\right)=O_{C^1\left(\mathcal{K}\right)}\left(h^{r+1}\right),\hfill \end{array}$$

(A327)

where $D_{qv}{\ell }_h^E=O_{C^1\left(K\right)}\left(h\right)$. This proves Equation (A322). At the $C^0$ order-counting level it parallels the standard variational-order mechanism [36], Theorem 2.3.1; the scaled $C^2\to C^1$ statement proved here is not a direct consequence of that theorem. For the fixed-time bound, the tube hypothesis makes the local estimate uniform on a compact neighborhood of $\mathcal{T}$, where $\parallel D{\mathsf{\Phi }}_h\parallel \le 1+C_{T}h$ since the exact flow has that bound. While $e_k$ is sufficiently small, the joining coordinate segments stay in this neighborhood, giving $e_{k+1}\le (1+C_{T}h)e_k+C_T{h}^{r+1}$. With $e_0=0$, discrete Grönwall gives $e_k=O\left(h^r\right)$, closing the small-h bootstrap uniformly over the admissible initial points and proving Equation (A323). □

For completeness, the scaled $C^2$ hypothesis also gives, for $a\in \{0,1\}$,

$$D_a(L_{d,h}-L_{d,h}^E)=O\left(h^r\right),D_0{D}_1(L_{d,h}-L_{d,h}^E)=O\left(h^{r-1}\right),$$

(A328)

whereas, for a general regular Lagrangian,

$$D_0{D}_1{L}_{d,h}^E=-\frac{1}{h}{L}_{vv}(q,v)+O\left(1\right).$$

(A329)

For the mechanical Lagrangian in Equation (754), this specializes to $-{\mu }_{\ast }{g}_q/h+O\left(1\right)$. The mixed-Hessian perturbation is therefore relatively $O\left(h^r\right)$; the incoming solve and the identity $P_1-P_0=D_q{\ell }_h$ recover the additional power in the phase-map defect.

The compact-uniform hypotheses of Corollary 52 allow K to contain uniformly in its interior the exact incoming inverses near the whole phase tube for small h. Applying the proposition with $r=1$ to Equation (A317) and with $r=2$ to Equations (A318) and (A319) proves Equations (770) and (773). The sharp directed term follows directly from the two incoming momentum relations

$$\begin{array}{cc}\hfill P_{0,h}^{\to }& ={\mu }_{\ast }{g}_q(v,·)+\frac{h}{2}\mathrm{d}{V}_q,\hfill \end{array}$$

(A330)

$$\begin{array}{cc}\hfill P_{0,h}^E& ={\mu }_{\ast }{g}_q(v,·)+h\left[\frac{{\mu }_{\ast }}{4}{\mathcal{C}}_q(v,v,·)+\frac{1}{2}\mathrm{d}{V}_q\right]+O\left(h^2\right).\hfill \end{array}$$

(A331)

For a common incoming $(q,p)$, put $u={\mu }_{\ast }^{-1}{g}_q^{♯}p$. Solving gives

$$v_h^{\to }=u-\frac{h}{2{\mu }_{\ast }}{grad}_{g}V,v_h^E=u-\frac{h}{4}{\mathcal{C}}_q^{♯}(u,u)-\frac{h}{2{\mu }_{\ast }}{grad}_{g}V+O\left(h^2\right).$$

(A332)

Multiplication by h proves Equation (772); in particular, the potential cancels from its leading coefficient.

Appendix G.9. Discrete Noether Representatives and Quantum Symmetry

For unitary conjugation, ${\xi }_M\left(\rho \right)=[\xi ,\rho ]$, and cyclicity gives

$$\langle J_d,\xi \rangle =Tr\left({\mathsf{A}}_1[\xi ,{\rho }_1]\right)=Tr\left([{\rho }_1,{\mathsf{A}}_1]\xi \right).$$

(A333)

Using $\langle H,\xi \rangle =\mathrm{i}Tr\left(H\xi \right)$ yields Equation (777).

For $V=V_{\beta }$, specialize Equation (711) at generic ${\kappa }_{\mathsf{\Delta }t}$, with the mechanical and fixed-scale interpretations distinguished in Remark 19:

$$\begin{array}{cc}\hfill L_{d,\beta }^{\to }({\rho }_0,{\rho }_1)& ={\kappa }_{\mathsf{\Delta }t}{D}_{\mathrm{U}}({\rho }_1\parallel {\rho }_0)-\frac{\mathsf{\Delta }t}{2\beta }\left[D_{\mathrm{U}}({\rho }_0\parallel {\tau }_{\beta })+D_{\mathrm{U}}({\rho }_1\parallel {\tau }_{\beta })\right],\hfill \\ \hfill D_0{D}_1{L}_{d,\beta }^{\to }& =-{\kappa }_{\mathsf{\Delta }t}{g}_{{\rho }_0}^{\mathrm{BKM}},\hfill \end{array}$$

(A334)

$$\begin{array}{cc}\hfill {\mathsf{A}}_1& \equiv {\kappa }_{\mathsf{\Delta }t}(log{\rho }_1-log{\rho }_0)-\frac{\mathsf{\Delta }t}{2\beta }(log{\rho }_1-log{\tau }_{\beta })(mod\mathbb{R}I).\hfill \end{array}$$

(A335)

Thus the floor changes the outgoing momentum but not Type-I regularity. For a generator that need not stabilize ${\tau }_{\beta }$, define the outgoing canonical charge

$$j_{k,\xi }^{+}:=D_1{L}_{d,\beta }^{\to }({\rho }_k,{\rho }_{k+1})\left([\xi ,{\rho }_{k+1}]\right).$$

(A336)

The discrete Euler-Lagrange equation identifies its step-to-step change with the diagonal variation of the current step and gives the exact balance

$$\begin{array}{cc}\hfill j_{k,\xi }^{+}-j_{k-1,\xi }^{+}& =\frac{\mathsf{\Delta }t}{2\beta }\left[Tr\left([\xi ,{\rho }_k]log{\tau }_{\beta }\right)+Tr\left([\xi ,{\rho }_{k+1}]log{\tau }_{\beta }\right)\right]\hfill \\ & =\frac{\mathsf{\Delta }t}{2}\left[Tr\left({\rho }_k[\xi ,H_{\mathrm{sys}}]\right)+Tr\left({\rho }_{k+1}[\xi ,H_{\mathrm{sys}}]\right)\right].\hfill \end{array}$$

(A337)

This is the trapezoidal discrete counterpart of Equation (781); it vanishes for stabilizer generators. Off the stabilizer, $j_{k,\xi }^{+}$ is a sourced endpoint charge rather than a conserved momentum map.

For $n=2$, write the faithful state in Bloch form and choose a traceless outgoing covector representative:

$${\rho }_1=\frac{1}{2}\left(I+{\mathit{r}}_1·\sigma \right),\left|{\mathit{r}}_1\right|<1,{\mathsf{A}}_1={\mathit{a}}_1·\sigma ,{\mathcal{J}}_1:=-\mathrm{i}[{\rho }_1,{\mathsf{A}}_1]=\left({\mathit{r}}_1\times {\mathit{a}}_1\right)·\sigma .$$

(A338)

For $\xi =-\mathrm{i}\mathbf{\theta }·\sigma /2$, the pairing gives $\langle {\mathcal{J}}_1,\xi \rangle =\mathbf{\theta }·({\mathit{r}}_1\times {\mathit{a}}_1)$. In the invariant case ${\mathcal{J}}_1$ represents $J_d$; off the stabilizer it represents the sourced outgoing charge $j^{+}$. It is internal $\mathrm{SU}\left(2\right)$ data and becomes spatial angular momentum only after an explicit identification with physical rotations; it is not the Lorentz boost charge.

Appendix H.1. Mass-Shell Generator Bounds and the Collinear Model

Theorem A4

(Conditional Poincaré boost-translation algebra and boost-generator-rapidity uncertainty). Let the ${\widehat{P}}^{\mu }$ be the strongly commuting self-adjoint spacetime-translation generators, and let ${\widehat{\mathsf{K}}}_i$ be the self-adjoint boost generators of a strongly continuous, positive-energy unitary representation of the proper orthochronous Poincaré group (or its universal cover). Fix an independent action normalization $\hslash >0$, put ${\widehat{P}}^0=\widehat{H}/c$, and assume that the joint spectral measure of ${\widehat{P}}^{\mu }$ is supported on the positive mass shell

$$p^0>0,{\left(p^0\right)}^2-{\left|\mathit{p}\right|}^2=m^2{c}^2,m>0.$$

(A339)

Equivalently, the mass-shell identity holds by joint spectral calculus and ${\widehat{P}}^0\ge mcI$. For a unit spatial vector $\mathit{n}$, set ${\widehat{P}}_{\mathit{n}}:=n^i{\widehat{P}}_i$ and ${\widehat{\mathsf{K}}}_{\mathit{n}}:=n^i{\widehat{\mathsf{K}}}_i$. Choose the boost orientation so that, on the Poincaré smooth-vector core ${\mathcal{D}}_{\mathrm{P}}$,

$$\begin{array}{cccc}\hfill [{\widehat{\mathsf{K}}}_{\mathit{n}},{\widehat{P}}^0]& =-\mathrm{i}\hslash {\widehat{P}}_{\mathit{n}},\hfill & \hfill [{\widehat{\mathsf{K}}}_{\mathit{n}},{\widehat{P}}_{\mathit{n}}]& =-\mathrm{i}\hslash {\widehat{P}}^0.\hfill \end{array}$$

(A340)

The single real Borel function

$$f_{\mathit{n}}(p^0,p_{\mathit{n}}):=\frac{1}{2}log\left(\frac{p^0+p_{\mathit{n}}}{p^0-p_{\mathit{n}}}\right)=artanh\left(\frac{p_{\mathit{n}}}{p^0}\right)$$

(A341)

defines a densely defined self-adjoint boost-adapted longitudinal rapidity ${\widehat{\eta }}_{\mathit{n}}:=f_{\mathit{n}}({\widehat{P}}^0,{\widehat{P}}_{\mathit{n}})$ by joint functional calculus. It agrees with the scalar principal-log rapidity in Equation (784) on the collinear orbit $\mathbf{P}\Vert \mathit{n}$; with transverse momentum it remains the additive coordinate of the chosen boost flow but is not a Cartesian component of the radial Lie log.

Set $U_{\mathit{n}}\left(s\right):=exp(-\mathrm{i}s{\widehat{\mathsf{K}}}_{\mathit{n}}/\hslash )$ and $V_{\mathit{n}}\left(t\right):=exp\left(\mathrm{i}t{\widehat{\eta }}_{\mathit{n}}\right)$. Poincaré covariance gives

$$U_{\mathit{n}}{\left(s\right)}^{†}{\widehat{\eta }}_{\mathit{n}}{U}_{\mathit{n}}\left(s\right)={\widehat{\eta }}_{\mathit{n}}+sI.$$

(A342)

There is a dense Weyl-smooth domain ${\mathcal{C}}_{\mathit{n}}$, constructed in Appendix H.1, such that ${\mathcal{C}}_{\mathit{n}}\subset Dom\left({\widehat{\eta }}_{\mathit{n}}{\widehat{\mathsf{K}}}_{\mathit{n}}\right)\cap Dom\left({\widehat{\mathsf{K}}}_{\mathit{n}}{\widehat{\eta }}_{\mathit{n}}\right)$ and, on this domain,

$$[{\widehat{\mathsf{K}}}_{\mathit{n}},{\widehat{\eta }}_{\mathit{n}}]=-\mathrm{i}\hslash I,[{\widehat{\eta }}_{\mathit{n}},{\widehat{\mathsf{K}}}_{\mathit{n}}]=\mathrm{i}\hslash I.$$

(A343)

Consequently, every normalized $\psi \in {\mathcal{C}}_{\mathit{n}}$ with finite second moments satisfies Robertson’s bound

$$\begin{array}{|c|}\hline {\mathsf{\Delta }}_{\psi }{\widehat{\eta }}_{\mathit{n}}{\mathsf{\Delta }}_{\psi }{\widehat{\mathsf{K}}}_{\mathit{n}}\ge \frac{1}{2}\left|{〈[{\widehat{\eta }}_{\mathit{n}},{\widehat{\mathsf{K}}}_{\mathit{n}}]〉}_{\psi }\right|=\frac{\hslash }{2}.\\ \hline\end{array}$$

(A344)

Because rapidity is dimensionless, ${\widehat{\mathsf{K}}}_{\mathit{n}}$ has units of action.

Proof of Theorem A4. 

The mass condition gives $p^0>\left|p_{\mathit{n}}\right|$ on the joint spectrum, while ${\left({\widehat{P}}^0\right)}^{-1}$ is bounded because ${\widehat{P}}^0\ge mcI$. Hence $f_{\mathit{n}}$ is finite on the spectral support and defines the stated, generally unbounded self-adjoint operator on its natural dense domain. Poincaré covariance gives

$$\begin{array}{cc}\hfill U_{\mathit{n}}{\left(s\right)}^{†}{\widehat{P}}^0{U}_{\mathit{n}}\left(s\right)& ={\widehat{P}}^{0}coshs+{\widehat{P}}_{\mathit{n}}sinhs,\hfill \end{array}$$

(A345)

$$\begin{array}{cc}\hfill U_{\mathit{n}}{\left(s\right)}^{†}{\widehat{P}}_{\mathit{n}}{U}_{\mathit{n}}\left(s\right)& ={\widehat{P}}_{\mathit{n}}coshs+{\widehat{P}}^{0}sinhs.\hfill \end{array}$$

(A346)

The light-cone combinations therefore scale by ${\mathrm{e}}^{\pm s}$, and joint functional calculus yields Equation (A342).

For the groups $U_{\mathit{n}},V_{\mathit{n}}$ already defined, Equation (A342) gives the Weyl relation $U_{\mathit{n}}{\left(s\right)}^{†}{V}_{\mathit{n}}\left(t\right)U_{\mathit{n}}\left(s\right)={\mathrm{e}}^{\mathrm{i}st}{V}_{\mathit{n}}\left(t\right)$. Let ${\mathcal{C}}_{\mathit{n}}$ be the linear span of

$${\int }_{{\mathbb{R}}^2}\varphi (s,t)U_{\mathit{n}}\left(s\right)V_{\mathit{n}}\left(t\right)\psi \mathrm{d}s\mathrm{d}t,\varphi \in C_c^{\infty }\left({\mathbb{R}}^2\right),\psi \in \mathcal{H}.$$

(A347)

The standard Gårding-domain argument makes ${\mathcal{C}}_{\mathit{n}}$ dense. Moving either one-parameter group through the integral and integrating by parts transfers generator derivatives to $\varphi$; repeating this twice gives the two product-domain inclusions stated in Theorem A4. Differentiating the Weyl relation on this domain proves Equation (A343) [111]; Robertson’s inequality then gives Equation (A344). □

The preceding Weyl argument has the following alternative longitudinal massless-sector consequence; it is not a specialization of the positive-mass hypothesis in Theorem A4.

Corollary A7

(Chiral null boost-log-momentum pair). Let ${\mathcal{H}}_{+}\ne \left\{0\right\}$ carry a strongly continuous positive-energy unitary representation of the longitudinal $1+1$-dimensional proper orthochronous Poincaré group ${\mathrm{ISO}}_0(1,1)$ (or its universal cover), with strongly commuting translation generators ${\widehat{P}}^0,{\widehat{P}}_{\mathit{n}}$, self-adjoint boost generator ${\widehat{\mathsf{K}}}_{\mathit{n}}$, and joint translation spectrum in the closed forward cone. Fix an independent $\hslash >0$ and set

$${\widehat{P}}_{\pm }:={\widehat{P}}^0\pm {\widehat{P}}_{\mathit{n}},U_{\mathit{n}}\left(s\right):=exp(-\mathrm{i}s{\widehat{\mathsf{K}}}_{\mathit{n}}/\hslash ).$$

(A348)

Assume on this right-moving sector that

$${\widehat{P}}_{-}=0,{\widehat{P}}_{+}\ge 0,E_{+}\left(\left\{0\right\}\right)=0,$$

(A349)

where $E_{+}$ is the spectral measure of ${\widehat{P}}_{+}$, and assume the operator covariance

$$U_{\mathit{n}}{\left(s\right)}^{†}{\widehat{P}}_{+}{U}_{\mathit{n}}\left(s\right)={\mathrm{e}}^s{\widehat{P}}_{+},s\in \mathbb{R}.$$

(A350)

For any reference momentum $p_{★}>0$, Borel functional calculus defines the densely defined self-adjoint operator

$${\widehat{\eta }}_{+}:=log({\widehat{P}}_{+}/p_{★}).$$

(A351)

It obeys, as an equality of self-adjoint operators and their domains,

$$U_{\mathit{n}}{\left(s\right)}^{†}{\widehat{\eta }}_{+}{U}_{\mathit{n}}\left(s\right)={\widehat{\eta }}_{+}+sI.$$

(A352)

Let $V_{+}\left(t\right):=exp\left(\mathrm{i}t{\widehat{\eta }}_{+}\right)$, and let ${\mathcal{C}}_{+}$ be the dense Weyl-smooth domain obtained from Equation (A347) by replacing $(U_{\mathit{n}},V_{\mathit{n}},\mathcal{H})$ with $(U_{\mathit{n}},V_{+},{\mathcal{H}}_{+})$. Then

$${\mathcal{C}}_{+}\subset Dom\left({\widehat{\eta }}_{+}{\widehat{\mathsf{K}}}_{\mathit{n}}\right)\cap Dom\left({\widehat{\mathsf{K}}}_{\mathit{n}}{\widehat{\eta }}_{+}\right),[{\widehat{\eta }}_{+},{\widehat{\mathsf{K}}}_{\mathit{n}}]=\mathrm{i}\hslash I\mathrm{on}{\mathcal{C}}_{+}.$$

(A353)

Consequently, every normalized $\psi \in {\mathcal{C}}_{+}$ satisfies

$${\mathsf{\Delta }}_{\psi }{\widehat{\eta }}_{+}{\mathsf{\Delta }}_{\psi }{\widehat{\mathsf{K}}}_{\mathit{n}}\ge \frac{\hslash }{2}.$$

(A354)

Replacing $p_{★}$ by $p_{★}^{\prime }>0$ adds $log(p_{★}/p_{★}^{\prime })I$ to ${\widehat{\eta }}_{+}$ and therefore changes neither its variance, the commutator, nor the uncertainty bound.

Proof. 

The zero-kernel condition makes the spectral cutoffs $E_{+}\left([p_{★}/j,j{p}_{★}]\right)$ converge strongly to I on ${\mathcal{H}}_{+}$. Hence the spectral logarithm is self-adjoint on its natural dense domain. Unitary covariance of Borel functional calculus gives

$$U_{\mathit{n}}{\left(s\right)}^{†}log({\widehat{P}}_{+}/p_{★})U_{\mathit{n}}\left(s\right)=log({\mathrm{e}}^s{\widehat{P}}_{+}/p_{★})={\widehat{\eta }}_{+}+sI,$$

(A355)

including equality of domains. Exponentiation yields $U_{\mathit{n}}{\left(s\right)}^{†}{V}_{+}\left(t\right)U_{\mathit{n}}\left(s\right)={\mathrm{e}}^{\mathrm{i}st}{V}_{+}\left(t\right)$. The Gårding-domain argument used in the proof of Theorem A4 gives the displayed product-domain inclusions; differentiation gives Equation (A353), and Robertson’s inequality gives Equation (A354).

Finally, Equation (A350) makes the nonempty spectrum of ${\widehat{P}}_{+}$ invariant under every positive rescaling. Together with Equation (A349), this forces $\sigma \left({\widehat{P}}_{+}\right)=[0,\infty )$ with zero in the continuous spectrum and $\sigma \left({\widehat{\eta }}_{+}\right)=\mathbb{R}$. Thus no positive spectral gap is being assumed. The reference-scale statement follows from the logarithm law. □

The hypotheses are nonvacuous: the standard right-moving realization on $L^2(\mathbb{R},\mathrm{d}{\eta }_{+})$ has ${\widehat{P}}_{+}=p_{★}{\mathrm{e}}^{{\eta }_{+}}$, ${\widehat{P}}_{-}=0$, and ${\widehat{\mathsf{K}}}_{\mathit{n}}=-\mathrm{i}\hslash {\partial }_{{\eta }_{+}}$ on the common core $C_c^{\infty }\left(\mathbb{R}\right)$.

Returning to the positive-mass setting of Theorem A4, the spinless collinear $1+1$-dimensional realization on $L^2(\mathbb{R},\mathrm{d}\eta )$ has

$${\widehat{P}}^0=mccosh\eta ,{\widehat{P}}_{\mathit{n}}=mcsinh\eta ,{\widehat{\mathsf{K}}}_{\mathit{n}}=-\mathrm{i}\hslash {\partial }_{\eta }$$

(A356)

on the common core $C_c^{\infty }\left(\mathbb{R}\right)$. The rapidity-boost pair also has the usual Schwartz core; Equation (A343) is the ordinary coordinate-generator relation, and real Gaussian rapidity packets attain the Robertson minimum.

Appendix H.2. Exactness, Periods, and Canonical Quotient-Bundle Triviality

Proof of Proposition 34. 

The factor ${\kappa }_A$ is arbitrary and, for every closed curve $\gamma$,

$${\oint }_{\gamma }{\iota }_{{\kappa }_A,p}^{\ast }{\lambda }_A={\kappa }_A{\oint }_{\gamma }\mathrm{d}{\psi }^{\left[p\right]}=0.$$

(A357)

Moreover, ${\omega }_A=\mathrm{d}{\lambda }_A$ is exact, so $Per\left({\omega }_A\right)=\left\{0\right\}$. The trivial line bundle therefore satisfies prequantization for every supplied nonzero ℏ and cannot select its value. Because the pulled-back Liouville form on the exact reference-centered graph is exact, the restricted prequantum connection has trivial holonomy: the graph is Bohr-Sommerfeld for every supplied ℏ and hence does not quantize it. On the present contractible phase space there is no nontrivial flat twist; on another topology such a twist would be additional data and still would not select an action scale. □

Proposition A9

(Topological triviality of the canonical quotient bundle). For $\pi :\mathrm{SL}(n,\mathbb{C})\to {\mathcal{P}}_n^1$, $\pi \left(g\right)=g{g}^{†}$, the square-root map $s\left(P\right)=P^{1/2}$ is a smooth global section and

$$\mathrm{SL}(n,\mathbb{C})\cong {\mathcal{P}}_n^1\times \mathrm{SU}\left(n\right),(P,U)⟼P^{1/2}U.$$

(A358)

Consequently, every associated vector bundle is topologically trivial and all Chern classes and Chern-Weil characteristic numbers on the base vanish.

Proof. 

The positive square root is smooth and satisfies $\pi \left(P^{1/2}\right)=P$; the displayed trivialization is the unique right polar decomposition. The characteristic-class statement follows from bundle triviality. □

Appendix H.3. Standard Quantization Conventions and Qualifications

For an action-valued symplectic manifold $(\mathcal{P},\omega )$, Kostant-Souriau prequantization requires

$$F_{\nabla }=-\mathrm{i}\omega /\hslash ,\left[\frac{\omega }{2\pi \hslash }\right]\in H^2(\mathcal{P},\mathbb{Z})$$

(A359)

[114,115,116,148]. It constrains the cohomology class rather than every loop action. The exact graph is Bohr-Sommerfeld for every supplied ℏ and therefore selects no scale (Proposition 34). Nonzero level data therefore require a different cycle or topology, a multivalued Hamilton-Jacobi phase, or a separately supplied compact phase sector.

For the rank-one ray orbit, the standard moment map, Hopf connection, and Kostant-Souriau orbit-quantization framework [114,117,118] give, for $k\in {\mathbb{Z}}_{>0}$,

$${\omega }_k=k\hslash {\mathsf{\Omega }}_{\mathrm{FS}},L_k\cong \mathcal{O}\left(k\right),{\mathcal{H}}_k\cong H^0\left({\mathbb{CP}}^{n-1},\mathcal{O}\left(k\right)\right)\cong {Sym}^k\left({\left({\mathbb{C}}^n\right)}^{\ast }\right),$$

(A360)

Here the last isomorphism is the elementary ${\mathbb{CP}}^{n-1}$ specialization of Borel-Weil: holomorphic sections of $\mathcal{O}\left(k\right)$ are homogeneous degree-k polynomials on ${\mathbb{C}}^n$. Orbit, level, polarization, and coupling to the mixed-state bundle are extra data; canonical mixed-state holonomy supplies none of them.

Finally, non-Abelian loop transport requires path (and, in surface formulations, surface) ordering and is not generally $exp(-{\int }_{\mathsf{\Sigma }}\mathcal{F})$ [149]. By Chern-Weil theory [82], the preceding trivialization forces the canonical quotient bundle’s characteristic classes to vanish, even though the separately proved connection has nonzero curvature and full holonomy; a scalar phase instead requires an integral $\mathrm{U}\left(1\right)$ line. The hypercharge $\mathrm{U}\left(1\right)$ in Theorem 20 is different: it is the relative block phase inside $K_{\mathrm{SM}}$ and is generated by the off-diagonal curvature (A147), not by a scalar projective line.