Relative-Entropy Geometry of Translations and Boosts: Minimal Realisation and Affine Rigidity

1. Introduction

Relative entropy induces a metric and dual connections by differentiation at coincidence. These local tensors do not determine comparisons between separated states. We ask what the complete two-point relative entropy of a prescribed quantum family fixes about an ambient positive Hessian realisation.

For a faithful mixed-state family in the positive-energy scalar representation of the $1+1$-dimensional Poincaré algebra, we compute the translation-boost contrast exactly and justify its logarithmic expectations. The contrast admits a three-dimensional positive Hessian realisation, and curvature proves minimality. Finite reference comparisons recover every minimal affine placement and the potential’s first jet. Global convex positivity is nevertheless impossible in every finite ambient dimension, although every bounded rapidity strip admits an exact convex realisation.

The distinction from existing embedding theory is finite-comparison preservation. Marugame treats local metric and connection realisation [1]; Nielsen studies nonlinear restrictions of a supplied Bregman divergence [2]. Convex extension of prescribed first-order jets is established [3]. Here the contrast determines the minimal placement and jet before constraining convex extension. This scalar-divergence realisation is not compression into a finite-dimensional quantum state model.

The same family separates commuting statistics from quantum distinguishability. A standard Gaussian dephasing channel preserves the complete energy-momentum law and the specified loop expectations, but erases translation sensitivity. Its contrast approaches the classical rapidity divergence without a trace-norm limiting state. This is consistent with covariance uncertainty: purity decreases and boost variance grows [4,5]. A calibrated preparation Hamiltonian gives an exact spectral-mismatch/ergotropy decomposition, sharp work-area bounds, and finite-resource entropy estimates. We use established Gibbs, passivity and Gaussian-entropy principles [6,7,8] to obtain these explicit consequences. The entropy-length bound is intrinsic to the chosen preparation family, not to unrestricted quantum states sharing its observable law.

Established tangent-bundle Born geometry [9] and the companion’s statistical shear [10] provide the geometric setting. We compute a genuine pullback of the commutator form, its invariant area remainder, and a global nonisomorphism of the dual tuples. Numerical invariants also identify the dimensionless preparation parameters. A discrete-mechanical application is stated in Section 8; auxiliary Born and mechanical derivations are collected in Appendices Appendix E and Appendix J. The general Hamilton-Jacobi and variational methods are established [11,12,13].

Spaces and normalisations.

$\mathcal{H}$ is the quantum Hilbert space; $\mathcal{M}={\mathbb{R}}^2$ is a fixed-preparation control surface with metric g; $\mathcal{N}$ is its three-dimensional Hessian realisation; and $\mathcal{P}$ is the four-parameter preparation family with metric $\mathfrak{g}$. The physical control form is $\omega$, while ${\Omega }_B$ denotes the exact form on a tangent carrier. Logarithms are natural, and entropy in physical units is $\mathcal{S}=-k_{\mathrm{B}}Tr\rho log\rho$.

2. Relativistic Controls and Faithful Preparations

2.1. Generators and the Finite Ordering Loop

A scalar phase has no density-operator tangent: $e^{\mathrm{i}\alpha }{\rho }_0{e}^{-\mathrm{i}\alpha }={\rho }_0$. Instead, calibrate a spatial displacement d by an action-valued label

$$A=mcd,\alpha =A/\hslash ,U_T\left(\alpha \right)=e^{-\mathrm{i}\alpha P/\left(mc\right)}.$$

(1)

Its phase on a momentum-p component is $-\alpha p/\left(mc\right)$. Neither A nor the endpoint action of Section 8 is identified with the semiclassical history action $-m{c}^2\int d\tau$.

On $\mathcal{H}=L^2(\mathbb{R},\mathrm{d}r)$, let r be particle rapidity and $\pi =-\mathrm{i}{\partial }_r$. The positive-energy massive scalar representation is [14]

$$P=mcsinhr,H=m{c}^{2}coshr,K=\hslash \pi ,m>0.$$

(2)

The self-adjoint operators satisfy $H^2-c^2{P}^2=m^2{c}^{4}id$ and, on the common invariant core $C_c^{\infty }\left(\mathbb{R}\right)$,

$$[K,P]=-\mathrm{i}\hslash H/c,[K,H]=-\mathrm{i}\hslash cP,[r,K]=\mathrm{i}\hslash id.$$

(3)

For $U_B\left(\eta \right)=e^{-\mathrm{i}\eta K/\hslash }$,

$$U_B{\left(\eta \right)}^{†}r{U}_B\left(\eta \right)=r+\eta id.$$

(4)

Thus $\Delta r\Delta K\ge \hslash /2$ on the product domain [15,16]; the control $\eta$ is not the observable r. An exact finite-dimensional representation with $H\ge m{c}^{2}id$ is impossible, since tracing $[K,P]$ would give $TrH=0$. We work directly in the infinite-dimensional representation.

Strong commutativity of $H,P$ gives the exact ordering loop

$$U_{\mathrm{loop}}\left(\eta \right):=U_B{U}_T\left(\alpha \right)U_B^{-1}{U}_T{\left(\alpha \right)}^{-1}=exp\left[-{\displaystyle \frac{\mathrm{i}d}{\hslash }}\left(P(cosh\eta -1)-{\displaystyle \frac{H}{c}}sinh\eta \right)\right].$$

(5)

For fixed d and $\psi \in C_c^{\infty }\left(\mathbb{R}\right)$, Taylor expansion of the multiplier uniformly on $supp\psi$ yields constants $C_{d,\psi }<\infty$ and ${\eta }_0>0$ such that

$${∥U_{\mathrm{loop}}\left(\eta \right)\psi -\psi -{\displaystyle \frac{\mathrm{i}d\eta }{\hslash c}}H\psi ∥}_{\mathcal{H}}\le C_{d,\psi }{\left|\eta \right|}^2\left(\right|\eta |\le {\eta }_0).$$

(6)

This is a vector-norm estimate, not an operator-norm expansion. Its linear term corresponds to $\delta t=-d\eta /c$ for $U_t=e^{-\mathrm{i}Ht/\hslash }$. The finite loop is generally operator-valued; Appendix D computes its phase and visibility in a specified controlled-reference protocol.

2.2. Preparation and Physical Encoding

For dimensionless $\beta ,s>0$, choose

$$Q_s=\frac{1}{2}(s^2{\pi }^2+r^2/s^2),{\rho }_0=Z_{\beta }^{-1}{e}^{-\beta Q_s},Z_{\beta }={[2sinh(\beta /2)]}^{-1}.$$

(7)

The oscillator spectrum makes ${\rho }_0$ trace class and faithful ($ker{\rho }_0=\left\{0\right\}$), with finite entropy. Appendix A specifies its closed-form realisation. A supplied Hamiltonian $H_{\mathrm{prep}}=E_{*}{Q}_s$ calibrates a temperature $E_{*}/\left(k_{\mathrm{B}}\beta \right)$; it is not the free relativistic H. Section 6 and Appendix H separate these notions. The rapidity marginal is Gaussian, with

$$v={\langle r^2\rangle }_0={\displaystyle \frac{s^2}{2}}coth(\beta /2),{\langle {\pi }^2\rangle }_0={\displaystyle \frac{1}{2s^2}}coth(\beta /2).$$

(8)

The controlled family is

$$\rho \left(q\right)=U\left(q\right){\rho }_{0}U{\left(q\right)}^{†},U(\alpha ,\eta )=e^{-\mathrm{i}\alpha sinhr}{e}^{-\mathrm{i}\eta \pi },q=(\alpha ,\eta ).$$

(9)

The preparation and encoding order are fixed inputs. Define

$$F\left(\eta \right)={\displaystyle \frac{1+e^{2v}cosh\left(2\eta \right)}{2}},M\left(\eta \right)=e^{v/2}cosh\eta .$$

(10)

They equal ${\langle {cosh}^{2}r\rangle }_{\rho }$ and ${\langle coshr\rangle }_{\rho }$, respectively, and

$${\langle H\rangle }_{\rho }=m{c}^{2}M\left(\eta \right),{\langle P\rangle }_{\rho }=mc{e}^{v/2}sinh\eta .$$

(11)

All moments used below are finite at finite controls. Appendix A proves the trace-norm smoothness and weighted Hilbert-Schmidt estimates needed for relative entropy.

3. Exact Relative Entropy and Intrinsic Geometry

Umegaki relative entropy is generally extended-valued [17]. For this family Appendix A establishes finite entropy and cross entropy separately; $\mathcal{D}(q\parallel q^{\prime })$ denotes their difference. All derivatives below are taken from its finite scalar formula. In Eguchi’s convention [18], primed derivatives act on the second endpoint and a vertical bar denotes diagonal evaluation:

$$\begin{array}{cc}\hfill g_{ab}& =-{\partial }_a{\partial }_{b^{\prime }}\mathcal{D}|,\hfill \end{array}$$

(12)

$$\begin{array}{cccc}\hfill {\Gamma }_{ab,c}& =-{\partial }_a{\partial }_b{\partial }_{c^{\prime }}\mathcal{D}|,\hfill & \hfill {\Gamma }_{ab,c}^{*}& =-{\partial }_{a^{\prime }}{\partial }_{b^{\prime }}{\partial }_c\mathcal{D}|.\hfill \end{array}$$

(13)

For $g>0$ these give torsion-free metric-dual connections, with midpoint ${\nabla }^{\mathrm{LC}}$ and symmetric cubic

$${\mathcal{C}}_{abc}={\Gamma }_{ab,c}^{*}-{\Gamma }_{ab,c}=g({({\nabla }^{*}-\nabla )}_{{\partial }_a}{\partial }_b,{\partial }_c).$$

(14)

The last index is lowered by g. The metric is the Bogoliubov-Kubo-Mori (BKM) metric of the family [19,20], not generally generator covariance. The cubic controls directed asymmetry:

$$\mathcal{D}(q\parallel q+\delta q)-\mathcal{D}(q+\delta q\parallel q)=\frac{1}{6}{\mathcal{C}}_{abc}\delta q^a\delta q^b\delta q^c+{O(\parallel \delta q\parallel }^4).$$

(15)

3.1. The Finite Relativistic Contrast

Theorem 1

(Exact relativistic contrast). For fixed $\beta ,s>0$, every finite pair of controls has

$$\begin{array}{|c|}\hline \mathcal{D}(q\parallel q^{\prime })={\displaystyle \frac{\beta }{2}}\left[{\displaystyle \frac{{({\eta }^{\prime }-\eta )}^2}{s^2}}+s^{2}F\left(\eta \right){({\alpha }^{\prime }-\alpha )}^2\right].\\ \hline\end{array}$$

(16)

The contrast is jointly smooth, the family is identifiable, and its BKM metric is everywhere positive definite.

Proof. 

Put $T={\rho }_0^{1/2}$ and $Q_q=U\left(q\right)Q_{s}U{\left(q\right)}^{*}$. Spectral calculus gives

$$-log\rho \left(q\right)=\beta Q_q+log{Z}_{\beta }id.$$

(17)

Lemma A1 defines finite form energies $\mathfrak{E}(q,q^{\prime })$ and proves $\mathcal{D}=\beta [\mathfrak{E}(q,q^{\prime })-\mathfrak{E}(q,q)]$. Write $\delta \alpha ={\alpha }^{\prime }-\alpha$, $\delta \eta ={\eta }^{\prime }-\eta$ and $C_{\eta }\left(r\right)=cosh(r+\eta )$. Since the kernel of T is real, that of $\pi T$ is imaginary and that of $C_{\eta }T$ is real. The even rapidity marginal therefore gives

$$\Re {\langle \pi T,C_{\eta }T\rangle }_{\mathrm{HS}}=0,{\langle rT,T\rangle }_{\mathrm{HS}}=0.$$

(18)

Moreover ${\parallel T\parallel }_2^2=1$ and $\parallel C_{\eta }{T\parallel }_2^2=F\left(\eta \right)$. Expanding the two squared norms in (A8) now gives

$$\begin{array}{cc}\hfill \parallel [\pi +\delta \alpha C_{\eta }]{T\parallel }_2^2-{\parallel \pi T\parallel }_2^2& =\delta {\alpha }^{2}F\left(\eta \right),\hfill \\ \hfill {\parallel (r-\delta \eta )T\parallel }_2^2-{\parallel rT\parallel }_2^2& =\delta {\eta }^2.\hfill \end{array}$$

This proves (16). Its positivity off the diagonal proves identifiability; its derivatives give the assertions about smoothness and the metric. □

For a fixed first endpoint this contrast is exactly quadratic in the displacement, but dependence of F on that endpoint makes it asymmetric.

3.2. Metric, Cubic Response and Curvature

Set $x=\alpha$, $y=\eta$, $a\left(y\right)=\beta s^{2}F\left(y\right)$ and $b=\beta /s^2$. The parameters $\beta ,s$ are fixed on each control surface. Differentiation gives

$$g=a\mathrm{d}{x}^2+b\mathrm{d}{y}^2,detg={\beta }^{2}F,g_{xy}=0.$$

(19)

The only nonzero lower connection coefficients are

$${\Gamma }_{xy,x}={\Gamma }_{yx,x}=a^{\prime },{\Gamma }_{xx,y}^{*}=-a^{\prime },$$

(20)

and the only potentially nonzero cubic components are

$${\mathcal{C}}_{xxy}=-a^{\prime }$$

(21)

and its permutations. Primes denote y derivatives. In particular,

$$\begin{array}{cc}\hfill \mathcal{D}(q\parallel q^{\prime })-\mathcal{D}(q^{\prime }\parallel q)& =\frac{1}{2}[a\left(y\right)-a\left(y^{\prime }\right)]\delta x^2\hfill \\ \hfill & =-\frac{1}{2}{a}^{\prime }\left(y\right)\delta x^2\delta y+{O(\parallel \delta q\parallel }^4),\hfill \end{array}$$

as required by (15). The forward cubic term in $\mathcal{D}(q\parallel q+\delta q)$ vanishes in these coordinates despite $\mathcal{C}\ne 0$; Appendix B verifies the cancellation.

For $E=e^{2v}>1$, the Gaussian curvature is

$$\begin{array}{cc}\hfill {\mathcal{K}}_g& =-{\displaystyle \frac{1}{b}}\left({\displaystyle \frac{F^{\prime \prime }}{2F}}-{\displaystyle \frac{{\left(F^{\prime }\right)}^2}{4F^2}}\right)\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & =-{\displaystyle \frac{1}{b}}\left[1+{\displaystyle \frac{E^2-1}{{(1+Ecosh2y)}^2}}\right],-{\displaystyle \frac{2E}{b(1+E)}}\le {\mathcal{K}}_g<-{\displaystyle \frac{1}{b}}.\hfill \end{array}$$

(23)

Thus

$${\mathcal{C}|}_{y=0}=0,{\mathcal{K}}_g{|}_{y=0}=-{\displaystyle \frac{2E}{b(1+E)}}\ne 0.$$

(24)

Pointwise equality of the dual connections is not local self-duality. With $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$,

$$R^{\nabla }({\partial }_x,{\partial }_y){\partial }_y=-{\displaystyle \frac{a^{\prime \prime }}{a}}{\partial }_x=-{\displaystyle \frac{F^{\prime \prime }}{F}}{\partial }_x\ne 0.$$

(25)

Metric duality gives $g(R^{\nabla }(X,Y)Z,W)=-g(Z,R^{{\nabla }^{*}}(X,Y)W)$, so both connections are curved. “Dual-affine” here means metric-dual affine connections, not dual flatness.

4. Minimal Realisation, Affine Rigidity and Convexity

4.1. Preserving the Entire Contrast

We first construct a positive potential for the prescribed contrast, rather than restrict a previously supplied Bregman divergence [2] or match only local statistical tensors [1].

Theorem 2

(Exact finite Hessian realisation). Let $a:\mathbb{R}\to (0,\infty )$ be smooth, $b>0$, and $\mathcal{D}(q\parallel q^{\prime })=a\left(y\right){(x-x^{\prime })}^2/2+b{(y-y^{\prime })}^2/2$. For any ${\lambda }_{\perp }>0$, define

$$\begin{array}{cc}\hfill \Phi (z,w,Y)& ={\displaystyle \frac{w^2}{2z}}+{\displaystyle \frac{b}{2}}{Y}^2+{\displaystyle \frac{{\lambda }_{\perp }}{2}}{[z-a\left(Y\right)]}^2,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \mathcal{N}& =\{(z,w,Y):z>0,b-{\lambda }_{\perp }[z-a\left(Y\right)]a^{\prime \prime }\left(Y\right)>0\},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill j(x,y)& =\left(a\right(y),a(y)x,y).\hfill \end{array}$$

(28)

Here $w,Y$ are unrestricted except for the displayed inequality. The Hessian $G={\partial }^2\Phi$ is positive on $\mathcal{N}$, which contains $j\left({\mathbb{R}}^2\right)$. Its Bregman function

$$B_{\Phi }(Z\parallel Z^{\prime })=\Phi \left(Z\right)-\Phi \left(Z^{\prime }\right)-\mathrm{d}{\Phi }_{Z^{\prime }}(Z-Z^{\prime })$$

(29)

is a contrast on convex coordinate neighbourhoods and satisfies

$$\begin{array}{|c|}\hline B_{\Phi }(j\left(q\right)\parallel j\left(q^{\prime }\right))=\mathcal{D}(q\parallel q^{\prime })\\ \hline\end{array}$$

(30)

for every finite physical endpoint pair. It induces exactly $g,\nabla ,{\nabla }^{*}$. The flat ambient dual connections give strongly integrable Born structures on $T\mathcal{N}$ [9].

Proof. 

Writing $\delta =z-a\left(Y\right)$ gives

$$G={\displaystyle \frac{1}{z}}{\left(\mathrm{d}w-{\displaystyle \frac{w}{z}}\mathrm{d}z\right)}^2+{\lambda }_{\perp }{(\mathrm{d}z-a^{\prime }\mathrm{d}Y)}^2+[b-{\lambda }_{\perp }\delta a^{\prime \prime }]\mathrm{d}{Y}^2.$$

(31)

The three one-forms are independent and all coefficients are positive on $\mathcal{N}$. On j, the penalty and its first derivative vanish:

$$\Phi \circ j=\frac{1}{2}a\left(y\right)x^2+\frac{b}{2}{y}^2{,\mathrm{d}\Phi |}_{j(x,y)}=(-x^2/2,x,by).$$

(32)

Substitution proves (30); diagonal differentiation gives the statistical tensors. Positive Hessian along segments in any small convex neighbourhood proves the local contrast property. The positive Hessian structure has flat dual connections, so the Hessian Born theorem applies [9]. □

The identity is global on the physical surface, but ambient nonnegativity is only local when $\mathcal{N}$ is nonconvex. “Realisation” does not mean metric completion: ${\lambda }_{\perp }$ is free, no ambient quantum state model or transverse dynamics is supplied, and the explicit metric has a finite-distance singular boundary (Appendix G).

Corollary 1

(Minimal Hessian dimension). For the relativistic contrast, any smooth pullback realisation of a local Bregman contrast on a positive Hessian manifold has dimension at least three. Theorem 2 attains this bound; the tangent carrier has minimum dimension six within this Hessian-tangent class.

Proof. 

Positive $j^{*}G$ forces immersion. A two-dimensional immersion is locally invertible, so it would pull a flat ambient statistical connection back to ∇, contradicting (25). Lower dimensions cannot carry the positive two-dimensional pullback metric. □

4.2. Finite-Contrast Rigidity and Global Convexity

A single-chart realisation is a smooth $i:{\mathbb{R}}^2\to \Omega \subset {\mathbb{R}}^d$ and smooth $\Psi$ with $B_{\Psi }(i\left(q\right)\parallel i\left(q^{\prime }\right))=\mathcal{D}(q\parallel q^{\prime })$ for all controls. The open domain need not be convex; positivity means ${\partial }^2\Psi >0$ there.

Theorem 3

(Affine rigidity and normal freedom). Let $b>0$. If $a>0$ is smooth and non-affine, every single-chart realisation of $\mathcal{D}=a\left(y\right){(x-x^{\prime })}^2/2+b{(y-y^{\prime })}^2/2$ has $d\ge 3$. For $d=3$, necessarily

$$i(x,y)=Tj(x,y)+c,T\in GL(3,\mathbb{R}),c\in {\mathbb{R}}^3.$$

(33)

After this affine coordinate change and subtraction of an affine function, the potential near the entire surface is

$${\Psi }_{♯}={\Phi }_0+\frac{1}{2}{[z-a\left(Y\right)]}^2\Lambda (z,w,Y),{\Phi }_0={\displaystyle \frac{w^2}{2z}}+{\displaystyle \frac{b}{2}}{Y}^2.$$

(34)

Its Hessian is positive definite on the surface exactly when ${\Lambda |}_j>0$; then it is positive definite on an open neighbourhood. Conversely, every such smooth normal term preserves all restricted comparisons. Thus finite comparisons fix the minimal affine placement and first jet, not the normal Hessian.

Proof. 

For $q_0=(0,0)$, $q_{\pm }=(\pm 1,0)$ and $q_y=(0,1)$, let $d_{\nu }\left(q\right)=\mathcal{D}(q\parallel q_{\nu })-\mathcal{D}(q\parallel q_0)$. Then

$$d_{+}=-ax+a/2,d_{-}=ax+a/2,d_y=-by+b/2.$$

(35)

Each Bregman difference is affine in $i\left(q\right)$ because its common first-endpoint potential cancels. Hence $j\left(q\right)=Li\left(q\right)+c_0$. The functions $1,a\left(y\right),a\left(y\right)x,y$ are linearly independent: varying x removes the third coefficient, and non-affineness removes the remaining relation. Thus L has rank three; in dimension three it is invertible.

Pull back by $T·+c$ and set $R=\Psi \circ (T·+c)-{\Phi }_0$ where $z>0$. Both potentials reproduce $\mathcal{D}$. Fixing $q_0$ shows $R\left(j\left(q\right)\right)=R\left(j\left(q_0\right)\right)+\mathrm{d}{R}_{j\left(q_0\right)}(j\left(q\right)-j\left(q_0\right))=:\ell \left(j\left(q\right)\right)$, where ℓ is affine. Varying the second endpoint gives $(\mathrm{d}{R}_{j\left(q^{\prime }\right)}-\mathrm{d}\ell )(j\left(q\right)-j\left(q^{\prime }\right))=0$ for all q. Affine spanning implies $\mathrm{d}{R}_{j\left(q^{\prime }\right)}=\mathrm{d}\ell$. After subtracting ℓ, both R and $\mathrm{d}R$ vanish on $z=a\left(Y\right)$. Taylor’s integral formula in $\delta =z-a\left(Y\right)$ gives $R={\delta }^2\Lambda /2$; the local quotients agree off the surface and extend smoothly across it.

On the surface the Hessian is ${\partial }^2{\Phi }_0+\Lambda \mathrm{d}{\delta }^2$. In the basis $j_{*}{\partial }_x,j_{*}{\partial }_y,n$, where $n=(1,x,0)$, its matrix is $diag(a,b,\Lambda )$. This proves positivity and its open extension, without a uniform-width claim. The normal term has zero value and first derivative on the surface, proving the converse. □

Theorem 4

(No globally convex positive realisation in finite dimension). Let $b>0$. If $a>0$ is smooth and $a\left(R\right)/R^2\to \infty$ as $R\to \infty$, then $\mathcal{D}(q\parallel q^{\prime })=a\left(y\right){(x-x^{\prime })}^2/2+b{(y-y^{\prime })}^2/2$ has no single-chart realisation in any finite dimension with open convex domain and everywhere positive-definite Hessian. In particular this holds for $a=\beta s^{2}F$.

Proof. 

Suppose $i:{\mathbb{R}}^2\to \Omega \subset {\mathbb{R}}^d$ and $\Psi$ give such a realisation. For $q_0=(0,0)$ and $q_{\pm }=(\pm 1,0)$, $\mathcal{D}(q\parallel q_{+})+\mathcal{D}(q\parallel q_{-})-2\mathcal{D}(q\parallel q_0)=a\left(y\right)$. Bregman differences are affine in the first endpoint, so

$$a\left(y\right)=\ell \left(i\right(x,y\left)\right)+c$$

(36)

for a constant nonzero covector ℓ and constant c. Subtract the tangent affine function at $p_0=i\left(q_0\right)$ to obtain $\overline{\Psi }\left(p_0\right)=0$, $\mathrm{d}{\overline{\Psi }}_{p_0}=0$, and

$$\overline{\Psi }\left(p_R\right)=\frac{b}{2}{R}^2,p_R=i(0,R).$$

(37)

The positive Hessian makes the gradient map locally invertible at $p_0$. For a sufficiently small $t>0$ there is therefore $p_t\in \Omega$ with $\mathrm{d}{\overline{\Psi }}_{p_t}=t\ell$. Its supporting plane gives

$$\frac{b}{2}{R}^2\ge \overline{\Psi }\left(p_t\right)+t\ell (p_R-p_t)=ta\left(R\right)+C_t.$$

Dividing by $a\left(R\right)$ contradicts $t>0$. No evenness or uniform Hessian lower bound is used. The relativistic coefficient satisfies the hypothesis because $F\left(R\right)\sim e^{2v+2R}/4$. □

The obstruction requires all finite comparisons on the unbounded control plane, one affine chart, and positivity in every ambient direction. It does not cover degenerate Hessians, infinite dimensions or general atlas-based constructions. Indeed, ${\Phi }_0$ already reproduces the whole contrast on $z>0$, but its Hessian has rank two.

Corollary 2

(Convex realisation on bounded rapidity strips). For the smooth $a>0$ and $b>0$ of Theorem 2, every compact rapidity interval I is contained in a bounded open interval J for which the contrast on $\mathbb{R}\times J$ has a positive globally convex three-dimensional realisation of the form (26). For the relativistic coefficient this dimension is minimal.

Proof. 

Choose bounded open $J\supset I$, and set $a_{min}={min}_{\overline{J}}a>0$, $a_{max}={max}_{\overline{J}}a$, and $B_2={max}_{\overline{J}}\left|a^{\prime \prime }\right|$. Take ${\lambda }_{\perp }>0$ with $2{\lambda }_{\perp }{a}_{max}{B}_2<b$; if $B_2=0$, any positive value works. The convex box $(a_{min}/2,2a_{max})\times \mathbb{R}\times J$ contains $j(\mathbb{R}\times J)$ and satisfies $b-{\lambda }_{\perp }(z-a)a^{\prime \prime }\ge b-2{\lambda }_{\perp }{a}_{max}{B}_2>0$. Equation (31) proves positivity; its line segments then give a global Bregman contrast. Restriction remains exact. For the relativistic a, $a^{\prime \prime }>0$ on J, so the local minimality proof still applies. □

Thus unbounded translations alone cause no convexity obstruction: bounded rapidity ranges can be realised exactly, with domain-dependent stiffness. Finite Hessian dimension is nevertheless not finite Hilbert-space dimension. Any finite-dimensional quantum family with all ordered relative entropies finite has common support. On that support a fixed reference ${\sigma }_0$ satisfies $D(\sigma \parallel {\sigma }_0)\le -log{\lambda }_{min}\left({\sigma }_0\right)$. Since (16) is unbounded against a fixed reference, no such finite-level family reproduces the entire contrast, even without the Poincaré algebra. This does not exclude regulated approximations or finite control sets.

5. Observable Statistics, Dephasing and Commutator Geometry

5.1. Commutator and Covariance Responses

For $G_a=\mathrm{i}\left({\partial }_{a}U\right)U^{†}$, one has ${\partial }_a\rho =-\mathrm{i}[G_a,\rho ]$ and

$$G_{\alpha }=sinhr,G_{\eta }=\pi +\alpha coshr,[G_{\alpha },G_{\eta }]=\mathrm{i}coshr.$$

(38)

The dimensionless orbit pairing [21] is

$${\omega }_{ab}=-\mathrm{i}Tr\rho [G_a,G_b],\omega =M\left(\eta \right)\mathrm{d}\alpha \wedge \mathrm{d}\eta ,{\omega }_{\alpha \eta }={\displaystyle \frac{\langle H\rangle }{m{c}^2}}.$$

(39)

It is closed and nondegenerate; its action-valued version is $\hslash \omega$. It is not an unspecified mixed-state phase curvature. The control surface is not a subgroup because its mixed bracket generates time translation. Appendix D supplies a phase protocol.

Generator covariance satisfies the Robertson-Schrödinger inequality [15,16]:

$$V_{ab}=\frac{1}{2}\langle \{G_a-\langle G_a\rangle ,G_b-\langle G_b\rangle \}\rangle ,V+\mathrm{i}\omega /2\ge 0,detV\ge {\omega }_{\alpha \eta }^2/4.$$

(40)

Undoing the encoding gives

$$V_{\alpha \alpha }=F-1-e^v{sinh}^2\eta ,V_{\eta \eta }=v/s^4,V_{\alpha \eta }=0.$$

(41)

This is not the BKM metric. The pure-state covariance construction [22] does not identify them for faithful mixed states.

The action-valued form has a canonical mean-momentum interpretation. At fixed preparation, write $\mathcal{Q}\left(\eta \right)=e^{v/2}sinh\eta$, $p=\langle P\rangle =mc\mathcal{Q}$ and $\mathcal{E}=\langle H\rangle$. Since ${\mathcal{Q}}^{\prime }=M>0$ and $\mathcal{Q}$ maps $\mathbb{R}$ onto itself,

$${\Omega }_{\mathrm{act}}:=\hslash \omega =\mathrm{d}A\wedge \mathrm{d}\mathcal{Q}=\mathrm{d}d\wedge \mathrm{d}p,\{d,p\}=1,\{A,\eta \}=M^{-1}.$$

(42)

The brackets use ${\iota }_{X_f}{\Omega }_{\mathrm{act}}=\mathrm{d}f$ and $\{f,g\}=X_{g}f$; they are control-space Poisson brackets, not operator commutators. The expectation values also give

$${\mathcal{E}}^2-c^2{p}^2=m^2{c}^4{e}^v,\mathrm{d}\mathcal{E}=ctanh\eta \mathrm{d}p.$$

(43)

The invariant mass of the mean four-momentum is $m{e}^{v/2}$, not a change of the particle’s operator mass. These unitary controls have constant entropy, so the energy variation is kinematic, not heat. Appendix I distinguishes this reduced canonical description from exact free quantum evolution.

5.2. Fixed Observable Laws and the Purity Resource

Theorem 5

(No universal entropy-metric area bound). No $c_0>0$ makes

$$detg\ge c_0{\omega }_{\alpha \eta }^2$$

(44)

valid for all preparations (7), even at fixed complete joint energy-momentum probability law.

Proof. 

At $\eta =0$, $F=e^{v}coshv$, $M^2=e^v$, and

$$detg/{\omega }_{\alpha \eta }^2={\beta }^{2}coshv.$$

(45)

Fix $v>0$ and choose $s^2=2vtanh(\beta /2)$. The ratio tends to zero as $\beta \downarrow 0$, while the joint law below is unchanged. Every member is faithful; no limiting density operator is needed. □

Both numerator and denominator acquire the same squared Jacobian under coordinate changes. Thus neither reparametrisation nor a constant rescaling of $\omega$ repairs the bound.

Corollary 3

(Observable law and purity-controlled bound). Along this fixed-v sequence the complete $H,P$ law at each fixed control, and every seed ordering-loop expectation, are independent of β. Within the preparation family, fixed purity $\mathfrak{p}=Tr{\rho }^2=tanh(\beta /2)$ instead gives

$${\displaystyle \frac{detg}{{\omega }_{\alpha \eta }^2}}=4{artanh}^2\left(\mathfrak{p}\right){\displaystyle \frac{\langle H^2\rangle }{{\langle H\rangle }^2}}\ge 4{artanh}^2\left(\mathfrak{p}\right).$$

(46)

For each $0<\mathfrak{p}<1$ this coefficient is the optimal infimum over widths and controls.

Proof. 

For every bounded Borel f, the joint law gives

$$Tr\rho \left(q\right)f(H,P)={\displaystyle \frac{1}{\sqrt{2\pi v}}}{\int }_{\mathbb{R}}{e}^{-{(r-\eta )}^2/\left(2v\right)}f(m{c}^{2}coshr,mcsinhr)\mathrm{d}r.$$

(47)

It depends only on $v,\eta$. The same holds for integrable moments and for the loop, a bounded function of $H,P$. The geometric eigenvalues give $\mathfrak{p}=tanh(\beta /2)$. Use $detg/{\omega }_{\alpha \eta }^2={\beta }^{2}F/M^2$ and $F\ge M^2$. At fixed $\beta$, taking $\eta =0$ and $s\downarrow 0$ gives $v\downarrow 0$ and $F/M^2=coshv\downarrow 1$, proving sharpness. □

Energy-momentum measurements are already blind to $\alpha$: translation changes off-diagonal phase coherence, not this commuting law. The sequence does not fix all resources: purity vanishes and boost variance diverges. Appendix C checks that covariance uncertainty still holds. The bounds in [4,5] concern covariance and quantum-Fisher expressions involving state-observable commutators, not the lower bound on BKM area by an observable-observable expectation pairing tested here.

5.3. Exact Dephasing and Its Classical Limit

Fix $v>0$ and write ${\rho }_{\beta ,v}\left(q\right)=U\left(q\right){\rho }_{0,\beta ,v}U{\left(q\right)}^{*}$ with

$$s_{\beta }^2=2vtanh(\beta /2),{\nu }_{\beta }={\langle {\pi }^2\rangle }_{0,\beta ,v}={\displaystyle \frac{1}{4v{tanh}^2(\beta /2)}}.$$

(48)

Define the trace-norm integral

$${\mathcal{E}}_{\tau }\left(T\right)={\displaystyle \frac{1}{\sqrt{2\pi \tau }}}{\int }_{\mathbb{R}}{e}^{-k^2/\left(2\tau \right)}{e}^{-\mathrm{i}kr}T{e}^{\mathrm{i}kr}\mathrm{d}k,\tau >0,{\mathcal{E}}_0=id.$$

(49)

This is a one-quadrature additive Gaussian-noise channel, canonically equivalent for $\tau >0$ to Holevo’s $B_1$ class [23]. The parameter is a phase-kick variance, not a physical clock or specified bath temperature. Its exact action on the nonlinear family is as follows.

Proposition 1

(Control-covariant dephasing). The maps ${\mathcal{E}}_{\tau }$ are completely positive, trace preserving, and satisfy ${\mathcal{E}}_{\tau }{\mathcal{E}}_{{\tau }^{\prime }}={\mathcal{E}}_{\tau +{\tau }^{\prime }}$. They commute with conjugation by the supplied spatial translations, time translations and boosts. For $0<{\beta }_2<{\beta }_1$,

$${\mathcal{E}}_{{\nu }_{{\beta }_2}-{\nu }_{{\beta }_1}}\left({\rho }_{{\beta }_1,v}\left(q\right)\right)={\rho }_{{\beta }_2,v}\left(q\right)\mathrm{for}\mathrm{every}q.$$

(50)

The Heisenberg adjoint fixes all bounded Borel functions of r, including all bounded Borel functions of $H,P$ and every finite ordering-loop operator.

Proof. 

Strong continuity of the phase unitaries implies trace-norm continuity of conjugation, first for finite rank and then by approximation. The integrand has constant trace norm. Its probability integral is completely positive and trace preserving; Gaussian convolution proves the semigroup law. Phase kicks commute with functions of r, while a boost shifts r by a scalar whose phase cancels in conjugation.

The seed kernel is

$${\rho }_{0,\beta ,v}(r,r^{\prime })={\displaystyle \frac{1}{\sqrt{2\pi v}}}exp\left[-{\displaystyle \frac{{(r+r^{\prime })}^2}{8v}}-{\displaystyle \frac{{\nu }_{\beta }}{2}}{(r-r^{\prime })}^2\right].$$

(51)

The channel multiplies it by $e^{-\tau {(r-r^{\prime })}^2/2}$, proving the seed identity. Covariance proves (50), including the non-Gaussian encoded states. The adjoint assertion follows by commutation; unchanged spectral measures give all finite moments. □

Corollary 4

(Classical contrast limit). Let $p_{v,\eta }$ be the Gaussian rapidity law of mean η and variance v. As $\beta \downarrow 0$, locally smoothly in the endpoints,

$$\begin{array}{cc}\hfill {\mathcal{D}}_{\beta ,v}(q\parallel q^{\prime })& ⟶D_{\mathrm{KL}}(p_{v,\eta }\parallel p_{v,{\eta }^{\prime }})={\displaystyle \frac{{({\eta }^{\prime }-\eta )}^2}{2v}},\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill g_{\beta ,v}& ⟶v^{-1}\mathrm{d}{\eta }^2.\hfill \end{array}$$

(53)

The exact nonnegative gap is

$$\begin{array}{cc}\hfill {\mathcal{D}}_{\beta ,v}-D_{\mathrm{KL}}& =v\beta tanh(\beta /2)F\left(\eta \right){\left(\delta \alpha \right)}^2\hfill \\ \hfill & +{\displaystyle \frac{\beta coth(\beta /2)-2}{4v}}{\left(\delta \eta \right)}^2.\hfill \end{array}$$

(54)

Every positive noise increment strictly decreases the contrast between distinct controls and strictly decreases the BKM metric in the quadratic-form order. No trace-norm limiting density operator exists.

Proof. 

Substitute (48) into (16). Both $\beta tanh(\beta /2)$ and $\beta coth(\beta /2)$ increase strictly: for the latter, ${\partial }_t(tcotht)=(sinhtcosht-t)/{sinh}^{2}t>0$. Also $tcotht>1$, proving positivity and strict contraction, consistent with data processing [24]. Taylor expansion gives

$$\begin{array}{cccc}\hfill a_{\beta }\left(\eta \right)& =vF\left(\eta \right){\beta }^2+O\left({\beta }^4\right),\hfill & \hfill b_{\beta }& =v^{-1}+{\displaystyle \frac{{\beta }^2}{12v}}+O\left({\beta }^4\right),\hfill \\ \hfill {\mathcal{D}}_{\beta ,v}-D_{\mathrm{KL}}& ={\beta }^2\left[{\displaystyle \frac{vF\left(\eta \right)}{2}}{\left(\delta \alpha \right)}^2+{\displaystyle \frac{{\left(\delta \eta \right)}^2}{24v}}\right]+O\left({\beta }^4\right).\hfill \end{array}$$

(55)

Remainders and endpoint derivatives of fixed order are uniform on compact control sets. Finally, $\parallel {\rho }_{\beta ,v}{\left(q\right)\parallel }_2^2=tanh(\beta /2)\to 0$ while $Tr\rho =1$. Trace-norm convergence would force both Hilbert-Schmidt convergence to zero and trace one, a contradiction. □

Displacements growing as $1/\beta$ need not lose contrast. At finite noise the convexity obstruction still applies; only the singular limit has the positive quotient potential ${\eta }^2/\left(2v\right)$ after forgetting translations.

Fixed-width heating is a different limit.

Holding $s,E_{*}>0$ fixed heats one Hamiltonian $E_{*}{Q}_s$ as $\beta \downarrow 0$. Then $v_{\beta }^{\left(s\right)}=s^2/\beta +O\left(\beta \right)$ and

$${\left.{\displaystyle \frac{detg}{{\omega }_{\alpha \eta }^2}}\right|}_{\eta =0}={\beta }^{2}cosh{v}_{\beta }^{\left(s\right)}\sim {\displaystyle \frac{{\beta }^2}{2}}{e}^{s^2/\beta }⟶\infty .$$

(56)

At fixed finite $\eta$, also $g_{\alpha \alpha }\sim (\beta s^2/2)e^{2s^2/\beta }cosh\left(2\eta \right)\to \infty$. These follow from (8) and (19); the fixed-v dephasing limit instead changes s. Its loss of translation response is not a general high-temperature effect.

At equal boost, the quantum Pinsker inequality gives

$$\parallel {\rho }_{\beta ,v}(\alpha ,\eta )-{\rho }_{\beta ,v}({\alpha }^{\prime },\eta ){\parallel }_1\le min\{2,\sqrt{a_{\beta }\left(\eta \right)}|{\alpha }^{\prime }-\alpha \left|\right\}=O\left(\beta \right)$$

(57)

for fixed displacement. Measuring the positive spectral projection of $\rho -\sigma$ reduces Pinsker to its classical form by data processing [24]. Thus all measurements lose translation distinguishability. No channel acting on the noisy system alone can exactly recover a distinct pair: data processing for recovery would contradict strict contraction.

6. Calibrated Temperature, Work and Entropy Capacity

Write $\mathcal{S}=-k_{\mathrm{B}}Tr\rho log\rho$ and

$$u_{\beta }=\frac{1}{2}coth(\beta /2),{\chi }_{\beta }={\displaystyle \frac{1}{4{sinh}^2(\beta /2)}},S_0\left(\beta \right)=\beta u_{\beta }+log{Z}_{\beta }.$$

(58)

Then $u_{\beta }^{\prime }=-{\chi }_{\beta }$, $S_0^{\prime }=-\beta {\chi }_{\beta }$, and every control state at fixed preparation has entropy $k_{\mathrm{B}}{S}_0\left(\beta \right)$. Its relative entropy with another control is not an entropy change of either state.

6.1. Temperature and Susceptibility

The following are standard Gibbs and equilibrium-thermometry identities [6,25,26], stated with the preparation energy scale explicit.

Proposition 2

(Calibrated preparation temperature). Supply $H_{\mathrm{prep}}=E_{*}{Q}_s$ with $E_{*}>0$. At fixed $s,E_{*}$, the seed is Gibbs and

$$T_{\mathrm{prep}}={\displaystyle \frac{E_{*}}{k_{\mathrm{B}}\beta }},U_{\mathrm{p}}=E_{*}{u}_{\beta },{\displaystyle \frac{\mathrm{d}\mathcal{S}}{\mathrm{d}{U}_{\mathrm{p}}}}={\displaystyle \frac{1}{T_{\mathrm{prep}}}}.$$

(59)

On this commuting thermal family, the BKM, classical and SLD Fisher metrics agree and satisfy

$$g_{\beta \beta }={\chi }_{\beta },{\mathcal{I}}_{TT}={\displaystyle \frac{Var\left(H_{\mathrm{prep}}\right)}{k_{\mathrm{B}}^2{T}_{\mathrm{prep}}^4}}={\displaystyle \frac{C_{\mathrm{p}}}{k_{\mathrm{B}}{T}_{\mathrm{prep}}^2}},C_{\mathrm{p}}=k_{\mathrm{B}}{\beta }^2{\chi }_{\beta }>0.$$

(60)

Allowing the gap to vary gives the equilibrium identity

$$\mathrm{d}{U}_{\mathrm{p}}=T_{\mathrm{prep}}\mathrm{d}\mathcal{S}+u_{\beta }\mathrm{d}{E}_{*}.$$

(61)

It also holds for the instantaneous Gibbs family of $E_{*}U\left(q\right)Q_{s}U{\left(q\right)}^{*}$; its isospectral $s,q$ variations have zero equilibrium mean generalized force.

Proof. 

$H_{\mathrm{prep}}/\left(k_{\mathrm{B}}{T}_{\mathrm{prep}}\right)=\beta Q_s$ identifies the Gibbs state. The partition function gives $-{\partial }_{\beta }log{Z}_{\beta }=u_{\beta }$ and ${\partial }_{\beta }^{2}log{Z}_{\beta }={\chi }_{\beta }$. Thus $\mathrm{d}\mathcal{S}=k_{\mathrm{B}}\beta \mathrm{d}{u}_{\beta }$ and $\mathrm{d}{U}_{\mathrm{p}}=E_{*}\mathrm{d}{u}_{\beta }+u_{\beta }\mathrm{d}{E}_{*}$. The eigenvalue score $u_{\beta }-(n+1/2)$ has variance ${\chi }_{\beta }$; ${\partial }_T\beta =-\beta /T$ gives (60). Finally $Z_{\beta }$ is independent of $s,q$, so differentiating it in these isospectral directions gives zero equilibrium mean force. □

Purity determines only the dimensionless preparation parameter:

$$\mathfrak{p}=tanh(\beta /2),T_{\mathrm{prep}}={\displaystyle \frac{E_{*}}{2k_{\mathrm{B}}artanh\mathfrak{p}}}.$$

(62)

The rescaling $(E_{*},T_{\mathrm{prep}})\mapsto (c_{*}{E}_{*},c_{*}{T}_{\mathrm{prep}})$, $c_{*}>0$, leaves $H_{\mathrm{prep}}/\left(k_{\mathrm{B}}{T}_{\mathrm{prep}}\right)=\beta Q_s$, and hence all state-derived geometry, unchanged. An independent gap, work or clock calibration is therefore necessary. The rest-energy scale $m{c}^2$ does not identify $E_{*}{Q}_s$ with the free H; Appendix H gives the equilibrium tests.

6.2. Full Preparation-Control Contrast and Extractable Work

Let the endpoints have parameters $({\beta }_i,{\ell }_i,{\alpha }_i,{\eta }_i)$, where $s_i=e^{{\ell }_i}$, $u_i=u_{{\beta }_i}$, $v_i=s_i^2{u}_i$, and $F_i=(1+e^{2v_i}cosh2{\eta }_i)/2$. Write ${\rho }_i={\rho }_{{\beta }_i,s_i}\left(q_i\right)$ and $\delta \alpha ={\alpha }_2-{\alpha }_1$, $\delta \eta ={\eta }_2-{\eta }_1$.

Proposition 3

(Exact four-parameter contrast). For every finite pair with ${\beta }_i,s_i>0$,

$$\begin{array}{cc}\hfill D({\rho }_1\parallel {\rho }_2)& =D_{\mathrm{sp}}({\beta }_1\parallel {\beta }_2)+{\beta }_2{u}_1[cosh(2{\ell }_2-2{\ell }_1)-1]\hfill \\ \hfill & +{\displaystyle \frac{{\beta }_2}{2}}\left[s_2^2{F}_1{\left(\delta \alpha \right)}^2+{\displaystyle \frac{{\left(\delta \eta \right)}^2}{s_2^2}}\right],\hfill \end{array}$$

(63)

where

$$D_{\mathrm{sp}}=({\beta }_2-{\beta }_1)u_1+log{Z}_{{\beta }_2}-log{Z}_{{\beta }_1}$$

(64)

is the classical relative entropy of the two geometric spectra. The full BKM metric is

$$\mathfrak{g}={\chi }_{\beta }\mathrm{d}{\beta }^2+4\beta u_{\beta }\mathrm{d}{\ell }^2+\beta s^{2}F\left(\eta \right)\mathrm{d}{\alpha }^2+{\displaystyle \frac{\beta }{s^2}}\mathrm{d}{\eta }^2.$$

(65)

Proof. 

The target logarithm and the same real-kernel cancellation as in Theorem 1 give

$${\langle U\left(q_2\right)Q_{s_2}U{\left(q_2\right)}^{*}\rangle }_{{\rho }_1}={\displaystyle \frac{1}{2}}\left[s_2^2\left({\displaystyle \frac{u_1}{s_1^2}}+F_1\delta {\alpha }^2\right)+{\displaystyle \frac{s_1^2{u}_1+\delta {\eta }^2}{s_2^2}}\right].$$

(66)

The Gaussian estimates of Appendix A are locally uniform also in positive preparation parameters. Thus cross entropy is finite; subtracting ${\beta }_1{u}_1+log{Z}_{{\beta }_1}$ proves (63). Its mixed diagonal derivatives give (65), with zero cross terms. □

This defines the statistical manifold $\mathcal{P}=(0,\infty )\times {\mathbb{R}}^3$, with coordinates $(\beta ,\ell ,\alpha ,\eta )$. Its thermal, squeezing and control directions are orthogonal, although their coefficients are coupled. Squeezing has nonzero length and unchanged entropy. The minimality and rigidity theorems concern fixed-preparation slices, not this larger family.

Fix $H_2=E_{*}U\left(q_2\right)Q_{s_2}U{\left(q_2\right)}^{*}$ and $T_2=E_{*}/\left(k_{\mathrm{B}}{\beta }_2\right)$. The Gibbs logarithm gives the standard identity [6,27]

$$k_{\mathrm{B}}{T}_{2}D({\rho }_1\parallel {\rho }_2)={\mathcal{F}}_{H_2,T_2}\left({\rho }_1\right)-{\mathcal{F}}_{H_2,T_2}\left({\rho }_2\right),{\mathcal{F}}_{H,T}\left(\rho \right)=Tr\rho H-T\mathcal{S}\left(\rho \right).$$

(67)

Define ergotropy by ${\mathcal{W}}_{H_2}\left(\rho \right)=Tr\rho H_2-{inf}_{V}Tr\left(V\rho V^{*}{H}_2\right)$, where the infimum ranges over unitaries with finite final energy. This is unrestricted cyclic-unitary work [28], not work restricted to the two control operations. Its relative-entropy representation is established [7]; here it is explicit.

Corollary 5

(Spectral mismatch and ergotropy). For these endpoints,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\mathcal{W}}_{H_2}\left({\rho }_1\right)}{E_{*}}}& =u_1[cosh(2{\ell }_2-2{\ell }_1)-1]\hfill \\ \hfill & +\frac{1}{2}[s_2^2{F}_1\delta {\alpha }^2+\delta {\eta }^2/s_2^2],\hfill \end{array}$$

(68)

and

$$\begin{array}{|c|}\hline D({\rho }_1\parallel {\rho }_2)=D_{\mathrm{sp}}({\beta }_1\parallel {\beta }_2)+{\displaystyle \frac{{\mathcal{W}}_{H_2}\left({\rho }_1\right)}{k_{\mathrm{B}}{T}_2}}.\\ \hline\end{array}$$

(69)

At equal β the spectral term vanishes. On a fixed-preparation control slice the work also equals the energy excess over the target Gibbs state, evaluated using $H_2$.

Proof. 

The state ${\sigma }_{*}=U\left(q_2\right){\rho }_{{\beta }_1,s_2}U{\left(q_2\right)}^{*}$ has the spectrum of ${\rho }_1$ and is Gibbs for $H_2$ at inverse scale ${\beta }_1/E_{*}$. Equal entropies and relative-entropy positivity imply, for finite-energy V,

$$0\le D(V{\rho }_1{V}^{*}\parallel {\sigma }_{*})={\displaystyle \frac{{\beta }_1}{E_{*}}}[Tr\left(V{\rho }_1{V}^{*}{H}_2\right)-E_{*}{u}_1].$$

Undoing $U\left(q_1\right)$, dilating the oscillator width from $s_1$ to $s_2$, and applying $U\left(q_2\right)$ attains the lower energy. Subtract $u_1$ from (66) and use (63). □

6.3. Work Susceptibility and Finite-Budget Control Area

Fix $q_0=({\alpha }_0,{\eta }_0)$ and its calibrated Hamiltonian $H_{q_0}=E_{*}U\left(q_0\right)Q_{s}U{\left(q_0\right)}^{*}$. Hold $\beta ,s,E_{*}>0$ and $H_{q_0}$ fixed while varying q. Corollary 5 gives

$$W(q\mid q_0):=Tr\rho \left(q\right)H_{q_0}-Tr\rho \left(q_0\right)H_{q_0}={\displaystyle \frac{E_{*}}{\beta }}\mathcal{D}(q\parallel q_0).$$

(70)

Thus work refers to the preparation Hamiltonian, not the free H. For any positive-definite symmetric bilinear form k, define ${\mathcal{A}}_k{(u,w)}^2=k(u,u)k(w,w)-k{(u,w)}^2$ with ${\mathcal{A}}_k\ge 0$.

Proposition 4

(Sharp work-susceptibility area). The source Hessian at $q=q_0$ is an energy-valued tensor $B^W\left(q_0\right)$. Letting the reference point vary gives the tensor field

$$B^W=k_{\mathrm{B}}{T}_{\mathrm{prep}}g=E_{*}{s}^{2}F\mathrm{d}{\alpha }^2+{\displaystyle \frac{E_{*}}{s^2}}\mathrm{d}{\eta }^2.$$

(71)

For independent $u,w$,

$${\displaystyle \frac{{\mathcal{A}}_{B^W}{(u,w)}^2}{\omega {(u,w)}^2}}=E_{*}^2{\displaystyle \frac{F}{M^2}}=E_{*}^2\left(1+{\displaystyle \frac{Var\left(H\right)}{{\langle H\rangle }^2}}\right)>E_{*}^2.$$

(72)

The coefficient $E_{*}$ in ${\mathcal{A}}_{B^W}\ge E_{*}\left|\omega \right|$ is optimal uniformly over preparations at fixed gap. For a fixed preparation, the optimal coefficient over control points is $E_{*}\sqrt{coshv}$.

Proof. 

The first source derivative of W vanishes at $q_0$, so its Hessian is coordinate covariant. Equation (70) gives $B^W=(E_{*}/\beta )g$ and $det{B}^W=E_{*}^{2}F$. The two-dimensional Gram determinant proves (72). The Gaussian rapidity law has $Var(coshr)>0$, while

$${\displaystyle \frac{F}{M^2}}=coshv+sinhv{tanh}^2\eta \ge coshv.$$

Equality in this last bound occurs at $\eta =0$. At fixed $\beta >0$, letting $s\downarrow 0$ makes $v\downarrow 0$ and $coshv\downarrow 1$ through faithful preparations, proving uniform optimality. □

At fixed $v,E_{*}$, multiplying by $k_{\mathrm{B}}{T}_{\mathrm{prep}}=E_{*}/\beta$ removes the explicit Gibbs-parameter factor responsible for the entropy-area collapse. The work area ratio stays fixed, but its eigenvalues become anisotropic and the reference width changes. This bounds neither each directional response nor a fixed Hamiltonian along the sequence; a closing gap removes the uniform energy coefficient.

Corollary 6

(Finite-work accessibility). For $W_0>0$, let ${\mathcal{U}}_{W_0}\left(q_0\right)=\{q:W(q\mid q_0)\le W_0\}$ on the fixed-preparation control plane, oriented by $\mathrm{d}\alpha \wedge \mathrm{d}\eta$. This set is compact and has positive physically calibrated symplectic area satisfying

$$0<{\int }_{{\mathcal{U}}_{W_0}\left(q_0\right)}\hslash \omega <{\displaystyle \frac{2\pi \hslash W_0}{E_{*}\sqrt{coshv}}}.$$

(73)

The coefficient is the optimal uniform bound over reference points and positive budgets for that preparation.

Proof. 

The global coordinates $z_1=s\sqrt{F\left(\eta \right)}(\alpha -{\alpha }_0)$ and $z_2=(\eta -{\eta }_0)/s$ have Jacobian $\sqrt{F\left(\eta \right)}>0$. Their smooth global inverse is $\eta ={\eta }_0+s{z}_2$, $\alpha ={\alpha }_0+z_1/\left[s\sqrt{F({\eta }_0+s{z}_2)}\right]$. They transform (70) into $W=E_{*}(z_1^2+z_2^2)/2$, so the sublevel set is the inverse image of a closed disk under a diffeomorphism and is compact. Moreover,

$$\hslash \omega =\hslash {\displaystyle \frac{M\left(\eta \right)}{\sqrt{F\left(\eta \right)}}}\mathrm{d}{z}_1\wedge \mathrm{d}{z}_2\le {\displaystyle \frac{\hslash }{\sqrt{coshv}}}\mathrm{d}{z}_1\wedge \mathrm{d}{z}_2.$$

Integrate over the disk of radius $\sqrt{2W_0/E_{*}}$. Equality of the densities is confined to $\eta =0$, a set of zero area, so the bound is strict for $W_0>0$. With ${\eta }_0=0$ and $W_0\downarrow 0$, continuity makes the area-to-budget ratio tend to the stated coefficient. □

This exact area bound concerns net preparation-energy excess, not transient or dissipated work. Its area has action units and counts neither quantum states nor entropy; the sublevel set is isospectral. Independently measuring $B^W$ calibrates $k_{\mathrm{B}}{T}_{\mathrm{prep}}=B^W(X,X)/g(X,X)$, $X\ne 0$; direction independence tests the stipulated Gibbs identification [29]. The Born metric is not identified with that measured Hessian.

6.4. Entropy Bounds and the Missing Resource

Every finite preparation has finite entropy, but

$$\mathcal{S}/k_{\mathrm{B}}=S_0\left(\beta \right)=1-log\beta +{\beta }^2/24+O\left({\beta }^4\right).$$

(74)

For a fixed gap this divergence requires $T_{\mathrm{prep}}\to \infty$. Bounds $T_{\mathrm{prep}}\le T_{max}<\infty$ and $E_{*}\ge E_{min}>0$, with $T_{max}>0$, give

$$\mathcal{S}\le k_{\mathrm{B}}{S}_0(E_{min}/\left(k_{\mathrm{B}}{T}_{max}\right)).$$

(75)

A closing gap can instead give unbounded entropy at finite temperature; Appendix H exhibits that changing-Hamiltonian limit. A temperature bound alone does not constrain arbitrary spectra.

Define

$$h_{\mathrm{osc}}\left(u\right)=(u+\frac{1}{2})log(u+\frac{1}{2})-(u-\frac{1}{2})log(u-\frac{1}{2}),u\ge \frac{1}{2},$$

(76)

with $0log0=0$.

Proposition 5

(Entropy from two quadratic resources). For any density operator on $L^2(\mathbb{R},\mathrm{d}r)$ satisfying $Tr\rho r^2\le v_{max}<\infty$ and $Tr\rho {\pi }^2\le {\nu }_{max}<\infty$, with positive budgets, put $u_{max}=\sqrt{v_{max}{\nu }_{max}}$. A nonempty constraint set requires $u_{max}\ge 1/2$, and

$$\begin{array}{|c|}\hline \mathcal{S}\left(\rho \right)\le k_{\mathrm{B}}{h}_{\mathrm{osc}}\left(u_{max}\right).\\ \hline\end{array}$$

(77)

For $u_{max}>1/2$ the bound is attained by the centered seed with $s^2=\sqrt{v_{max}/{\nu }_{max}}$ and $\gamma =log[(u_{max}+1/2)/(u_{max}-1/2)]$.

This is the Gaussian maximum-entropy principle [8] for raw second-moment budgets. Appendix C.1 supplies a self-contained proof that establishes finite entropy before using the Gibbs bound, and verifies sharpness. For $u_{max}>1/2$, its maximizing seed ${\sigma }_{*}$ also obeys

$$\begin{array}{cc}\hfill h_{\mathrm{osc}}\left(u_{max}\right)-\mathcal{S}\left(\rho \right)/k_{\mathrm{B}}& =D(\rho \parallel {\sigma }_{*})+\gamma (u_{max}-Tr\rho Q_s)\hfill \\ \hfill & \ge \frac{1}{2}{\parallel \rho -{\sigma }_{*}\parallel }_1^2.\hfill \end{array}$$

(78)

The identity is the Gibbs logarithm; the inequality is Pinsker. Thus near-saturation certifies trace-norm proximity to the specified Gaussian, not merely a similar entropy value.

Corollary 7

(Relativistic energy and boost budget). If $Tr\rho H\le E_{max}<\infty$ and $Tr\rho K^2\le K_{max}^2<\infty$, a nonempty constraint set obeys (77) with

$$v_{max}={arcosh}^2\left({\displaystyle \frac{E_{max}}{m{c}^2}}\right),{\nu }_{max}={\displaystyle \frac{K_{max}^2}{{\hslash }^2}}.$$

(79)

The resulting finite bound is not asserted sharp for these physical constraints.

Proof. 

First $coshr\ge 1+r^2/2$ ensures a finite second moment. The function $\phi \left(t\right)=cosh\sqrt{t}$ is increasing and convex for $t\ge 0$, as its power series shows. Jensen’s inequality for the spectral measure of r gives $\langle H\rangle /\left(m{c}^2\right)\ge cosh\sqrt{\langle r^2\rangle }$. Together with $K^2={\hslash }^2{\pi }^2$ this proves (79). □

This improves the elementary estimate $v_{max}=2(E_{max}/m{c}^2-1)$. Fixing the entire $H,P$ law alone still permits unbounded entropy because ${\nu }_{\beta }\sim 1/\left(v{\beta }^2\right)$ diverges. For the fixed-v seed family an additional ${\nu }_{\beta }\le {\nu }_{max}$, with $v{\nu }_{max}>1/4$, yields

$$\beta \ge {\beta }_{min}=2artanh{\displaystyle \frac{1}{2\sqrt{v{\nu }_{max}}}},{\displaystyle \frac{detg}{{\omega }_{\alpha \eta }^2}}\ge {\beta }_{min}^{2}coshv>0.$$

(80)

Equality holds at $\eta =0$ and ${\langle {\pi }^2\rangle }_0={\nu }_{max}$. This metric estimate is family-specific. For encoded states, seed resource formulas require undoing the known controls rather than inserting raw covariances of a non-Gaussian state.

6.5. Noise Energy and the Rate of Information Loss

Keep the initial centered preparation Hamiltonian $H_1=E_{*}{Q}_{s_1}$ fixed. Set $v=s_1^2{u}_1$, ${\nu }_1=u_1/s_1^2$, ${\rho }_{\tau }={\mathcal{E}}_{\tau }\left({\rho }_{{\beta }_1,s_1}\right)$, and $u_{\tau }=\sqrt{v({\nu }_1+\tau )}$.

Corollary 8

(Exact noise-energy accounting). For finite $\tau \ge 0$,

$$\begin{array}{cc}\hfill Tr{\rho }_{\tau }{H}_1-E_{*}{u}_1& =E_{*}{s}_1^2\tau /2,\hfill \\ \hfill {\mathcal{W}}_{H_1}\left({\rho }_{\tau }\right)& =E_{*}(u_1+s_1^2\tau /2-u_{\tau }),\hfill \\ \hfill \mathcal{S}\left({\rho }_{\tau }\right)& =k_{\mathrm{B}}{h}_{\mathrm{osc}}\left(u_{\tau }\right).\hfill \end{array}$$

(81)

For $\tau >0$ the extractable fraction of added energy is

$${\displaystyle \frac{{\mathcal{W}}_{H_1}\left({\rho }_{\tau }\right)}{E_{*}{s}_1^2\tau /2}}={\displaystyle \frac{\sqrt{1+\tau /{\nu }_1}-1}{\sqrt{1+\tau /{\nu }_1}+1}}.$$

(82)

Proof. 

Only $\langle {\pi }^2\rangle$ increases, by $\tau$, giving the first formula. The output has oscillator spectral parameter $u_{\tau }$, so its passive energy for $H_1$ is $E_{*}{u}_{\tau }$ by Corollary 5. Subtracting this gives the work; the spectrum gives the entropy. Use $u_{\tau }=u_1\sqrt{1+\tau /{\nu }_1}$ for the fraction. □

The supplied anisotropic noise adds both entropy and ergotropy, not work from a single equilibrium bath. At small noise, ${\mathcal{W}}_{H_1}=E_{*}{s}_1^4{\tau }^2/\left(8u_1\right)+O\left({\tau }^3\right)$; at large noise, $\mathcal{S}/k_{\mathrm{B}}=1+\frac{1}{2}log\left(v\tau \right)+o\left(1\right)$. Free relativistic energy stays fixed, but the boost moment and fixed-$H_1$ energy do not.

A further exact connection is the one-quadrature de Bruijn identity [30]. Put ${\beta }_{\tau }=log[(u_{\tau }+1/2)/(u_{\tau }-1/2)]$, $s_{\tau }^2=v/u_{\tau }$, and define the BKM response to a phase kick by

$$J_r\left(\rho \right)={\left.{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{k}^2}}D(\rho \parallel e^{-\mathrm{i}kr}\rho e^{\mathrm{i}kr})\right|}_{k=0}.$$

(83)

For the noise trajectory, including its controlled versions,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}\mathcal{S}\left({\rho }_{\tau }\right)}{\mathrm{d}\tau }}& ={\displaystyle \frac{k_{\mathrm{B}}}{2}}{J}_r\left({\rho }_{\tau }\right)={\displaystyle \frac{k_{\mathrm{B}}{\beta }_{\tau }v}{2u_{\tau }}},\hfill \\ \hfill g_{\alpha \alpha ,\tau }\left(\eta \right)& ={\displaystyle \frac{2F\left(\eta \right)}{k_{\mathrm{B}}}}{\displaystyle \frac{\mathrm{d}\mathcal{S}}{\mathrm{d}\tau }}.\hfill \end{array}$$

(84)

Indeed $u_{\tau }^{\prime }=v/\left(2u_{\tau }\right)$ and $h_{\mathrm{osc}}^{\prime }\left(u\right)=\beta \left(u\right)$. Conjugating the oscillator logarithm by $e^{-\mathrm{i}kr}$ shifts $\pi$ by k, giving $D={\beta }_{\tau }{s}_{\tau }^2{k}^2/2$ for the centered seed; control covariance gives the same result after encoding. Finally $g_{\alpha \alpha ,\tau }={\beta }_{\tau }{s}_{\tau }^{2}F$. Since $\beta \left(u\right)/u$ decreases, so does the positive entropy rate. Translation susceptibility tracks this rate, not accumulated entropy. The derivative is per noise variance, not per physical second.

6.6. Intrinsic Entropy Length and Constrained Dephasing

Lengths in this subsection are intrinsic to $(\mathcal{P},\mathfrak{g})$ in (65), not infima over arbitrary density-operator paths. This information geometry is related to thermodynamic length [31], but no relaxation dynamics or dissipation metric is supplied.

Proposition 6

(Entropy gradient and exact thermal distance). On $\mathcal{P}$,

$${grad}_{\mathfrak{g}}\mathcal{S}=-k_{\mathrm{B}}\beta {\partial }_{\beta },{\parallel \mathrm{d}\mathcal{S}\parallel }_{{\mathfrak{g}}^{-1}}^2=k_{\mathrm{B}}{C}_{\mathrm{p}}<k_{\mathrm{B}}^2.$$

(85)

Every piecewise $C^1$ path $\gamma :[0,1]\to \mathcal{P}$ with endpoint inverse parameters ${\beta }_0,{\beta }_1>0$ obeys

$$\left|\Delta \mathcal{S}\right|\le {\int }_{\gamma }\sqrt{k_{\mathrm{B}}{C}_{\mathrm{p}}}d{s}_{\mathfrak{g}}\le k_{\mathrm{B}}{L}_{\mathfrak{g}}\left(\gamma \right).$$

(86)

Set $\zeta \left(\beta \right)=-logtanh(\beta /4)$. Then

$$L_{\mathfrak{g}}\left(\gamma \right)\ge |\zeta \left({\beta }_1\right)-\zeta \left({\beta }_0\right)|.$$

(87)

For endpoints agreeing in $\ell ,\alpha ,\eta$, this is their exact Riemannian distance, attained by monotone variation of β alone. The uniform coefficient $k_{\mathrm{B}}$ in (86) is optimal. The boundary $\beta \downarrow 0$ is infinitely distant intrinsically in $\mathcal{P}$.

Proof. 

Use $\mathrm{d}\mathcal{S}=-k_{\mathrm{B}}\beta {\chi }_{\beta }\mathrm{d}\beta$ and the inverse of the diagonal metric. This gives (85), with $C_{\mathrm{p}}/k_{\mathrm{B}}={\beta }^2{\chi }_{\beta }={[\beta /(2sinh(\beta /2))]}^2<1$. Cauchy-Schwarz and integration prove (86). Moreover $d{s}_{\mathfrak{g}}\ge \sqrt{{\chi }_{\beta }}\left|\mathrm{d}\beta \right|$ and ${\zeta }^{\prime }=-\sqrt{{\chi }_{\beta }}$, giving (87). With the other coordinates constant, the monotone thermal path attains it; any nonzero motion in another direction adds positive squared speed. For fixed $c>1$, the thermal path from $\beta =ϵ$ to $cϵ$ has entropy change tending in magnitude to $k_{\mathrm{B}}logc$ and length tending to $logc$ as $ϵ\downarrow 0$. This proves optimality. Finally $\zeta \to \infty$ at $\beta \downarrow 0$. □

Equivalently,

$$u_{\beta }=\frac{1}{2}{cosh\zeta ,\mathfrak{p}=sech\zeta ,\mathfrak{g}|}_{\ell ,\alpha ,\eta \mathrm{fixed}}=\mathrm{d}{\zeta }^2.$$

(88)

The entropy slope tends to $k_{\mathrm{B}}$ at large $\zeta$. Apart from stationary segments, equality in the weighted bound requires monotone thermal motion with $\ell ,\alpha ,\eta$ fixed; the uniform bound is strict for every nonconstant finite path. The pure-state boundary $\beta \to \infty$ instead has finite thermal distance. Neither limit is a physical-time statement.

Corollary 9

(Length of the fixed-observable dephasing path). On a fixed-v preparation path with fixed controls,

$$d{s}_v^2={\chi }_{\beta }\left(1+{\displaystyle \frac{\beta }{sinh\beta }}\right)\mathrm{d}{\beta }^2.$$

(89)

For monotone β between distinct finite endpoints,

$$\left|\Delta \zeta \right|<L_v<\sqrt{2}\left|\Delta \zeta \right|.$$

(90)

Along the dephasing trajectory $\nu ={\nu }_1+\tau$, as $\tau \to \infty$,

$$L_v\left(\tau \right)={\displaystyle \frac{log\tau }{\sqrt{2}}}+O\left(1\right),{\displaystyle \frac{\mathcal{S}\left({\rho }_{\tau }\right)-\mathcal{S}\left({\rho }_0\right)}{k_{\mathrm{B}}{L}_v\left(\tau \right)}}⟶{\displaystyle \frac{1}{\sqrt{2}}}.$$

(91)

Proof. 

Since $s^2=v/u_{\beta }$, $\mathrm{d}\ell /\mathrm{d}\beta ={\chi }_{\beta }/\left(2u_{\beta }\right)$. Substitution into (65) gives (89), using ${\chi }_{\beta }/u_{\beta }=1/sinh\beta$. The inequalities follow from $0<\beta /sinh\beta <1$ by integration. Near zero, $d{s}_v/\left|\mathrm{d}\beta \right|=\sqrt{2}/\beta +O\left(\beta \right)$. Also ${\beta }_{\tau }=u_{\tau }^{-1}+O\left(u_{\tau }^{-3}\right)$ and $u_{\tau }=\sqrt{v({\nu }_1+\tau )}$, so $L_v=(log\tau )/\sqrt{2}+O\left(1\right)$. Equation (81) gives the entropy change $(k_{\mathrm{B}}/2)log\tau +O\left(1\right)$. □

Within $\mathcal{P}$, fixed v forces squeezing. The comparison is with the thermal projection, not the distance between the same endpoints. Monotone $\beta$ minimizes length at fixed $v,q$. The factor $\sqrt{2}$ measures this prescribed path’s geometric overhead, not work, irreversible entropy production, or a constraint on all states with the same law.

For either induced almost-Born metric ${\mathbf{h}}_c$ on $T\mathcal{P}$, ${\mathbf{h}}_c((X,U),(X,U))=\mathfrak{g}(X,X)+\mathfrak{g}(U,U)$. Thus a lifted path $\Gamma$ satisfies $|\Delta \left({\pi }_T^{*}\mathcal{S}\right)|\le k_{\mathrm{B}}{L}_{{\mathbf{h}}_c}\left(\Gamma \right)$; the statistical shear preserves this pointwise comparison and the base projection. This is the eight-dimensional tangent construction on $\mathcal{P}$, not the minimal six-dimensional Hessian tangent carrier of a control slice. Finite-time dissipation geometry additionally depends on dynamics [32]. Only with a supplied physical speed bound $\parallel \dot{\lambda }{\parallel }_{\mathfrak{g}}\le V_{max}$ does $\Delta t\ge \left|\Delta \zeta \right|/V_{max}$ follow. No such speed bound or clock is selected here, even for the closing-gap calibration. The intrinsic boundary statement neither bounds the admissible entropies nor excludes shorter paths outside $\mathcal{P}$.

Remark 1

(Paths outside the preparation family). For centered endpoints ${\rho }_i={\rho }_{{\beta }_i,v}(0,0)$, the mixtures ${\rho }_t=(1-t){\rho }_0+t{\rho }_1$ preserve the same complete energy-momentum law but generally leave $\mathcal{P}$. Monotonicity under a binary prepare-and-forget channel [24,33] gives

$${\mathsf{g}}_{{\rho }_t}^{\mathrm{BKM}}({\rho }_1-{\rho }_0,{\rho }_1-{\rho }_0)\le {\displaystyle \frac{1}{t(1-t)}},L_{\mathrm{BKM}}\left({\rho }_t\right)\le \pi .$$

(92)

Appendix C.3 gives a direct infinite-rank proof. Thus arbitrarily large finite entropy differences admit bounded-length paths outside $\mathcal{P}$, even at the same measured law. This limits the interpretation, not the validity, of Proposition 6 and Corollary 9. No limiting density operator or bounded-time or bounded-work protocol is constructed.

7. Born Lifts, Invariant Area and Global Transport

7.1. The Statistical Metric and the Physical Form

The two responses involve different energy moments:

$$g_{\alpha \alpha }=\beta s^2{\displaystyle \frac{\langle H^2\rangle }{m^2{c}^4}},{\omega }_{\alpha \eta }={\displaystyle \frac{\langle H\rangle }{m{c}^2}},{\mathcal{C}}_{\alpha \alpha \eta }=-{\partial }_{\eta }{g}_{\alpha \alpha }.$$

(93)

The endomorphism defined by $\omega (X,Y)=g(\mathcal{J}X,Y)$ satisfies

$${\mathcal{J}}^2=-{\displaystyle \frac{M^2}{{\beta }^{2}F}}id.$$

(94)

For $v>0$, $F/M^2=e^v-sinhv{sech}^2\eta$ is not constant on any open interval. Hence no constant rescaling makes $g,\omega$ an orthogonally compatible complex pair. A doubled carrier supplies such compatibility, with the actual pullback determined below.

7.2. Statistical Shear and Global Nonisomorphism

For $(\mathcal{M},g,\nabla ,{\nabla }^{*})$ with $g>0$ and torsion-free metric-dual connections, set

$$D_0=\frac{1}{2}(\nabla +{\nabla }^{*})={\nabla }^{\mathrm{LC}},L=\frac{1}{2}(\nabla -{\nabla }^{*}),g(L_{X}Y,Z)=-\frac{1}{2}\mathcal{C}(X,Y,Z).$$

(95)

For $c=\nabla$ or ${\nabla }^{*}$, use the horizontal-vertical splitting of $T_u\left(T\mathcal{M}\right)$ and identify $X^{H,c}+U^V$ with $(X,U)$. The induced tensors and endomorphisms are

$$\begin{array}{cc}\hfill h_c((X,U),(Y,V))& =g(X,Y)+g(U,V),\hfill \\ \hfill {\eta }_c((X,U),(Y,V))& =g(X,V)+g(U,Y),\hfill \\ \hfill {\Omega }_c((X,U),(Y,V))& =g(X,V)-g(U,Y),\hfill \end{array}$$

(96)

$$I_c(X,U)=(-U,X),J_c(X,U)=(U,X),K_c(X,U)=(X,-U).$$

(97)

They form an almost-Born tuple, with $-I_c^2=J_c^2=K_c^2=id$ and $I_c{J}_c{K}_c=-id$ [9]. Other Born connections require their own hypotheses [34]. The following shear is recalled from [10]; its statistical hypotheses do not require flatness.

Proposition 7

(Statistical shear [10]). For the tangent projection ${\pi }_T$, let ${\left({\theta }_g\right)}_u\left(A\right)=g(u,{\left({\pi }_T\right)}_{*}A)$. Then

$${\Omega }_{\nabla }={\Omega }_{{\nabla }^{*}}=-\mathrm{d}{\theta }_g=:{\Omega }_B.$$

(98)

The bundle map

$$N_{u}A={\left(2L_{{\left({\pi }_T\right)}_{*}A}u\right)}^V,S=id+N,N^2=0$$

(99)

covers ${id}_{T\mathcal{M}}$ and obeys

$$\begin{array}{cccccc}\hfill S{I}_{\nabla }& =I_{{\nabla }^{*}}S,\hfill & \hfill S{J}_{\nabla }& =J_{{\nabla }^{*}}S,\hfill & \hfill S{K}_{\nabla }& =K_{{\nabla }^{*}}S,\hfill \\ \hfill S^{*}{h}_{{\nabla }^{*}}& =h_{\nabla },\hfill & \hfill S^{*}{\eta }_{{\nabla }^{*}}& ={\eta }_{\nabla },\hfill & \hfill S^{*}{\Omega }_B& ={\Omega }_B.\hfill \end{array}$$

(100)

Here $S^{*}t(A,B)=t(SA,SB)$ is a pointwise tensor pullback, not the pullback of a diffeomorphism.

The component change proves these identities without flatness; Appendix E.1 records the verification in these conventions. In our model, at $u=\xi {\partial }_x+\zeta {\partial }_y$,

$$\begin{array}{cc}\hfill {\Omega }_B& =a\mathrm{d}x\wedge \mathrm{d}\xi +b\mathrm{d}y\wedge \mathrm{d}\zeta +a^{\prime }\xi \mathrm{d}x\wedge \mathrm{d}y,\hfill \end{array}$$

(101)

$$\begin{array}{cc}\hfill 2L_{X}u& ={\displaystyle \frac{a^{\prime }}{a}}(X^x\zeta +X^y\xi ){\partial }_x+{\displaystyle \frac{a^{\prime }}{b}}{X}^x\xi {\partial }_y.\hfill \end{array}$$

(102)

At $y=0$, $S=id$ on the fiber, not necessarily to first order. Its effect on thermodynamic functions and the closure of temperature-dependent rescalings are distinguished in Appendix I.

For a torsion-free connection,

$$[X^{H,c},Y^{H,c}]-{[X,Y]}^{H,c}=-{\left(R^c(X,Y)u\right)}^V.$$

(103)

The intrinsic horizontal distributions are Lagrangian but not integrable on any open tangent-bundle neighbourhood, since (25) is nonzero. Its contraction vanishing on the zero section is insufficient. The general midpoint curvature identity is

$$R^{D_0}(X,Y)=\frac{1}{2}(R^{\nabla }+R^{{\nabla }^{*}})(X,Y)-[L_X,L_Y].$$

(104)

Here the averaged affine curvature accounts for nonzero midpoint curvature where $L=0$; it is absent only in the flat-pair specialisation.

Proposition 8

(Global nonisomorphism). The relativistic g and ∇ are geodesically complete, but ${\nabla }^{*}$ is incomplete. No global affine diffeomorphism identifies the dual connections, and no diffeomorphism identifies their full induced almost-Born tuples.

Proof. 

The lower bound $g\ge min\{a\left(0\right),b\}(\mathrm{d}{x}^2+\mathrm{d}{y}^2)$ gives metric and geodesic completeness. The ∇ equations are $\ddot{y}=0$ and ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left(a{\left(y\right)}^2\dot{x}\right)=0$; their solutions extend for all t because $a\ge a\left(0\right)>0$. For the dual connection choose $p\ne 0$, $y_0>0$, and

$$x=x_0+pt,tanh{\displaystyle \frac{y\left(t\right)}{2}}=tanh{\displaystyle \frac{y_0}{2}}{e}^{kt},k=\sqrt{2}\left|p\right|s^2{e}^v.$$

(105)

Then $\dot{y}=ksinhy$ and $\ddot{y}=(a^{\prime }/b)p^2$, the dual geodesic equation. Escape to $y=\infty$ occurs at $t_{*}=k^{-1}logcoth(y_0/2)<\infty$. Affine diffeomorphisms preserve completeness. A diffeomorphism preserving the full induced tuple must be a tangent lift of a connection-preserving base isometry followed by a target-parallel fiber translation ([9] Propositions 4.22, 4.23 and 4.29), giving the last claim. □

This is an explicit statistical-completeness obstruction [35], not a physical-time singularity. Appendix E gives specified tuple-preserving transports conjugated pointwise by S.

7.3. The Physical Pullback and Invariant Area

Using the mean-momentum primitive from (42), define

$$\mathcal{Q}\left(y\right)=e^{v/2}sinhy,{\mathcal{Q}}^{\prime }=M,\sigma (x,y)={\displaystyle \frac{\mathcal{Q}\left(y\right)}{a\left(y\right)}}{\partial }_x.$$

(106)

Since ${\sigma }^{*}{\theta }_g=\mathcal{Q}\mathrm{d}x$,

$${\sigma }^{*}{\Omega }_B=-\mathrm{d}\left(\mathcal{Q}\mathrm{d}x\right)=M\mathrm{d}x\wedge \mathrm{d}y=\omega .$$

(107)

The connection gives ${\nabla }_x\sigma =0$ and ${\nabla }_y\sigma =(M/a){\partial }_x$. The normal calculation in Appendix E.2 gives $\mathrm{II}(X,\sigma )=0$. Thus the genuine map $s_B=Tj\circ \sigma =(a,ax,y;0,\mathcal{Q},0)$ has

$$\begin{array}{cc}\hfill s_B^{*}{\overline{\Omega }}_B& =\omega ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill h_{\mathrm{ind}}:=s_B^{*}{\overline{h}}_{\overline{\nabla }}& ={\sigma }^{*}{h}_{\nabla }=a\mathrm{d}{x}^2+(b+M^2/a)\mathrm{d}{y}^2,\hfill \end{array}$$

(108)

$$\begin{array}{cc}\hfill det{h}_{\mathrm{ind}}& =ab+M^2.\hfill \end{array}$$

(109)

The standard pointwise Wirtinger bound $\left|\Omega (A,B)\right|\le {\mathcal{A}}_h(A,B)$ holds for a positive Hermitian pair, with equality on an independent plane exactly when it is I-invariant [36]. Here the specified lift gives the stronger exact remainder below.

Theorem 6

(Invariant area and normal-potential independence). For the specified lift,

$$\begin{array}{|c|}\hline {\mathcal{A}}_{h_{\mathrm{ind}}}{(u,w)}^2={\mathcal{A}}_g{(u,w)}^2+\omega {(u,w)}^2.\\ \hline\end{array}$$

(110)

For independent $u,w$ the inequality is strict and the scalar

$$\mathcal{R}={\displaystyle \frac{{\mathcal{A}}_{h_{\mathrm{ind}}}{(u,w)}^2}{\omega {(u,w)}^2}}=1+{\displaystyle \frac{ab}{M^2}}>1$$

(111)

is basis- and coordinate-independent. After affine normalisation, the same pullback pair and identity hold for every positive minimal single-chart potential in Theorem 3.

Proof. 

For $J=u^x{w}^y-u^y{w}^x$, the two-dimensional Gram determinant gives ${\mathcal{A}}_k^2=(detk)J^2$, while $\omega (u,w)=MJ$. Equation (109) proves the identity and strictness. Both numerator and denominator acquire the same squared determinant under basis or coordinate changes.

Write $e_x=j_{*}{\partial }_x$ and $e_y=j_{*}{\partial }_y$. Every allowed potential has $G(e_x,e_x)=a$, $G(e_x,e_y)=0$, $G(e_y,e_y)=b$ on j. In its flat affine splitting,

$$d{s}_B\left({\partial }_x\right)=(e_x,0),d{s}_B\left({\partial }_y\right)=(e_y,(M/a)e_x).$$

Both components are tangent to $j\left(\mathcal{M}\right)$, so the Born metric and two-form give the same pair independently of $\Lambda$ or its normal derivatives. Affine changes carry the result to any minimal placement. □

Corollary 10

(Energy fluctuations and sharpness). For the relativistic family,

$$\mathcal{R}=1+{\beta }^2{\displaystyle \frac{\langle H^2\rangle }{{\langle H\rangle }^2}}=1+{\beta }^2\left(e^v-sinhv{sech}^{2}y\right).$$

(112)

It equals $1+{\beta }^{2}coshv$ at $y=0$ and tends to $1+{\beta }^2{e}^v$ as $\left|y\right|\to \infty$. The coefficient one in the area inequality is optimal over faithful preparations, even at fixed joint energy-momentum law.

Proof. 

Use $ab={\beta }^{2}F$ and the energy moments. The fixed-v sequence of Theorem 5 gives $\mathcal{R}\left(0\right)\downarrow 1$ while preserving that law. Every finite member remains strict; infinite boost at fixed preparation is not the saturation limit. □

Equivalently, $F/M^2=1+Var\left(H\right)/{\langle H\rangle }^2$: the normalised excess is set by relative energy fluctuations and the preparation parameter.

Proposition 9

(Geometric identification of the preparation). With the prescribed contrast and two-form normalisation, set $\mathfrak{r}=\mathcal{R}-1$, $r_{-}:=inf\mathfrak{r}$ and $r_{+}:=sup\mathfrak{r}$ over the control plane. Then

$$\beta ={\left[r_{+}\left(2r_{-}{r}_{+}\right)\right]}^{1/4},v=\frac{1}{2}log{\displaystyle \frac{r_{+}}{2r_{-}{r}_{+}}}.$$

(113)

Alternatively, let $k_{-}:=inf(-{\mathcal{K}}_g)$, $k_{+}:=sup(-{\mathcal{K}}_g)$ and $\xi =k_{+}/k_{-}$. The curvature range fixes

$$b=k_{-}^{-1},v=\frac{1}{2}log{\displaystyle \frac{\xi }{2-\xi }},bv=\frac{\beta }{2}coth(\beta /2),s^2=\beta /b.$$

(114)

The equation for $\beta >0$ has a unique solution. A global isometry between two such metrics, or an isomorphism of their full intrinsic almost-Born tuples, therefore preserves $\beta ,s$ and entropy.

Proof. 

Equation (112) gives $r_{-}={\beta }^{2}coshv$ and $r_{+}={\beta }^2{e}^v$; $2coshv-e^v=e^{-v}$ proves (113). Equation (23) gives $k_{-}=1/b$ and $k_{+}=2e^{2v}/\left[b(1+e^{2v})\right]$. Finally $bv=\beta u_{\beta }$, and $tcotht$ increases strictly from one to infinity for $t>0$. Isometries preserve the curvature range; the intrinsic Born classification used in Proposition 8 supplies the base isometry. □

When the symmetric line and a nonzero finite rapidity ${\eta }_1$ are calibrated, global extrema are unnecessary. Put ${\delta }_r=[\mathfrak{r}\left({\eta }_1\right)-\mathfrak{r}\left(0\right)]/{tanh}^2{\eta }_1={\beta }^{2}sinhv$. Then

$$\beta ={[\mathfrak{r}{\left(0\right)}^2-{\delta }_r^2]}^{1/4},v=artanh{\displaystyle \frac{{\delta }_r}{\mathfrak{r}\left(0\right)}}.$$

(115)

The finite reconstruction requires $\mathfrak{r}\left(0\right)>{\delta }_r>0$; out-of-domain measurements do not specify a preparation by these formulas. These are structural identities, not robust noisy-data estimators: small differences and near-cancellation can be ill-conditioned. They fix entropy and $k_{\mathrm{B}}{T}_{\mathrm{prep}}/E_{*}$, not the energy gap. With independent energy moments, local data give ${\beta }^2=(\mathcal{R}-1)/(1+Var\left(H\right)/{\langle H\rangle }^2)$. An unmarked ambient completion or mere existence of a pointwise shear is not covered by the global identification statement.

Remark 2

(Singular sharpness limit). At fixed v and controls,

$$\begin{array}{cccc}\hfill {\left(h_{\beta }\right)}_{\alpha \alpha }& =vF{\beta }^2+O\left({\beta }^4\right),\hfill & \hfill {\left(h_{\beta }\right)}_{\eta \eta }& ={\displaystyle \frac{M^2}{vF}}{\beta }^{-2}+O\left(1\right),\hfill \\ \hfill det{h}_{\beta }& =M^2+{\beta }^{2}F.\hfill \end{array}$$

Thus the area ratio has a finite limit, but the metric has no finite positive-definite tensor limit under fixed control identification. A fixed regular coordinate change does not alter this; a preparation-dependent rescaling changes the comparison.

For relatively compact oriented U of positive area,

$${Area}_{h_{\mathrm{ind}}}\left(U\right)>{\int }_U\left|\omega \right|\ge \left|{\int }_U\omega \right|,\mathrm{d}{A}_{h_{\mathrm{ind}}}=\sqrt{\mathcal{R}}\left|\omega \right|.$$

(116)

The whole surface has infinite area: a width-L strip to rapidity Y has area greater than $L{e}^{v/2}(sinhY-sinh{y}_0)$. Adding a closed one-form to $\mathcal{Q}\mathrm{d}x$ preserves the symplectic pullback but may change its metric. Conversely, ${\sigma }^{*}{h}_{\nabla }=g$ forces $\nabla \sigma =0$ and hence ${\sigma }^{*}{\Omega }_B=0$. A nonzero pairing requires a changed section metric, not a dynamically selected section. The action-valued comparison is $\hslash {\mathcal{A}}_{h_{\mathrm{ind}}}\ge |\hslash \omega |$; it is neither a minimum spatial area nor restoration of information lost to dephasing.

8. Entropy Endpoints and Discrete Kinetic Mechanics

Contrast-based Hamilton-Jacobi constructions [11,12] and variational generating functions [13] are established. Here the exact contrast gives a global map. Supply $\kappa >0$ and set

$$S_{\kappa }(q_0,q_1)=\kappa \mathcal{D}(q_1\parallel q_0)=\frac{\kappa }{2}[a\left(y_1\right){(x_1-x_0)}^2+b{(y_1-y_0)}^2].$$

Use ${\omega }_{\mathrm{can}}=\mathrm{d}x\wedge \mathrm{d}{p}_x+\mathrm{d}y\wedge \mathrm{d}{p}_y$ and ${\iota }_{X_{\mathcal{H}}}{\omega }_{\mathrm{can}}=\mathrm{d}\mathcal{H}$.

Proposition 10

(Global exact symplectic map). The generating equations

$$p_0=-{\partial }_{q_0}{S}_{\kappa },p_1={\partial }_{q_1}{S}_{\kappa }$$

(117)

define a global exact symplectic diffeomorphism:

$$\begin{array}{cccc}\hfill y_1& =y_0+{\displaystyle \frac{p_{y0}}{\kappa b}},\hfill & \hfill x_1& =x_0+{\displaystyle \frac{p_{x0}}{\kappa a\left(y_1\right)}},\hfill \\ \hfill p_{x1}& =p_{x0},\hfill & \hfill p_{y1}& =p_{y0}+{\displaystyle \frac{a^{\prime }\left(y_1\right)p_{x0}^2}{2\kappa a{\left(y_1\right)}^2}}.\hfill \end{array}$$

(118)

For $\kappa =\mu /\epsilon$ with supplied $\mu ,\epsilon >0$, it is exactly the Lie-Trotter splitting

$$F_{\epsilon }^{+}={\phi }_{{\mathcal{H}}_x}^{\epsilon }\circ {\phi }_{{\mathcal{H}}_y}^{\epsilon },{\mathcal{H}}_x={\displaystyle \frac{p_x^2}{2\mu a\left(y\right)}},{\mathcal{H}}_y={\displaystyle \frac{p_y^2}{2\mu b}}.$$

(119)

Reversing the contrast reverses the order of subflows.

The generating equations, global inverse and split flows are verified in Appendix J; the symplectic argument is the standard generating-function construction [13]. For the canonical bracket and $v_p=g^{-1}p/\mu$,

$$\{{\mathcal{H}}_x,{\mathcal{H}}_y\}=-{\displaystyle \frac{a^{\prime }{p}_x^2{p}_y}{2{\mu }^2{a}^{2}b}}={\displaystyle \frac{\mu }{6}}\mathcal{C}(v_p,v_p,v_p).$$

(120)

The last equality uses the three permutations of ${\mathcal{C}}_{xxy}=-a^{\prime }$. Thus directed contrast asymmetry also measures noncommutation of this specified kinetic split. The subflows commute identically exactly when a is constant. The split uses the physical control coordinates and is not the translation-boost operator commutator. It preserves $p_x$, but is generally not the exact combined kinetic flow and need not conserve its energy exactly. Symmetric half-step composition gives standard Strang splitting. The scale and clock remain inputs; the fixed-$q_0$ Lagrangian graph ${\mathrm{d}}_{q_1}{S}_{\kappa }$ cannot coincide on an open set with the non-Lagrangian physical section; Appendix I gives the exterior-derivative obstruction.

9. Discussion and Conclusions

The prescribed finite contrast has a minimal positive Hessian realisation. Finite comparisons determine its minimal affine placement and first jet, while leaving normal-Hessian freedom. Positive global convexity is impossible in every finite dimension on the whole control plane, but every bounded rapidity strip admits a convex realisation. These statements concern preservation of a scalar contrast, not finite-level quantum compression or metric completion.

The dephasing family separates unchanged commuting statistics from lost translation distinguishability. Its limit is classical at the level of the specified contrast, not a trace-norm limiting density operator. Unlike fixed-width heating, this sequence changes the preparation width. Calibrating the preparation Hamiltonian makes relative entropy a free-energy difference and yields the explicit spectral/ergotropy decomposition, work-susceptibility bound and finite-work control-area estimate. Bounding both relativistic energy and the boost second moment bounds entropy; closing the preparation gap can instead permit unbounded entropy at finite limiting temperature.

Entropy change is bounded by length intrinsically in $\mathcal{P}$. The fixed-v path has an additional squeezing contribution, with an asymptotic factor $\sqrt{2}$ relative to its thermal projection. Neither statement holds for all paths preserving the observable law: mixtures outside $\mathcal{P}$ connect the same finite endpoints with BKM length at most $\pi$. The restriction on the preparation family, rather than the observable law alone, is essential.

The chosen Born lift has an exact area identity independent of the allowed normal potential, not of the lift itself. Its numerical invariants identify the dimensionless preparation but not an absolute temperature; its sharpness limit is anisotropically singular. The dual tuples are pointwise shear-related but globally inequivalent by affine completeness. With supplied kinetic scales, the contrast also generates an exact symplectic splitting, not generally the exact continuous kinetic flow. Auxiliary transport and ambient-boundary results concern the specified metrics.

9.1. Scope and Remaining Questions

No statistical, affine or noise parameter is identified with physical time. Nor is statistical area or curvature identified with spacetime geometry. Gravitational localisation estimates ${\ell }_{\mathrm{q}}\sim \hslash c/E_{\mathrm{loc}}$ and $r_{\mathrm{g}}\sim G_{\mathrm{N}}{E}_{\mathrm{loc}}/c^4$ require concentrated physical energy and gravitational dynamics [37,38]; a passive boost does not create a trapped surface. The massive model has no direct photon limit. Appendix H separates preparation, detector, modular and gravitational temperature assignments.

Remaining questions concern which further conditions select the normal potential, which larger restricted control domains admit convex realisations, and whether the ambient coordinates admit an independent quantum-state model. Regulated finite-level approximations require explicit energy and control cutoffs. None is assumed in the exact results above.

Appendix C.1. Finite-Entropy Gibbs Bound

Proof 

(Proof of Proposition 5). The specified s gives $Tr\rho Q_s\le u_{max}$. In its oscillator basis let $p_n=\langle h_n,\rho h_n\rangle$ and $q_n^{\left(\gamma \right)}=(1-e^{-\gamma })e^{-\gamma n}$. The log-sum inequality on $\{0,\dots ,N,>N\}$, whose reference tail is $e^{-\gamma (N+1)}$, gives

$$H(p_0,\dots ,p_N,p_{>N})\le \gamma \sum _{n\ge 0}n{p}_n-log(1-e^{-\gamma }).$$

Increasing N yields the diagonal Shannon entropy. Concavity of $-tlogt$ applied to the spectral resolution of $\rho$, followed by Tonelli’s theorem, bounds its von Neumann entropy by that diagonal entropy. Thus, without assuming finiteness in advance,

$$\mathcal{S}\left(\rho \right)/k_{\mathrm{B}}\le \gamma Tr\rho Q_s+log{Z}_{\gamma }\le \gamma u_{max}+log{Z}_{\gamma }(\gamma >0).$$

(C1)

Minimising sets $u_{\gamma }=u_{max}$ and gives the claimed bound; that seed saturates both budgets. If $u_{max}=1/2$, positivity of $Q_s-1/2$ forces support in its one-dimensional kernel, hence entropy zero. A smaller budget contradicts the ground-energy bound. □

Appendix C.2. Covariance at the Symmetric Preparation

At $\eta =0$, equations (39) and (41) give

$${\displaystyle \frac{detV}{{\omega }_{\alpha \eta }^2}}={\displaystyle \frac{vsinhv}{s^4}}={\displaystyle \frac{sinhv}{4v}}{coth}^2(\beta /2)\ge {\displaystyle \frac{1}{4}}.$$

(C2)

The last step uses $sinhv\ge v$ and $coth(\beta /2)>1$. Along the fixed-v sequence in theorem 5, the covariance ratio diverges as $\beta \downarrow 0$, while the entropy-metric ratio tends to zero. This makes the coexistence of ordinary uncertainty and the failed entropy-area bound explicit.

For fixed s, the limit $\beta \to \infty$ approaches a pure oscillator state. The BKM metric in equation (19) is not a finite pure-state metric in that limit. Distinct rank-one projectors have infinite Umegaki relative entropy when their supports are not included in one another; replacing this singular limit with pure-state covariance geometry would change the divergence under discussion.

Appendix C.3. Mixture Paths and Unrestricted BKM Length

For the endpoints of Remark 1, set $\Delta ={\rho }_1-{\rho }_0$ and $R={\rho }_t$, $0<t<1$. All are trace class, R is faithful, and each path for finite endpoints has finite quadratic moments and entropy. The inequalities

$$-{\displaystyle \frac{R}{1-t}}\le \Delta \le {\displaystyle \frac{R}{t}}$$

make $B=R^{-1/2}\Delta R^{-1/2}$ a bounded self-adjoint form extension. With $a={(1-t)}^{-1}$, $b=t^{-1}$, functional calculus gives $B^2\le (b-a)B+abI$. Since $TrRB=Tr\Delta =0$, $TrR{B}^2\le ab$.

In an eigenbasis of R, with eigenvalues ${\lambda }_i>0$, the BKM metric [19,33] is the nonnegative spectral sum

$$\begin{array}{cc}\hfill {\mathsf{g}}_R^{\mathrm{BKM}}(\Delta ,\Delta )& =\sum _{i,j}{\left|B_{ij}\right|}^2{\lambda }_i{\lambda }_j{\displaystyle \frac{log{\lambda }_i-log{\lambda }_j}{{\lambda }_i-{\lambda }_j}}\hfill \\ \hfill & \le \sum _{i,j}{\left|B_{ij}\right|}^2{\displaystyle \frac{{\lambda }_i+{\lambda }_j}{2}}=TrR{B}^2\le {\displaystyle \frac{1}{t(1-t)}}.\hfill \end{array}$$

(C3)

The quotient takes its continuous diagonal value. The scalar inequality uses the logarithmic mean’s lower bound by the geometric mean, followed by arithmetic-geometric mean. Nonnegative partial sums and bounded B justify the infinite sum. Integration gives the improper length ${\int }_0^1{\left[t(1-t)\right]}^{-1/2}\mathrm{d}t=\pi$.

The observable law is preserved by linearity. For distinct ${\beta }_i$, a strict mixture of the two seed kernels contains two different $e^{-{\nu }_i{(r-r^{\prime })}^2/2}$ factors and is not a single oscillator Gaussian. It therefore leaves $\mathcal{P}$. Taking ${\beta }_1\downarrow 0$ yields arbitrarily large finite endpoint entropy differences without violating an intrinsic bound on $\mathcal{P}$. The bound is also the metric consequence of the binary-channel data-processing inequality

$$D({\rho }_t\parallel {\rho }_{t+h})\le (1-t)log{\displaystyle \frac{1-t}{1-t-h}}+tlog{\displaystyle \frac{t}{t+h}},0<t,t+h<1.$$

Appendix E.1. Verification of the Statistical Shear

Proof 

(Proof of Proposition 7). Statistical symmetry gives

$$\mathrm{d}{\theta }_g(X^{H,c},Y^{H,c})=\left(c_{X}g\right)(u,Y)-\left(c_{Y}g\right)(u,X)=0.$$

The mixed pairing is $-\mathrm{d}{\theta }_g(X^{H,c},V^V)=g(X,V)$, and the vertical-vertical pairing vanishes. This proves (98) and nondegeneracy. The identity $X^{H,{\nabla }^{*}}-X^{H,\nabla }={\left(2L_{X}u\right)}^V$ shows that S preserves split components; the constant matrices prove all intertwining relations. Since N is vertical and vanishes on vertical vectors, $S^{-1}=id-N$. □

Appendix E.2. Midpoint Mechanics and Normal Restriction Terms

For $E\left(u\right)=g(u,u)/2$, define ${\vartheta }_u\left(A\right)=g(L_{u}u,{\left({\pi }_T\right)}_{*}A)$. The geodesic sprays satisfy

$$\begin{array}{cccc}\hfill Z_{\nabla }& =Z_{D_0}-{\left(L_{u}u\right)}^V,\hfill & \hfill Z_{{\nabla }^{*}}& =Z_{D_0}+{\left(L_{u}u\right)}^V,\hfill \\ \hfill {\iota }_{Z_{D_0}}{\Omega }_B& =\mathrm{d}E,\hfill & \hfill {\iota }_{Z_{\nabla }}{\Omega }_B& =\mathrm{d}E+\vartheta ,\hfill & \hfill {\iota }_{Z_{{\nabla }^{*}}}{\Omega }_B& =\mathrm{d}E-\vartheta .\hfill \end{array}$$

(E1)

These follow from the spray coefficients and ${\iota }_{W^V}{\Omega }_B=-g(W,\mathrm{d}q)$. With ${\iota }_{X_H}{\Omega }_B=\mathrm{d}H$, the midpoint is Hamiltonian with $H=E$; the dual sprays have opposite symplectic defects $\pm \mathrm{d}\vartheta$. Here $\vartheta =a^{\prime }\xi \zeta \mathrm{d}x+(a^{\prime }/2){\xi }^2\mathrm{d}y$.

For the constant-${\lambda }_{\perp }$ realisation, set $e_x=(0,a,0)$, $e_y=(a^{\prime },a^{\prime }x,1)$ and $n=(1,x,0)$ on j. Then $G(n,e_a)=0$, $G(n,n)={\lambda }_{\perp }$ and $j^{*}G=g$. Decomposing the ambient dual derivatives into tangent and normal parts,

$${\overline{\nabla }}_{j_{*}X}{j}_{*}Y=j_{*}\left({\nabla }_{X}Y\right)+\mathrm{II}(X,Y)$$

(and similarly for the dual), gives only

$${\mathrm{II}}_{yy}=a^{\prime \prime }n,{\mathrm{II}}_{xx}^{*}=-{\lambda }_{\perp }^{-1}n.$$

(E2)

Indeed, $j_{xx}=0$, $j_{xy}=(a^{\prime }/a)e_x$, $j_{yy}=a^{\prime \prime }n$. Differentiating $G(n,e_a)=0$ using metric duality and ${\partial }_{x}n=e_x/a$, ${\partial }_{y}n=0$ gives the dual normal coefficient. The flat ambient statistical Gauss equation is

$$\begin{array}{cc}\hfill g(R^{\nabla }(X,Y)Z,W)& =G(\mathrm{II}(Y,Z),{\mathrm{II}}^{*}(X,W))\hfill \\ \hfill & -G(\mathrm{II}(X,Z),{\mathrm{II}}^{*}(Y,W)).\hfill \end{array}$$

(E3)

It follows from the tangent part of $\overline{R}=0$ and dual orthogonality. For $X=W={\partial }_x$, $Y=Z={\partial }_y$, it gives $-a^{\prime \prime }$, recovering (25); the normal stiffness cancels.

The split injection and genuine tangent differential are different:

$$\begin{array}{cc}\hfill {\Xi }_u(X^{H,\nabla }+U^V)& ={\left(j_{*}X\right)}^{H,\overline{\nabla }}+{\left(j_{*}U\right)}^V,\hfill \end{array}$$

(E4)

$$\begin{array}{cc}\hfill T\left(Tj\right)(X^{H,\nabla }+U^V)& ={\Xi }_u(X^{H,\nabla }+U^V)+\mathrm{II}{(X,u)}^V.\hfill \end{array}$$

(E5)

The first preserves the tuple pointwise; the second has normal defects. For $A=X^{H,\nabla }+U^V$, $B=Y^{H,\nabla }+V^V$, orthogonality gives

$$\begin{array}{cccc}\hfill {\left(Tj\right)}^{*}{\overline{\Omega }}_B& ={\Omega }_B,\hfill & \hfill {\left(Tj\right)}^{*}{\overline{\eta }}_{\overline{\nabla }}& ={\eta }_{\nabla },\hfill \\ \hfill {\left(Tj\right)}^{*}{\overline{h}}_{\overline{\nabla }}(A,B)& =h_{\nabla }(A,B)+G(\mathrm{II}(X,u),\mathrm{II}(Y,u)).\hfill \end{array}$$

(E6)

For example, $T\left(Tj\right)I_{\nabla }A-{\overline{I}}_{\overline{\nabla }}T\left(Tj\right)A=\mathrm{II}{(X,u)}^H-\mathrm{II}{(U,u)}^V$. The dual formulas use ${\mathrm{II}}^{*}$. These terms obstruct descent of the full strongly integrable ambient tuple.

Appendix E.3. Transport Preserving the Full Tuple

The splitting ${\chi }_c(X^{H,c}+U^V)=(X,U)$ identifies $T\left(T\mathcal{M}\right)$ with ${\pi }_T^{*}T\mathcal{M}\oplus {\pi }_T^{*}T\mathcal{M}$. Define

$${\mathcal{A}}^c={\chi }_c^{-1}({\pi }_T^{*}{D}_0\oplus {\pi }_T^{*}{D}_0){\chi }_c.$$

(E7)

Since $D_{0}g=0$ and the Born endomorphisms have constant split matrices, this connection preserves the full tuple. The identity ${\chi }_{{\nabla }^{*}}S={\chi }_{\nabla }$ gives

$${\mathcal{P}}_{\gamma }^{{\nabla }^{*}}{S}_{\gamma \left(0\right)}=S_{\gamma \left(1\right)}{\mathcal{P}}_{\gamma }^{\nabla },$$

(E8)

where ${\mathcal{P}}_{\gamma }^c$ denotes its parallel transport along a piecewise smooth curve in $T\mathcal{M}$. Its curvature is ${\chi }_c^{-1}diag(R^{D_0}(X,Y),R^{D_0}(X,Y)){\chi }_c$, with $X,Y$ the projected arguments. Hence its holonomy is the diagonal midpoint holonomy; for this model it is $SO\left(2\right)$, by Appendix F.

This transport generally has torsion. For $c=D_0+\epsilon L$, $\epsilon =\pm 1$, the lift brackets give

$$\begin{array}{cc}\hfill T^{{\mathcal{A}}^c}(X^{H,c},Y^{H,c})& ={\left(R^c(X,Y)u\right)}^V,\hfill \\ \hfill T^{{\mathcal{A}}^c}(X^{H,c},Y^V)& =-\epsilon {\left(L_{X}Y\right)}^V,T^{{\mathcal{A}}^c}(X^V,Y^V)=0.\hfill \end{array}$$

(E9)

It is generally distinct from the Levi-Civita connection of $h_c$. Thus the exact conjugacy concerns a specified pair of tuple-preserving transports; it neither restores horizontal integrability nor identifies the affine sprays.

Appendix F.1. An Exact Comparison of the Base and Lifted Metrics

Compare $h:=h_{\mathrm{ind}}$ with g through their positive metric matching map [10]. Write $D^h$ for its Levi-Civita connection and, in $(x,y)$ coordinates, set

$$Q=diag\left(1,\sqrt{1+M^2/\left(ab\right)}\right),\widehat{D}=Q{D}^h{Q}^{-1},B_{\mathrm{rel}}=D_0-\widehat{D}.$$

(F1)

Since $g(QX,QY)=h(X,Y)$, both compared connections are g-metric. Fix the oriented orthonormal frame $e_1=a^{-1/2}{\partial }_x$, $e_2=b^{-1/2}{\partial }_y$ and quarter-turn $J_0{e}_1=e_2$. Define

$$f_g={\displaystyle \frac{a^{\prime }}{2\sqrt{ab}}},f_h={\displaystyle \frac{a^{\prime }}{2\sqrt{ab+M^2}}},d=f_g-f_h.$$

(F2)

The connection matrices are $-f_g{J}_0\mathrm{d}x$ and $-f_h{J}_0\mathrm{d}x$. Hence

$$\begin{array}{cc}\hfill B_{\mathrm{rel}}& =-d{J}_0\mathrm{d}x,F_{\mathrm{rel}}:=R^{D_0}-Q{R}^{D^h}{Q}^{-1}=d^{\prime }{J}_0\mathrm{d}x\wedge \mathrm{d}y,\hfill \end{array}$$

(F3)

$$\begin{array}{cc}\hfill U_{\gamma }& :={\left({\widehat{P}}_{\gamma }\right)}^{-1}{P}_{\gamma }^{D_0}=exp\left[J_0{\int }_{\gamma }d\left(y\right)\mathrm{d}x\right].\hfill \end{array}$$

(F4)

Here P denotes parallel transport and $J_0$ in the last expression acts at the starting point. These formulas follow by subtraction and integration: $\mathfrak{so}\left(2\right)$ is Abelian, so $B_{\mathrm{rel}}\wedge B_{\mathrm{rel}}=0$ and no path ordering remains. For $\gamma =\partial \Sigma$ positively oriented, its angle is ${\int }_{\Sigma }({\mathcal{K}}_g\mathrm{d}{A}_g-{\mathcal{K}}_h\mathrm{d}{A}_h)$, since ${\mathcal{K}}_g\mathrm{d}{A}_g=-f_g^{\prime }\mathrm{d}x\wedge \mathrm{d}y$ and likewise for h.

With $a_0=a\left(0\right)$, $M_0=M\left(0\right)$,

$$d\left(0\right)=0,d^{\prime }\left(0\right)={\displaystyle \frac{a^{\prime \prime }\left(0\right)}{2}}\left({\displaystyle \frac{1}{\sqrt{a_{0}b}}}-{\displaystyle \frac{1}{\sqrt{a_{0}b+M_0^2}}}\right)>0.$$

(F5)

Thus the connection defect vanishes on the symmetric line while its curvature defect survives. Nonzero ${\mathcal{K}}_g$ gives full midpoint holonomy $SO\left(2\right)$; variable rectangular loops in equation (A33) also realize every relative rotation. This intrinsic rank-two transport is distinct from the rank-three ambient transport considered next.

Appendix F.2. Spin Holonomy of the Explicit Completion Family

Proposition A1

(Holonomy of the explicit Hessian family). Fix $\beta ,s>0$. For each ${\lambda }_{\perp }>0$, let $(\mathcal{N},G)$ be the positive Hessian manifold in theorem 2, with $a\left(Y\right)=\beta s^{2}F\left(Y\right)$ and $b=\beta /s^2$. Its Levi-Civita connection has holonomy $SO\left(3\right)$, and its geometric spin lift has holonomy $Spin\left(3\right)\simeq SU\left(2\right)$.

Proof. 

At $j(0,0)$ use the G-orthonormal basis $E_0={\lambda }_{\perp }^{-1/2}{\partial }_z$, $E_1=\sqrt{a_0}{\partial }_w$, $E_2=b^{-1/2}{\partial }_Y$. Put $r={\left(a_0\sqrt{{\lambda }_{\perp }}\right)}^{-1}$ and $t=\sqrt{{\lambda }_{\perp }}{a}^{\prime \prime }\left(0\right)/b$, both positive. The only nonzero components of the Hessian cubic $\overline{C}={\partial }^3\Phi$ are ${\overline{C}}_{011}=-r$, ${\overline{C}}_{022}=-t$ and their permutations. The flat-pair identity $R_{ij}^{\mathrm{LC}}=-[{\overline{C}}_i^{♯},{\overline{C}}_j^{♯}]/4$ yields

$$R_{01}^{\mathrm{LC}}=-\frac{r^2}{4}{J}_{01},R_{02}^{\mathrm{LC}}=-\frac{t^2}{4}{J}_{02},R_{12}^{\mathrm{LC}}=\frac{rt}{4}{J}_{12},$$

(F6)

where $J_{ij}{E}_i=E_j$, $J_{ij}{E}_j=-E_i$ and it vanishes on the remaining basis vector. These curvature values span $\mathfrak{so}\left(3\right)$, proving the holonomy claim by Ambrose-Singer [39]. Since $a^{\prime \prime }>0$, the domain is $0<z<a\left(Y\right)+b/\left({\lambda }_{\perp }{a}^{\prime \prime }\left(Y\right)\right)$, with $w,Y$ unrestricted, and is contractible. Its oriented frame bundle therefore has a spin lift; the lifted curvature spans $\mathfrak{su}\left(2\right)$, giving full $SU\left(2\right)$ holonomy. □

This calculation concerns the explicit one-parameter family in theorem 2; it does not classify the holonomy of all dimension-minimal contrast-preserving Hessian realisations. The spin lift of $T\mathcal{N}$ supplies a geometric spin connection and an associated complex doublet. Identifying this doublet with particle spin or an internal interaction requires an independent physical identification and coupling.

Appendix H.1. Stationarity and Constrained Entropy Slopes

For $H_{\zeta }=m{c}^{2}cosh(r-\zeta )$, every finite preparation is nonstationary. Its kernel is

$$\rho \left(q\right)(r,r^{\prime })={\displaystyle \frac{e^{-\mathrm{i}\alpha (sinhr-sinh{r}^{\prime })}}{\sqrt{2\pi v}}}exp\left[-{\displaystyle \frac{{(r+r^{\prime }-2\eta )}^2}{8v}}-{\displaystyle \frac{\nu }{2}}{(r-r^{\prime })}^2\right],\nu =u_{\beta }/s^2.$$

(H1)

It is everywhere nonzero; multiplication by either $cosh(r-\zeta )$ or $cosh(r^{\prime }-\zeta )$ preserves square integrability. The nonzero Hilbert-Schmidt commutator kernel is $m{c}^2[cosh(r-\zeta )-cosh(r^{\prime }-\zeta )]\rho \left(q\right)(r,r^{\prime })$. Nonstationarity excludes Gibbs or KMS equilibrium for $H_{\zeta }$.

The operator $e^{-\gamma H}$ is not trace class: on a bounded interval its multiplier has a positive lower bound, giving a divergent diagonal sum in any infinite orthonormal family supported there. The finite integral $\int e^{-\gamma m{c}^{2}coshr}\mathrm{d}r$ is therefore a classical partition integral, not a quantum trace. Confinement changes the problem.

At fixed $\eta$, the independent preparation coordinates $(\beta ,v)$ give

$$\begin{array}{cccc}\hfill E_H& =m{c}^2{e}^{v/2}cosh\eta ,\hfill & \hfill \mathrm{d}{E}_H& =\frac{1}{2}{E}_H\mathrm{d}v,\hfill \\ \hfill \mathrm{d}\mathcal{S}& =-k_{\mathrm{B}}\beta {\chi }_{\beta }\mathrm{d}\beta ,\hfill & \hfill \mathrm{d}{E}_H\wedge \mathrm{d}\mathcal{S}& =\frac{1}{2}{k}_{\mathrm{B}}{E}_H\beta {\chi }_{\beta }\mathrm{d}\beta \wedge \mathrm{d}v\ne 0.\hfill \end{array}$$

(H2)

Thus no scalar T gives $\mathrm{d}{E}_H=T\mathrm{d}\mathcal{S}$ for all preparation variations. The fixed-s slope is $E_H{s}^2/\left(2k_{\mathrm{B}}\beta \right)$, while the fixed-v slope is zero; neither establishes equilibrium for H. A first law with work terms is not excluded.

Dephasing violates the fixed-$s_1$ Gibbs covariance condition $v/s_1^2=s_1^2\nu$. Moreover, a trace-class fixed point for $\tau >0$ would have a Hilbert-Schmidt kernel supported on the measure-zero diagonal, since the channel multiplies it by $e^{-\tau {(r-r^{\prime })}^2/2}$. It must vanish. Rewriting the output as Gibbs at a different width therefore supplies no bath temperature.

Appendix H.2. Gap and Low-Rapidity Calibrations

For the seed, or after undoing its known control,

$$u_{\beta }=\sqrt{v\nu },s^2=\sqrt{v/\nu },\beta =log{\displaystyle \frac{\sqrt{v\nu }+1/2}{\sqrt{v\nu }-1/2}}.$$

(H3)

These covariance formulas require the seed quadratures, unlike the control-invariant purity. A known gap still supplies $E_{*}$.

An optional oscillator calibration uses $p_{\mathrm{lin}}=mcr$ and $x_{\mathrm{lin}}=-\hslash \pi /\left(mc\right)$, so $[x_{\mathrm{lin}},p_{\mathrm{lin}}]=\mathrm{i}\hslash$. For $s^2=\hslash \Omega /\left(m{c}^2\right)$ and $E_{*}=\hslash \Omega$,

$$\hslash \Omega Q_s={\displaystyle \frac{p_{\mathrm{lin}}^2}{2m}}+{\displaystyle \frac{m{\Omega }^2{x}_{\mathrm{lin}}^2}{2}}.$$

(H4)

The auxiliary identity is exact, but the relativistic comparison is approximate. For the centered Gaussian,

$$〈H-m{c}^2-{\displaystyle \frac{p_{\mathrm{lin}}^2}{2m}}〉=m{c}^2(e^{v/2}-1-v/2)=m{c}^2{v}^2/8+O\left(v^3\right).$$

(H5)

In the regime $s^2\ll \beta \ll 1$, $v\simeq s^2/\beta \ll 1$ and $k_{\mathrm{B}}{T}_{\mathrm{prep}}\simeq m{c}^{2}v$. This expectation estimate does not equate the full Gibbs states of a relativistic trap and the oscillator.

Along the fixed-v sequence the optional calibration gives

$$\begin{array}{cc}\hfill E_{*}\left(\beta \right)& =2m{c}^{2}vtanh(\beta /2)\to 0,\hfill \\ \hfill T_{\mathrm{prep}}\left(\beta \right)& ={\displaystyle \frac{2m{c}^{2}vtanh(\beta /2)}{k_{\mathrm{B}}\beta }}\to {\displaystyle \frac{m{c}^{2}v}{k_{\mathrm{B}}}},U_{\mathrm{p}}=m{c}^{2}v.\hfill \end{array}$$

(H6)

Entropy diverges because the confining gap closes and the auxiliary position variance ${\hslash }^2{\nu }_{\beta }/\left(m^2{c}^2\right)$ diverges. The small-v approximation remains controlled, but the Hamiltonian changes: $T_{\mathrm{prep}}\mathrm{d}\mathcal{S}=-u_{\beta }\mathrm{d}{E}_{*}$ by (61).

Appendix H.3. Operational and Fitted Temperatures

Stationary weak-coupling, long-time detector thermometry uses [40]

$${\displaystyle \frac{{\Gamma }_{\uparrow }}{{\Gamma }_{\downarrow }}}=e^{-\Delta E/\left(k_{\mathrm{B}}T\right)},T={\displaystyle \frac{\Delta E}{k_{\mathrm{B}}log({\Gamma }_{\downarrow }/{\Gamma }_{\uparrow })}}.$$

(H7)

A common temperature requires consistent readings across gaps and couplings. The present model supplies no detector rates.

A different construction fits the rest-frame rapidity law using the specified reference measure $\mathrm{d}r$:

$$p_v\left(r\right)={\displaystyle \frac{e^{-r^2/\left(2v\right)}}{\sqrt{2\pi v}}},t_z\left(r\right)={\displaystyle \frac{e^{-zcoshr}}{Z_c\left(z\right)}},Z_c\left(z\right)=2K_0\left(z\right),z>0.$$

(H8)

Here $K_j$ denotes the modified Bessel function. Up to a z-independent constant, $D_{\mathrm{KL}}(p_v\parallel t_z)=z{e}^{v/2}+log{Z}_c\left(z\right)$. Its derivatives are $e^{v/2}-{\langle coshr\rangle }_{t_z}$ and ${Var}_{t_z}(coshr)>0$. The mean $K_1/K_0$ decreases from infinity to one: concentration at $r=0$ gives the large-z limit; as $z\downarrow 0$, divergent normalisation sends the probability of every compact interval to zero. Thus a unique minimizer satisfies

$${\displaystyle \frac{K_1\left(z_v\right)}{K_0\left(z_v\right)}}=e^{v/2},T_{\mathrm{fit}}={\displaystyle \frac{m{c}^2}{k_{\mathrm{B}}{z}_v}}.$$

(H9)

Equality of the two laws would require $r/v=z_{v}sinhr$ for every r, which is impossible. Since $K_1/K_0=1+1/\left(2z\right)+O\left(z^{-2}\right)$, $k_{\mathrm{B}}{T}_{\mathrm{fit}}\sim m{c}^{2}v$ at small v. The estimator $T_{\mathrm{kin}}=2m{c}^2(e^{v/2}-1)/k_{\mathrm{B}}$ agrees only to leading order. The fit is fixed at fixed v, depends on the reference measure ($\mathrm{d}r$ rather than $\mathrm{d}p$), and does not establish quantum equilibrium.

For a supplied lower-bounded self-adjoint $\mathsf{H}$ with finite $Z\left(\gamma \right)=Tr{e}^{-\gamma \mathsf{H}}$ on an open positive interval, set ${\sigma }_{\gamma }=e^{-\gamma \mathsf{H}}/Z\left(\gamma \right)$. A source with finite entropy and energy obeys, inside that interval,

$${\partial }_{\gamma }D(\rho \parallel {\sigma }_{\gamma })={\langle \mathsf{H}\rangle }_{\rho }-{\langle \mathsf{H}\rangle }_{{\sigma }_{\gamma }},{\partial }_{\gamma }^{2}D={Var}_{{\sigma }_{\gamma }}\left(\mathsf{H}\right)>0.$$

(H10)

For nonconstant $\mathsf{H}$ and a source mean in the Gibbs range, strict convexity gives a unique energy-matched reference temperature, not necessarily equilibrium of $\rho$. A regulator must specify its Hamiltonian and state map.

Appendix H.4. Modular Calibration and Gravitational Comparison

For the seed, $-log\rho =\beta Q_s+log{Z}_{\beta }$. The oriented modular automorphism ${\tilde{\sigma }}_u\left(A\right)={\rho }^{-\mathrm{i}u}A{\rho }^{\mathrm{i}u}$ equals $e^{\mathrm{i}u\beta Q_s}A{e}^{-\mathrm{i}u\beta Q_s}$. With the calibrated preparation Hamiltonian it agrees with physical Heisenberg evolution at

$$t={\displaystyle \frac{\hslash \beta }{E_{*}}}u={\displaystyle \frac{\hslash }{k_{\mathrm{B}}{T}_{\mathrm{prep}}}}u.$$

(H11)

A gap or clock converts the dimensionless modular flow to seconds [41]; reversing its orientation reverses u, not temperature. Its generator is neither K nor H, so the boost control supplies no horizon temperature or $\beta =2\pi$ identification.

At fixed work variables an equilibrium branch $T\left(E\right)>0$ obeys

$$\mathcal{S}\left(E\right)-\mathcal{S}\left(E_0\right)={\int }_{E_0}^E{\displaystyle \frac{\mathrm{d}{E}^{\prime }}{T\left(E^{\prime }\right)}}.$$

(H12)

For $T\left(E\right)\sim A{E}^p$, $A>0$, the upper-energy integral converges only for $p>1$; $p=1$ gives logarithmic growth. Imposing $T_{\mathrm{prep}}=f\left(E_H\right)$ at fixed $E_H,E_{*}$ fixes $\beta$ and excludes the original entropy-divergent sequence. Thus constraints on the equation of state change the admissible family.

For comparison, the four-dimensional asymptotically flat Schwarzschild family has [42,43,44]

$$T_H={\displaystyle \frac{\hslash c^3}{8\pi G_{\mathrm{N}}{k}_{\mathrm{B}}{M}_{\mathrm{BH}}}},{\mathcal{S}}_{\mathrm{BH}}={\displaystyle \frac{4\pi k_{\mathrm{B}}{G}_{\mathrm{N}}{M}_{\mathrm{BH}}^2}{\hslash c}}.$$

(H13)

Here $M_{\mathrm{BH}}$ is not the statistical function $M\left(\eta \right)$. These imply $\mathrm{d}\left(M_{\mathrm{BH}}{c}^2\right)=T_H\mathrm{d}{\mathcal{S}}_{\mathrm{BH}}$ and $C_{\mathrm{BH}}=-8\pi k_{\mathrm{B}}{G}_{\mathrm{N}}{M}_{\mathrm{BH}}^2/(\hslash c)<0$. Thus the Schwarzschild branch differs from a fixed-Hamiltonian normal Gibbs family, whose finite-variance heat capacity is $C=Var\left(\mathsf{H}\right)/\left(k_{\mathrm{B}}{T}^2\right)\ge 0$. Its entropy is finite at fixed mass, but unbounded over increasing masses. Charges, rotation and boundaries require other ensembles; none of these external relations is derived from the statistical realisation.