Protected Reconstruction from Codazzi Defects: Equianharmonic Quantization and a Minimal Standard-Model Carrier

1. Introduction

Geometric encodings of interactions often enlarge the ambient structure, as in Kaluza-Klein mechanisms [1], Eisenhart-Duval lifts [2], and Randers-Finsler descriptions [3]; finite-resolution curvature defects provide a local comparison class [4]. Here the inverse question is which finite internal data are forced by a resolved codimension-three timelike defect when only protected boundary observables, source grading, and closure relations are retained. Its oriented rank-three normal bundle has link $S^2\simeq {\mathbb{CP}}_{\mathsf{\Gamma }}^1$, and after removal of the scalar trace the second-order normal source has the order-one current and order-two trace-free Codazzi components used below. Structural reconstruction, one-defect realization, and low-energy completion are kept separate.

The organizing principle is self-reconstruction:

$$\mathsf{\Phi }⟼\left({\mathcal{A}}_{\mathsf{\Phi }},\mathfrak{S}\left({\mathcal{A}}_{\mathsf{\Phi }}\right),{\delta }_{\mathsf{\Phi }},{\mathrm{Sec}}_{\mathrm{iso}}\left({\mathcal{A}}_{\mathsf{\Phi }}\right),{\nabla }_{\mathsf{\Phi }}\right)⟼\widehat{\mathsf{\Phi }}.$$

(1)

Protected least-sufficient reconstruction (PLSR) acts on the retained algebraic, spectral, dynamical, and transport data and their recovery maps, with equality taken on the physical quotient after gauge, diffeomorphism, and unitary equivalence.

Section 2 applies PLSR to the $V_1\oplus V_2$ source types fixed by standard $SU\left(2\right)$ representation theory [5]. Gauss/Borel-Weil separation selects $C\oplus W=E_3\oplus E_2$ and its primitive multiplication; determinant/exterior functoriality and one norm-seven CM triality then organize the gauge and family packets, with pure-spinor/twistor [6] and finite permutation/Clifford [7] constructions retained as comparisons. Section 4 realizes this packet in one Alena-Codazzi Parent. Selection is organized by ${\chi }_Q=dQ/d\mathsf{\Omega }$: directional seed bounds and Fredholm-Feshbach reduction supply full-operator branch certificates, while a validated overlapping-box chain from the physical stable anchor identifies the first protected fold. The same fold exports the centered coordinate $u_{*}=\mathcal{U}\left({\mathsf{\Phi }}_{*}\right)$ and, independently, the exact critical residue reconstructed from stable-side response data. Section 5 expresses the fixed-Q family shells as $A_2$ circles, retains the marked $H_R$ and Wilson data as microscopic holdouts, and attaches the passive Schur-Herglotz spectral completion. Absolute normalization and continuum scattering remain completion layers.

2. Protected Least-Sufficient Reconstruction

The loop (1) is used with structural closure preceding microscopic attachment and Schur transfer. At depth i retain $X_{\le i}\left(\mathsf{\Phi }\right)=({\mathcal{A}}_{\mathsf{\Phi }};{\mathfrak{A}}_{\le i}^{\mathrm{rec}},{\pi }_i,{\mathcal{H}}_i,{\mathcal{M}}_{\le i})$, with $\mathcal{R}\mathcal{O}=\mathrm{id}$ on the protected quotient and Gram Hessian $H_i^{\mathrm{Gram}}=D{R}_i^{†}{G}_{i}D{R}_i$. Spectral/Fredholm conventions follow [8,9,10]; curvature and superselection follow [11,12]; determinant/index conventions follow [13,14,15].

For active principal types ${\mathsf{\Lambda }}_{\mathrm{pr}}$ define the marked central sector map

$${\zeta }_F:\underset{a\in {\mathsf{\Lambda }}_{\mathrm{pr}}}{⨁}\mathbb{C}{e}_a⟶Z\left({\mathfrak{A}}_F\right),{\zeta }_F\left(e_a\right)=p_a,$$

(2)

and, for equivariant coefficient/recovery maps ${\iota }_{F,a},{\rho }_{F,a}$,

$${\delta }_{\mathrm{fid}}\left(F\right)=\dim \ker {\zeta }_F+\sum _{a\in {\mathsf{\Lambda }}_{\mathrm{pr}}}\left(\dim \ker {\iota }_{F,a}+{\parallel {\rho }_{F,a}{\iota }_{F,a}-{\mathrm{id}}_{V_a}\parallel }_{\mathrm{HS}}^2\right)\ge 0.$$

(3)

Reconstructive completeness is

$${\pi }_i{\left({\mathfrak{A}}_{\le i}^{\mathrm{rec}}\right)}^{\prime }={\pi }_i\left(Z\left({\mathfrak{A}}_{\le i}^{\mathrm{rec}}\right)\right).$$

(4)

At a protected interface $X{⪯}_{\mathrm{pp}}Y$ means marking-preserving factorization of the observation of X through that of Y, commuting with all retained operations; mutual factorization through retained isomorphisms defines protected equivalence.

Definition 1

(Protected least-sufficient reconstruction). A realization is sufficient when marking-preserving observation and recovery satisfy $\mathcal{R}\mathcal{O}=\mathrm{id}$ on the protected physical quotient. It is least-sufficient when every sufficient realization descends to it in ${⪯}_{\mathrm{pp}}$ while preserving the declared grading, */Real or integral structure, generated products, exact kernels/cokernels, retained dynamics, and gapped functional calculus. Only canonically generated source-target morphisms are retained in the graded dagger correspondence envelope; unprotected multiplicities, spectral refinements, finite-index enlargements, and flat twists are removed whenever sufficiency is preserved.

Proposition 1

(Protected reduction, provenance, and metric recovery). For a split recovery pair $({\mathcal{O}}_X,{\mathcal{R}}_X)$, $e_X:={\mathcal{O}}_X{\mathcal{R}}_X$ represents the retained Karoubi image. Exact non-expansive recovery makes $\mathcal{O}$ isometric and $\mathcal{O}\mathcal{R}$ the orthogonal projector onto $\mathrm{ran}\mathcal{O}$. If ${\parallel \mathcal{R}\mathcal{O}-\mathrm{id}\parallel }_{\mathrm{op}}\le \epsilon <1$, then

$$(1-\epsilon )\parallel v\parallel \le \parallel \mathcal{O}v\parallel \le \parallel v\parallel ,\parallel \mathrm{id}-{\mathcal{O}}^{†}{\mathcal{O}\parallel }_{\mathrm{op}}\le 2\epsilon -{\epsilon }^2.$$

(5)

For a retained response line the exact positive Real gain is one after common protected normalization, while equality of independently normalized line norms remains only a metric checksum unless the corresponding source-target morphism is protected.

If ${\delta }_{\mathrm{fid}}\left(F\right)<1$, the integer kernel terms vanish and polar decomposition gives ${\parallel T-U\parallel }_{\mathrm{op}}\le \sqrt{{\delta }_{\mathrm{fid}}\left(F\right)}$ for a normalized retained embedding. Protected finite words and rational operators inherit construction-dependent Lipschitz bounds, propagated through a retained heavy gap by Proposition 36.

Definition 2

(Finite reconstruction core). The finite reconstruction core is the faithful marked representation of the least protected extension satisfying (4), modulo marking-preserving unitary equivalence.

Proposition 2

(Finite-core uniqueness and protected descent). For $\mathfrak{A}\simeq {⨁}_a{M}_{n_a}\left(\mathbb{C}\right)$ with labeled multiplicities $m_a\ge 1$ in a faithful marked representation,

$${\Delta }_{\mathrm{mult}}\left(\pi \right):={\dim }_{\mathbb{C}}\pi {\left(\mathfrak{A}\right)}^{\prime }-{\dim }_{\mathbb{C}}\pi \left(Z\left(\mathfrak{A}\right)\right)=\sum _a(m_a^2-1)\ge 0.$$

(6)

Hence (4) is equivalent to $m_a=1$ for every labeled simple block; the faithful finite core is unique up to marking-preserving unitary equivalence and source-free duplicates are removed whenever admissible.

Remark 1

(Integral reconstruction refinement). When a canonical integral or cyclotomic form is present, admissibility may be imposed before complexification; a commensurate downstream corner retains the least admissible finite-index enlargement.

For a graded source $Q={⨁}_{\ell \in \mathsf{\Lambda }}{Q}_{\ell }$ and reconstructed level operator $N_F$, degree fidelity is

$$R_{\mathrm{fil}}(F,\zeta )={\left(\zeta \left(e_{\ell }\right)-{\mathbf{1}}_{\{\ell \}}\left(N_F\right)\right)}_{\ell \in \mathsf{\Lambda }}=0.$$

(7)

For a finite typed residual ledger $\mathcal{L}={\left\{R_{\alpha }\right\}}_{\alpha \in A}$,

$$\mathfrak{C}\left(\mathcal{L}\right):=\bigcap _{\alpha \in A}\{R_{\alpha }=0\}=\left\{\sum _{\alpha \in A}{w}_{\alpha }{\parallel R_{\alpha }\parallel }^2=0\right\},w_{\alpha }>0.$$

(8)

On a smooth one-dimensional Parent branch, a regular transverse residual of rank k has expected local intersection dimension $1-k$; an isolated stable selector is therefore generically one scalar condition. Equalities at the selected point are counted separately only when their provenance supplies an independent condition. Below ${\mathcal{L}}_{\mathrm{Parent}}$ contains microscopic source, response, spectral, integral, and marked-transport data, whereas ${\mathcal{L}}_{\mathrm{red}}$ denotes the reduced finite-response ledger. Identifying ${\mathcal{L}}_{\mathrm{red}}$ with the image of a passed Parent certificate is itself a realization condition. Derived consequences, typed incidences, independent holdouts, and completion data are distinguished by provenance. Metrics, norms, and arclengths of differently typed retained objects are not identified without an explicit incidence; stationary-state and response-path lengths remain distinct. Projectors/Gram representatives are used for phase-free rank and metric tests, with phases and connections added only when transport is retained.

The comparison class includes the Robertson-Schrödinger bound [16,17], conditional expectations [18], data processing [19,20], and quantum sufficiency [21,22,23]. Approximate/subalgebra recovery is represented by [24,25,26,27]. These frameworks constrain recovery but do not select a Parent term or normalization.

3. Canonical Protected Geometry of the Primitive Defect

The structural reconstruction is organized around one primitive packet. The projective link and the marked grades $V_1\left[1\right]\oplus V_2\left[2\right]$ first determine the separated source datum; PLSR and Borel-Weil minimality then fix $W=E_2$, $C=E_3\simeq {\mathrm{Sym}}^{2}W$, and the ordered multiplication $\mu :{\mathrm{Sym}}^{2}W\to C$. The port geometry, determinant/exterior calculus, and Eisenstein enrichment used below are functorial readings of this packet. Their microscopic realization is deferred to Section 4; independently reconstructed readings are identified only through explicitly typed incidences.

3.1. Primitive $A_2$ Source Package and Marked Separation

Let $\mathsf{\Gamma }$ be a smooth embedded future-directed timelike worldline in an oriented time-oriented Lorentzian four-branch. Its oriented Euclidean normal bundle has rank three and the real blow-up has boundary fiber

$$S_{\mathsf{\Gamma }}^2\simeq {\mathbb{CP}}_{\mathsf{\Gamma }}^1.$$

(9)

The two-eigenvalue Codazzi comparison is represented by [28], with Lorentzian optical readings in [29,30]; spinor/twistor conventions follow [31,32,33]. The local linking class gives a positive line $L_{\mathsf{\Gamma }}\simeq \mathcal{O}\left(n\right)$,

$$c_1\left(L_{\mathsf{\Gamma }}\right)=n,L_{\mathsf{\Gamma }}\simeq \mathcal{O}\left(n\right),n>0,$$

(10)

whose atomic positive class is

$$c_1\left(L_{\mathsf{\Gamma }}\right)=1,L_{\mathsf{\Gamma }}\simeq \mathcal{O}\left(1\right).$$

(11)

Higher n are composite filtered link data. Regularity conventions are represented by [34,35]. After scalar traces are removed, the transverse source through second normal order is

$${\mathrm{gr}}_{>0}{\mathcal{J}}_{\perp ,\mathrm{ns}}^{\le 2}\simeq V_1\left[1\right]\oplus V_2\left[2\right].$$

(12)

Spherical-tensor and multipole conventions follow [36,37]. The relative Gauss-local algebra contains the marked Casimir projections and source-free annulus observables commuting with the boundary charge; the corresponding superselection comparison is [38].

Theorem 1

(Principal-$A_2$ source envelope and marked separation). With $V_2\simeq {\mathrm{Sym}}_0^2\left(N_{\mathsf{\Gamma }}\right)$, the compact and split readings are

$${\mathsf{\Psi }}_{\mathsf{\Gamma }}:V_1\oplus V_2⟶\mathfrak{su}\left(N_{\mathsf{\Gamma }}^{\mathbb{C}}\right),{\mathsf{\Psi }}_{\mathsf{\Gamma }}(u,B)={\left[u\right]}_{\times }+iB,$$

(13)

and ${\mathfrak{sl}}_3\left(\mathbb{R}\right)=\mathfrak{so}\left(N_{\mathsf{\Gamma }}\right)\oplus {\mathrm{Sym}}_0^2\left(N_{\mathsf{\Gamma }}\right)$. Their common complexification is ${\mathfrak{sl}}_3\left(\mathbb{C}\right)$ with $[V_1,V_1]\subset V_1$, $[V_1,V_2]\subset V_2$, $[V_2,V_2]\subset V_1$. Under principal $SU\left(2\right)$ the adjoint is $V_1\oplus V_2$ [39], selecting $A_2$ among simple rank-two systems. The trace-free second Lorentz jet has the same $V_0\oplus V_1\oplus V_2$ restriction, so the retained non-scalar grades are exactly (12).

For unit $e\in N_{\mathsf{\Gamma }}$ put

$$Q_e=e\otimes e-{\displaystyle \frac{1}{3}}\mathbf{1},\mathcal{V}\left(e\right)=\sqrt{{\displaystyle \frac{3}{2}}}{Q}_e,\mathcal{V}(-e)=\mathcal{V}\left(e\right),{\mathcal{V}}^{*}{g}_{S^4}=3g_{S^2}.$$

(14)

For $I_2=\mathrm{Tr}\left(B^2\right)$ and $I_3=\mathrm{Tr}\left(B^3\right)$,

$${\chi }_B\left(\lambda \right)={\lambda }^3-{\displaystyle \frac{I_2}{2}}\lambda -{\displaystyle \frac{I_3}{3}},{\Delta }_{A_2}={\displaystyle \frac{1}{2}}{I}_2^3-3I_3^2,\det {\mathcal{G}}_{\mathrm{inv}}=12{\Delta }_{A_2}.$$

(15)

Thus $(I_2,I_3)$ fixes the $SO\left(3\right)$ orbit and ${\Delta }_{A_2}=0$ is the repeated-eigenvalue/invariant-rank-loss locus. The normalized determinant cubic is the standard $g=3$ Cartan isoparametric system; (14) is its extremal focal orbit, with minimality represented by [40]. This source geometry does not fix a physical transverse stiffness.

For the simply connected compact form, the Coxeter-compatible maximal-torus lattice is

$${\mathsf{\Lambda }}_E:=Q\left(A_2\right)\simeq \mathbb{Z}\left[\omega \right],E_{\omega }:=\mathbb{C}/{\mathsf{\Lambda }}_E.$$

(16)

The source envelope precedes finite marked reconstruction and does not merge the two protected labels.

Lemma 1

(Marked Gauss-local two-label separation). The marked Casimir projectors are

$${\mathsf{\Pi }}_1={\displaystyle \frac{6\mathbf{1}-{\mathcal{C}}_{\partial }}{4}},{\mathsf{\Pi }}_2={\displaystyle \frac{{\mathcal{C}}_{\partial }-2\mathbf{1}}{4}}.$$

(17)

They are central in a source-free relative annulus. Nonzero recovered charges with protected left inverses therefore give two orthogonal central labels carrying faithful marked $V_1$ and $V_2$ copies; the source Euler operator is ${\mathsf{N}}_{\partial }=({\mathcal{C}}_{\partial }+2\mathbf{1})/4$.

Corollary 1

(Exclusion of the one-block alias). A single $M_3\left(\mathbb{C}\right)$ block cannot represent both nonzero protected labels because its center has no pair of nonzero orthogonal projections.

The primitive input is therefore $L_{\mathsf{\Gamma }}\simeq \mathcal{O}\left(1\right)$, oriented $N_{\mathsf{\Gamma }}$, the two grades (12), and Gauss-local recovery of their marked labels. The one-channel and higher/composite branches remain rigidity comparisons; they are not used to infer the primitive $3+2$ carrier.

3.2. Primitive Multiplication and Finite Carrier Readings

Let ${\mathcal{R}}_{\mathsf{\Gamma }}={⨁}_{k\ge 0}{H}^0({\mathbb{CP}}_{\mathsf{\Gamma }}^1,L_{\mathsf{\Gamma }}^k)$ and $E_q=H^0\left(L_{\mathsf{\Gamma }}^{q-1}\right)\simeq {\mathrm{Sym}}^{q-1}{\mathbb{C}}^2$. Borel-Weil gives $\dim E_q=q$ [41]; finite Toeplitz conventions follow [42,43]. The two retained grades therefore give $W=E_2$, $C=E_3\simeq {\mathrm{Sym}}^{2}W$ and, by Corollary 1 and finite-core uniqueness,

$$V_{\mathsf{\Gamma }}=C_{\mathsf{\Gamma }}\oplus W_{\mathsf{\Gamma }}=E_3\oplus E_2,{\mathfrak{A}}_{\mathsf{\Gamma }}^{\mathrm{src}}\simeq M_3\left(\mathbb{C}\right)\oplus M_2\left(\mathbb{C}\right).$$

(18)

$${\mathfrak{A}}_{\mathsf{\Gamma }}^{\mathrm{src}}:=C^{*}(p_C,p_W,{\iota }_2\left(V_2\right),{\iota }_1\left(V_1\right))=\mathrm{End}\left(C_{\mathsf{\Gamma }}\right)\oplus \mathrm{End}\left(W_{\mathsf{\Gamma }}\right).$$

(19)

Theorem 2

(Source-generated primitive packet). Equations (18)-(19) are the least marked support and generated fixed-center algebra on the primitive branch. The ordered section-ring product supplies the unique $\mu :{\mathrm{Sym}}^{2}W\to C$; a marked line $L\subset W$ determines $L^{\otimes 2}\subset C$, so the degree-one line, degree-two line, oriented axis, and Veronese source form one typed moment tower. The homogeneous split twist is $L_{\mathsf{\Gamma }}^3\oplus L_{\mathsf{\Gamma }}^2$, with determinant $L_{\mathsf{\Gamma }}^5$ and standard twisted chiral kernel $E_3\oplus E_2$; the weight difference one excludes an $SU\left(2\right)$-equivariant zeroth-order chiral mixing.

The pair $L_{\mathsf{\Gamma }}\oplus L_{\mathsf{\Gamma }}^2$ is the standard exceptional-pair/Kronecker realization on ${\mathbb{P}}^1$ [44]: its dimension vector is $(2,3)$, Tits value one [45], and King-stable endomorphism algebra $\mathbb{C}$ with vanishing self-${\mathrm{Ext}}^1$ [46]. The curried multiplication is the unique $E_2$ summand of $\mathrm{Hom}(W,C)\simeq E_2\oplus E_4$.

Proposition 3

(Primitive port projector packet). The multiplication globalizes to

$$0⟶L_{\mathsf{\Gamma }}^{-1}\otimes W\stackrel{j}{\to }\mathcal{O}\otimes C⟶L_{\mathsf{\Gamma }}^2⟶0,$$

(20)

and, after dualization and tensoring by $L_{\mathsf{\Gamma }}$,

$$0⟶L_{\mathsf{\Gamma }}^{-1}⟶L_{\mathsf{\Gamma }}^{\oplus 3}⟶{\left(L_{\mathsf{\Gamma }}^2\right)}^{\oplus 2}⟶0.$$

(21)

For unit $s\in W$, $P_s=|s\rangle \langle s|$ and $X=\sqrt{\alpha }{j}_s$, the nonzero squared singular values are $\{\alpha ,\alpha /2\}$. With $K=X{X}^{†}$,

$$P_D={\mathbf{1}}_C-X{\left(X^{†}X\right)}^{-1}{X}^{†}={\mathbf{1}}_C-K{K}^{+},L_{\mathrm{dark}}=\mathrm{Ran}{P}_D={\mathrm{Sym}}^2\left(s^{\perp }\right),$$

(22)

and for the tautological $S\simeq \mathcal{O}(-1)$ and quotient $Q_W=\underline{W}/S\simeq \mathcal{O}\left(1\right)$,

$$L_{\mathrm{dark}}\simeq {\mathrm{Sym}}^2{Q}_W\simeq Q_W^{\otimes 2}\simeq L_{\mathsf{\Gamma }}^2.$$

(23)

The rank-two port is one-step cyclic, so its compressed Hermitian resolvent has minimal realization dimension five.

The exact sequence, singular packet, dark line, and minimal dilation are therefore one primitive-port reading. The image in (20) is $\mathcal{O}{(-1)}^{\oplus 2}$, excluding a global fixed-frame minimal square root; $Q_W$ is the canonical bundle root.

Proposition 4

(Dark Veronese projector geometry). For $g\left(P\right)=\mathrm{Tr}\left(dPdP\right)/2$ and $\mathsf{\Omega }\left(P\right)=-i\mathrm{Tr}(PdP\wedge dP)$, let $P_Q$ project onto $S^{\perp }$ and $P_D$ be (22). Then

$$g_D=2g_Q,{\mathsf{\Omega }}_D=2{\mathsf{\Omega }}_Q,\mathrm{Tr}\left(P_D\left(s\right)P_D\left(t\right)\right)={\left[\mathrm{Tr}\left(P_Q\left(s\right)P_Q\left(t\right)\right)\right]}^2.$$

(24)

The image is the degree-two Veronese conic; it saturates the metric-curvature inequality and, for $c_1\left(L_{\mathrm{dark}}\right)=2$, has QGT area $2\pi$.

This complex conic and the real Cartan-Veronese surface (14) are distinct typed readings of the same symmetric-square representation.

With $D_{\mathsf{\Gamma }}=2P_C+P_W$, put $H_{\mathsf{\Gamma }}^{\mathrm{deg}}=2P_C-3P_W$. For an orthonormal basis $(s_1,s_2)$ of W and $A_{i}w=P_{\mathrm{sym}}(s_i\otimes w)$,

$$\sum _i{A}_i{A}_i^{†}={\mathbf{1}}_C,\sum _i{A}_i^{†}{A}_i={\displaystyle \frac{3}{2}}{\mathbf{1}}_W,{\mu }_K=P_C-{\displaystyle \frac{3}{2}}{P}_W={\displaystyle \frac{1}{2}}{H}_{\mathsf{\Gamma }}^{\mathrm{deg}}.$$

(25)

The induced completely positive adjoint pair $\mathsf{\Phi }\left(Y\right)={\sum }_i{A}_{i}Y{A}_i^{†}$, $\mathsf{\Psi }\left(X\right)={\displaystyle \frac{2}{3}}{\sum }_i{A}_i^{†}X{A}_i$ is the lower adjacent-spin covariant channel [47]; equivalently it is the symmetric $1\to 2$ cloning endpoint [48]. Exact coherent-symbol recovery selects the rank-two endpoint, for which

$$K_C:=\mathsf{\Phi }\mathsf{\Psi }=P_{V_0}+{\displaystyle \frac{2}{3}}{P}_{V_1},{\mathcal{L}}_K=\mathbf{1}-K_C={\displaystyle \frac{1}{3}}{P}_{V_1}+P_{V_2},$$

(26)

$$r_K=\sqrt{{\displaystyle \frac{2}{3}}},G_K:={\mathcal{L}}_K^{+}=3P_{V_1}+P_{V_2}.$$

(27)

The balanced Kempf-Ness quotient gives the same metric ray. Finally, left/right multiplication splits as

$${\mathsf{J}}_H={\lambda }_H+{\rho }_H,{\mathsf{K}}_H=i({\lambda }_H-{\rho }_H),{\lambda }_H={\displaystyle \frac{1}{2}}({\mathsf{J}}_H-i{\mathsf{K}}_H),$$

(28)

fixing the Jordan/Lie normalization used in the Parent response.

3.3. Determinant-Dual Gauge and Clifford Readings

Proposition 5

(Centered marked grading calculus). For a marked Hermitian carrier $V={⨁}_j{V}_j$ with ranks $d_j$ and distinct integral levels ${\ell }_j$, center $r{\ell }_j-{\sum }_k{d}_k{\ell }_k$ and divide by the common divisor. For a two-block split of levels $(2,1)$ and ranks $c>w\ge 2$, with $g=\gcd (c,w)$,

$$H_{c,w}^{\mathrm{prim}}={\displaystyle \frac{w}{g}}{P}_C-{\displaystyle \frac{c}{g}}{P}_W,{\lambda }_{c,w}^{\mathrm{prim}}\left(U\left(1\right)\right)\cap (SU\left(c\right)\times SU\left(w\right))\simeq {\mu }_{cw/g}.$$

(29)

On the coprime branch the rational degree has weights $-1/c,1/w$, ${\mathrm{Tr}}_V\left(H^2\right)=cw(c+w)$, $\dim {\mathsf{\Lambda }}^{\mathrm{even}}V=2^{c+w-1}$, and Abelian/non-Abelian trace ratio $2(c+w)/\left(cw\right)$.

For $(c,w)=(3,2)$ one has $(r,T,T^{*},cw)=(5,8,7,6)$ and

$${\widehat{R}}_{\mathsf{\Gamma }}\left(t\right)={\left(\det t^{D_{\mathsf{\Gamma }}}\right)}^{-1/5}{t}^{D_{\mathsf{\Gamma }}}=t^{H_{\mathsf{\Gamma }}^{\mathrm{deg}}/5}.$$

(30)

The retained volume morphism gives $S\left(U\right(C)\times U(W\left)\right)$; the same tensor automorphism group follows from Doplicher-Roberts reconstruction [38] after saturation of the auxiliary rigid envelope. The primitive intersection is ${\mu }_6$, hence

$$S(U\left(3\right)\times U\left(2\right))\simeq {\displaystyle \frac{SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y}{{\mathbb{Z}}_6}}.$$

(31)

Line-operator and cocharacter comparisons are represented by [49,50]; the filtered-fiber-functor comparison of (30) is [51]. Descent is $2t_3+3t_2+y\equiv 0\left(\mathrm{mod}6\right)$ for $y=6Y$.

The same split has the standard pure-spinor/exterior reading. For $0\ne \beta \in {\mathsf{\Lambda }}^{2}W$, the line $\mathbb{P}\langle 1,\beta \rangle$ lies in the pure-spinor variety, and its retained determinant-one stabilizer is $S\left(U\right(C)\times U(W\left)\right)$; the orbit of $\left[\beta \right]$ is ${\mathrm{Gr}}_2\left(V\right)$ with tangent $\mathrm{Hom}(W,C)$. Thus the gauge group is the flag stabilizer, while the positive AIII degree is retained as a correspondence rather than promoted to gauge degree zero.

Proposition 6

(Determinant-dual Clifford and weak-direct calculus). On $F_{\mathsf{\Gamma }}={\mathsf{\Lambda }}^{•}V$, exterior multiplication and contraction satisfy the canonical CAR [52]. Put

$$F_{\mathsf{\Gamma }}^{+}={\mathsf{\Lambda }}^{\mathrm{even}}V,F_{\mathsf{\Gamma }}^{-}={\mathsf{\Lambda }}^{\mathrm{odd}}V,\dim F_{\mathsf{\Gamma }}^{\pm }=16.$$

(32)

The $\left|1\right|$-grading of $\mathfrak{sl}\left(V\right)$ has degree-zero Levi $C\oplus W$ and positive part $\mathrm{Hom}(W,C)$; the primitive multiplication selects its protected $E_2$ slice. With the marked volume duality and $h\in W$, the four multiplicity-one color-singlet weak-direct blocks are

$$\begin{array}{cccc}\hfill T_{\nu }\left(h\right)& =\epsilon \left(h\right)P_{{\mathsf{\Lambda }}^{0}V},\hfill & \hfill T_u\left(h\right)& =\epsilon \left(h\right)P_{{\mathsf{\Lambda }}^{2}C},\hfill \\ \hfill T_e\left(h\right)& =\iota \left(h^{*}\right)P_{{\mathsf{\Lambda }}^{2}W},\hfill & \hfill T_d\left(h\right)& =\iota \left(h^{*}\right)P_{{\mathsf{\Lambda }}^{2}C\otimes {\mathsf{\Lambda }}^{2}W}.\hfill \end{array}$$

(33)

They preserve $B-L$ and are odd for the exterior grading; a spacetime fermionic-field interpretation is represented by [53] and is not used to determine the finite packet.

The normalized CAR trace is

$${\displaystyle \frac{1}{8}}{\mathrm{Tr}}_{F_{\mathsf{\Gamma }}^{+}}\left(\epsilon {\left(e_i\right)}^{†}\epsilon \left(e_j\right)\right)={\displaystyle \frac{1}{8}}{\mathrm{Tr}}_{F_{\mathsf{\Gamma }}^{+}}\left(\iota {\left(e_i^{*}\right)}^{†}\iota \left(e_j^{*}\right)\right)={\delta }_{ij},$$

(34)

with vanishing mixed trace. The induced Real/Hodge structure has the KO-6 signs used in the finite noncommutative-geometric comparison [54]; moreover $F_{\mathsf{\Gamma }}^{+}\simeq \mathbf{1}\oplus \mathbf{10}\oplus \overline{\mathbf{5}}$ is the standard $SU\left(5\right)$ restriction of one chiral Spin$\left(10\right)$ half-spin module [55]. These are ambient bookkeeping readings; order-one/Real compatibility of a microscopic weak-direct block remains a Parent test.

The centered charge is

$$Y=d\mathsf{\Gamma }\left(-{\displaystyle \frac{1}{6}}{H}_{\mathsf{\Gamma }}^{\mathrm{deg}}\right)=-{\displaystyle \frac{1}{3}}{N}_C+{\displaystyle \frac{1}{2}}{N}_W,$$

(35)

with $Q=T_3+Y$, $X_{\mathrm{F}}=5\mathbf{1}-2(N_C+N_W)=5(B-L)-4Y$, and $B-L=1-2N_C/3$. The even exterior character is

$${\chi }_{+}(u,v)={\displaystyle \frac{1}{2}}\left({(1+u)}^c{(1+v)}^w+{(1-u)}^c{(1-v)}^w\right).$$

(36)

Table 1. Protected exterior coefficient package and canonical Abelian gradings.

Summand	$\mathit{S}\mathit{U}\left(3\right)\times \mathit{S}\mathit{U}\left(2\right)$ type	Y	$\mathit{B}-\mathit{L}$	${\mathit{X}}_{\mathbf{F}}$
${\mathsf{\Lambda }}^{0}V$	$(1,1)$	0	1	5
${\mathsf{\Lambda }}^{2}C$	$(\overline{3},1)$	$-{\displaystyle \frac{2}{3}}$	$-{\displaystyle \frac{1}{3}}$	1
$C\otimes W$	$(3,2)$	${\displaystyle \frac{1}{6}}$	${\displaystyle \frac{1}{3}}$	1
${\mathsf{\Lambda }}^{2}W$	$(1,1)$	1	1	1
${\mathsf{\Lambda }}^{2}C\otimes {\mathsf{\Lambda }}^{2}W$	$(\overline{3},1)$	${\displaystyle \frac{1}{3}}$	$-{\displaystyle \frac{1}{3}}$	$-3$
${\mathsf{\Lambda }}^{3}C\otimes W$	$(1,2)$	$-{\displaystyle \frac{1}{2}}$	$-1$	$-3$

Table 1. Protected exterior coefficient package and canonical Abelian gradings.

Summand	$\mathit{S}\mathit{U}\left(3\right)\times \mathit{S}\mathit{U}\left(2\right)$ type	Y	$\mathit{B}-\mathit{L}$	${\mathit{X}}_{\mathbf{F}}$
${\mathsf{\Lambda }}^{0}V$	$(1,1)$	0	1	5
${\mathsf{\Lambda }}^{2}C$	$(\overline{3},1)$	$-{\displaystyle \frac{2}{3}}$	$-{\displaystyle \frac{1}{3}}$	1
$C\otimes W$	$(3,2)$	${\displaystyle \frac{1}{6}}$	${\displaystyle \frac{1}{3}}$	1
${\mathsf{\Lambda }}^{2}W$	$(1,1)$	1	1	1
${\mathsf{\Lambda }}^{2}C\otimes {\mathsf{\Lambda }}^{2}W$	$(\overline{3},1)$	${\displaystyle \frac{1}{3}}$	$-{\displaystyle \frac{1}{3}}$	$-3$
${\mathsf{\Lambda }}^{3}C\otimes W$	$(1,2)$	$-{\displaystyle \frac{1}{2}}$	$-1$	$-3$

The CAR AIII sums give the coefficient-side quadratic signature $(0,4,3,6,2,3)$ on $(N,U^c,Q,E^c,D^c,L)$ and the three connected-pair matrix-element ratio $1:2:1$; physical widths additionally require the Riesz residue and common kinematics. Evaluation of (36) gives

$$I_3=I_2=2,{\mathrm{Tr}}_{F_{\mathsf{\Gamma }}^{+}}\left(Y^2\right)={\displaystyle \frac{10}{3}},k_Y={\displaystyle \frac{5}{3}}={\displaystyle \frac{2r}{N}}.$$

(37)

For the induced exterior representation define

$$B_F(X,Y):=-{\mathrm{Tr}}_{F_{\mathsf{\Gamma }}^{+}}\left({\rho }_{*}\left(X\right){\rho }_{*}\left(Y\right)\right).$$

(38)

Its ray is fixed by (37); the three-state family multiplicity rescales but does not rotate this ray. For a normalized conditional expectation onto scalars, convexity gives the structural index target

$$d_F=16,\underset{\rho }{\min }\mathrm{Ind}\left(E_{\rho }\right)=d_F^2=256.$$

(39)

Proposition 7

(Chirality-independent anomaly-selected central cocharacter). For $Q_{a,b}=a{N}_C+b{N}_W$ on either chiral half,

$$\begin{array}{cccc}\hfill \mathrm{Tr}{Q}_{a,b}& =8(3a+2b),\hfill & \hfill {\mathcal{A}}_{SU{\left(3\right)}^{2}Q}& ={\mathcal{A}}_{SU{\left(2\right)}^{2}Q}=3a+2b,\hfill \\ \hfill \mathrm{Tr}{Q}_{a,b}^3& =4(3a+2b)(9a^2+6ab+5b^2).\hfill \end{array}$$

(40)

Hence a nontrivial central gauged $U\left(1\right)$ is locally anomaly-free exactly on $3a+2b=0$, with primitive generator $(-2,3)$ and

$${\mathfrak{g}}_{\mathsf{\Gamma }}=s(u\left(C\right)\oplus u\left(W\right)).$$

(41)

The same exterior character gives the vanishing Standard-Model local anomalies and the ${\mathbb{Z}}_6$ kernel in (31); the one-block $I\left(J\right)=\left\{2\right\}$ alternative retains a nonzero cubic $SU{\left(3\right)}^3$ coefficient.

A connection-compatible trivialization of the chiral determinant line removes its local vertical curvature obstruction and gauge-loop holonomy. The determinant-family framework is represented by [13]; characteristic and residual flat/torsion conventions by [56,57].

3.4. Equianharmonic Family and Integral $A_2$ Data

The arithmetic reading uses the integral $A_2$ package (16). On $E_{\omega }:y^2=x^3-1$ with Coxeter action $\rho (x,y)=(\omega x,y)$ and quotient $q_3=(1:y)$, the standard degree-three theta realization has invariant section ranks $2,3$ in degrees $3,6$, and $\rho$-character multiplicities $(2,1,0)$, $(3,2,1)$, $(3,2,2)$ in degrees $3,6,7$ [58]. A projective marking identifies the quotient line with $L_{\mathsf{\Gamma }}$; no normal $SU\left(2\right)$ action is transferred to the theta spaces. For a nonprimitive $L_{\mathsf{\Gamma }}\simeq \mathcal{O}\left(n\right)$ the two block ranks are $(2n+1,n+1)$ and

$${\mu }_{N_n}\simeq {\mu }_{2n+1}\times {\mu }_{n+1},N_n=(2n+1)(n+1).$$

(42)

Corollary 2

(Parity-exhausted degree comparison). If the W-side factor in (42) is exhausted by an order-two parity factor, then $n=1$ and the primary factors are ${\mu }_3,{\mu }_2$.

The tubular orbifold $[E_{\omega }/{\mu }_6]$ has signature $(6,3,2)$ [59]; its order-two root is understood in the standard root-stack sense [60]. For a coprime two-block split the tubular identity gives $(c-1)(w-1)=2$ and hence $(c,w)=(3,2)$; this is a rank-order checksum, not an identification of $S_3$ with ${\mu }_6$.

Proposition 8

(Root-line family orbit and relative-spin witness). The six $A_2$ roots are the Eisenstein units, and

$$\mathsf{\Phi }\left(A_2\right)/\{\pm 1\}\simeq {\mu }_6/{\mu }_2\simeq {({\mathsf{\Lambda }}_E/2{\mathsf{\Lambda }}_E)}^{\times }\simeq E{\left[2\right]}^{\times }.$$

(43)

For $P\in {\mathrm{Pic}}^0\left(E\right)\left[2\right]$ and $N_P=q_3^{*}{L}_{\mathsf{\Gamma }}\otimes P$, the trivial twist has $q_{3,*}{N}_0\simeq \mathcal{O}\left(1\right)\oplus \mathcal{O}\oplus \mathcal{O}(-1)$, while every $P\ne 0$ has

$$q_{3,*}{N}_P\simeq {\mathcal{O}}^{\oplus 3}.$$

(44)

Thus the degree-zero polystable branch is exactly $E{\left[2\right]}^{\times }$, on which the Coxeter action is simply transitive. For its regular cycle $G_{\mathrm{fam}}$,

$$H_F={\displaystyle \frac{G_{\mathrm{fam}}-G_{\mathrm{fam}}^{†}}{i\sqrt{3}}},K_F=-{\displaystyle \frac{G_{\mathrm{fam}}+G_{\mathrm{fam}}^{†}}{3}}=H_F^2-{\displaystyle \frac{2}{3}}\mathbf{1},H_F^3=H_F.$$

(45)

The cycle fixes character lines but not a mass filtration. Parity induction yields the two-arrow Kronecker heart; the integral regular family lattice is the nonsplit extension of its invariant line by the $A_2$ augmentation lattice. The ramified mod-three line remains distinct from the three root lines. For the selected split, $(c,w,r,T,T^{*},N)=(3,2,5,8,7,6)$ and $(f,d_F,k_Y)=(3,16,5/3)$ are dependent checks of the same rank cell.

3.4.1. Finite Torsor Response and the Primitive Norm-Seven Class

The projective family torsor is ${\mathcal{T}}_{\mathsf{\Gamma }}=E{\left[2\right]}^{\times }$ with ${\mathcal{H}}_{\mathrm{cen}}\simeq \mathbb{C}\left[{\mathbb{Z}}_3\right]$. Graph/bundle-Laplacian conventions are represented by [61,62]; finite spectral-module comparisons by [63]. For marked cycle transport ${\mathcal{W}}_{\mathsf{\Gamma }}$ and center expectation ${\mathbb{E}}_{Z,\mathsf{\Gamma }}$, retain

$$R_{\mathrm{tor}}^{\mathsf{\Gamma }}={\mathcal{W}}_{\mathsf{\Gamma }}-{\mathbb{E}}_{Z,\mathsf{\Gamma }}\left({\mathcal{W}}_{\mathsf{\Gamma }}\right).$$

(46)

On a scalar cycle-trivial frame, $d_{\theta }=S-e^{i{\theta }_{\mathsf{\Gamma }}}\mathbf{1}$, ${\Delta }_{\theta }=d_{\theta }^{†}{d}_{\theta }$, and

$$\sum _k{\mu }_k=6,\sum _{i<j}{\mu }_i{\mu }_j=9,\prod _k{\mu }_k={|1-{\mathcal{W}}_{\mathsf{\Gamma }}|}^2.$$

(47)

The Gram spectrum fixes Wilson magnitudes but leaves conjugation as pre-Gram orientation data. The integral $A_2$ lattice identifies the nontrivial rational group-ring factor with $\mathbb{Z}\left[\omega \right]$.

Proposition 9

(Primitive commensurate torsor spectrum). If the squared singular values are primitive commensurate, $p^2/n,q^2/n,r^2/n$, then (47) gives $r=p+q$, $n=(p^2+pq+q^2)/3$, and on the integral branch $p\equiv q\left(\mathrm{mod}3\right)$ with

$$\xi ={\displaystyle \frac{p-q\omega }{1-\omega }}\in \mathbb{Z}\left[\omega \right],N_E\left(\xi \right)=n.$$

(48)

The oriented phase is $e^{i{\theta }_{\mathsf{\Gamma }}}=-{\omega }^2\xi /\overline{\xi }$. Under $0<p<q<r$, the first simple integral class has norm seven and

$$\mathrm{spec}{\Delta }_{\theta }=\left\{{\displaystyle \frac{1}{7}},{\displaystyle \frac{16}{7}},{\displaystyle \frac{25}{7}}\right\}.$$

(49)

Proposition 10

(CM triality and the norm-seven theta packet). For $\xi =2-\omega$, multiplication on $E_{\omega }=\mathbb{C}/\mathbb{Z}\left[\omega \right]$ has determinant $N_E\left(\xi \right)=7$, satisfies $\left[\overline{\xi }\right]\circ \left[\xi \right]=\left[7\right]$, and pulls $\mathsf{\Theta }={\mathcal{O}}_E\left(O\right)$ back to $L_7={\mathcal{O}}_E\left(7O\right)$. Since $\xi \equiv \omega \left(\mathrm{mod}2\right)$, the same CM unit induces the regular order-three action on $E{\left[2\right]}^{\times }$; modulo ξ, $\omega \equiv 2$, so the six nonzero characters split into $\{1,2,4\}$ and $\{3,6,5\}$. Hence $H^0(E,L_7)=1\oplus 3\oplus \overline{3}$, restricting to marked $C_3$ multiplicities $3\left|2\right|2$. The regular cycle, norm-seven triplets, theta line, and conjugate arithmetic orientation are therefore one CM packet. Physical CP orientation still requires the same-defect determinant/holonomy incidence.

Put $H_R=3H_F/2-K_F/2$ in the arithmetic order. Its spectrum is $\{4/3,1/3,-5/3\}$ and, for spectral projectors $P_{+},P_0,P_{-}$,

$$G_{\mathrm{fam}}=P_0+\omega P_{+}+{\omega }^2{P}_{-}.$$

(50)

Thus $H_R$ reconstructs the regular cycle by polynomial functional calculus; the associated marked finite envelope of order 189 is used only as an arithmetic frame organizer.

Corollary 3

(Rank-one Eisenstein and Galois rigidity). For ${\xi }_m=1+m(1-\omega )$, $N_E\left({\xi }_m\right)=3m^2+3m+1$. The first nonunit simple-spectrum pair is $m=1,-2$, giving the conjugate norm-seven orientations and signed spectrum $\{1,4,-5\}$. If a commensurate class has $\mathrm{Tr}\xi =c+w$, $N_E\left(\xi \right)=c+2w$ with $c>w\ge 2$, then $(c,w)=(3,2)$; for $(\mathrm{Tr}\xi ,N_E\left(\xi \right))=(5,7)$ the class is unique up to conjugation.

For odd trace t,

$$t_{n+1}={\displaystyle \frac{t_n^2+3}{4}},5⟼7⟼13⟼43.$$

(51)

Lemma 2

(Minimal-trace cyclic companion). If $A\in M_2\left(\mathbb{Z}\right)$ has odd trace t and determinant $(t^2+3)/4$, a primitive cyclic basis may be chosen in which A, up to orientation and transpose, is $\left(\begin{array}{cc}0& -n\\ 1& t\end{array}\right)$.

For $L_7$, the theta-group realization [58] gives $3\left|2\right|2$ under the common $C_3$. In a compatible Schrödinger frame choose $Z{e}_a={\omega }^a{e}_a$, $ZS=\omega SZ$ and ${\mathcal{A}}_a=\mathrm{span}\left\{Z^a{S}^b\right\}$; the associated conditional-expectation Hessian uses the convention of [24]. The Hermitian generators are

$$K_Z={\displaystyle \frac{Z-Z^{†}}{2i}},H_S\left(\vartheta \right)=e^{-i\vartheta }S+e^{i\vartheta }{S}^{†},G={\displaystyle \frac{S-S^{\mathsf{T}}}{\sqrt{3}}}.$$

(52)

The CM action, Weyl pair, theta module and orientation are treated as one frame; microscopic filtration and physical orientation remain Parent incidences. The retained coefficient phase is

$$j_7:={\displaystyle \frac{3+2\omega }{\sqrt{7}}},e^{i{\delta }_7}:=-j_7^6.$$

(53)

For the comparison curve $E_{\omega }$, $\#E_{\omega }\left({\mathbb{F}}_7\right)=4$ and $P_7\left(T\right)=T^2-4T+7$, so $3+2\omega$ is a split $p=7$ CM Frobenius root associate to $2-\omega$; the sixth power removes the ${\mu }_6$ generator ambiguity. Conjugation defines the arithmetic orientation, while a physical orientation additionally requires the marked protected cycle. The elliptic-surface comparison is represented by [64,65]; it adds no gauge or family selector.

3.5. Rigid-Eisenstein Inverse Reconstruction and First-Cell Selection

The forward PLSR derivation above fixes $3+2$ before the family attachment. A complementary inverse comparison starts from a general two-block rank vector $(c,w)$ that is typed simultaneously to the recovered two-arrow Kronecker heart and to a trace-one Eisenstein family coefficient. The rigid Schur condition gives $c=w+1$. If the family norm is identified with the reversed carrier degree $T^{*}=c+2w$, then for an integral trace-one coefficient $|q_E|=2\ell +1$ one obtains

$$w=\ell (\ell +1),c=\ell (\ell +1)+1,T_{\ell }^{*}=3{\ell }^2+3\ell +1.$$

(54)

The remaining readings are $T_{\ell }=T_{\ell }^{*}+1$, $r_{\ell }=(2T_{\ell }^{*}+1)/3$, and $N_{\mathrm{sh},\ell }=(T_{\ell }^{*}-1)(T_{\ell }^{*}+2)/9$. Equivalently, with the principal spin-ℓ module, $|q_E|=\dim V_{\ell }$ and $w=C_2\left(V_{\ell }\right)$; this Casimir rewriting is algebraic and becomes a source-family identification only after a typed microscopic incidence. The first cells are $(c,w,T^{*})=(3,2,7),(7,6,19),(13,12,37),\dots$.

There are two logically distinct first-cell selectors. First, (29) shows that primitive order six alone allows $(c,w)=(3,2),(6,2),(6,3)$ for $c>w\ge 2$. If the same rank vector is a rigid Schur dimension vector of the recovered two-arrow heart, then ${(c-w)}^2=1$ and only $(3,2)$ remains. Second, if the carrier and family neutral intercepts are typed as the same response coefficient, then

$${\mathcal{R}}_{B0}:={\displaystyle \frac{cw}{{\left(T^{*}\right)}^2}}-{\displaystyle \frac{6}{{\left(T^{*}\right)}^2}}={\displaystyle \frac{(\ell -1)(\ell +2)({\ell }^2+\ell +3)}{{(3{\ell }^2+3\ell +1)}^2}}.$$

(55)

Hence ${\mathcal{R}}_{B0}=0$ exactly at $\ell =1$, while $\ell \ge 2$ gives the discrete gap ${\mathcal{R}}_{B0}\ge 36/361$. This selector does not use the primitive order-six loop. On the same diagonal tower the gauge trace ratio satisfies

$${\displaystyle \frac{2(c+w)}{cw}}-{\displaystyle \frac{5}{3}}=-{\displaystyle \frac{(\ell -1)(\ell +2)(5{\ell }^2+5\ell +3)}{3\ell (\ell +1)({\ell }^2+\ell +1)}},$$

(56)

so $k_Y=5/3$ also isolates the first cell when computed independently. After the carrier has already fixed $k_Y$ through (37), this equality is a checksum rather than an additional selector. Likewise the first-cell identity $N_{\mathrm{sh}}=T^{*}-1$ explains the special pair $(6,7)$ without assigning an independent physical role to residue-field cardinality.

3.6. Structural Synthesis and Realization Boundary

Theorem 3

(Primitive protected finite geometry). On the primitive two-grade branch, Gauss-local realization of both nonzero charges and PLSR recovery of their marked images determine the principal $A_2$ source datum, the carrier (18), and the algebra (19). The remaining structural data are organized by three objects. Proposition 3 gives the linking, exact sequence, fixed singular packet, dark projector, minimal dilation, and adjacent-spin Green geometry. The determinant/exterior reading gives the pure-spinor flag, whose compact stabilizer is (31); the same packet carries the chiral coefficients, weak-direct calculus, protected gauge metric, $k_Y$, and index target, while $A_4\subset D_5$ remains ambient. The integral $A_2$ package gives the three-state family orbit, and Propositions 9 and 10 organize the norm-seven theta/Heisenberg packet, its conjugate orientation, and theta line. These derived data are denoted collectively by ${\mathfrak{P}}_{\mathsf{\Gamma }}$.

The package ${\mathfrak{P}}_{\mathsf{\Gamma }}$ is the forward normalized structural target. Section 3.5 supplies inverse conditional checks of the same first cell; these checks do not enter the PLSR proof and count as physical selectors only after their typed readings are realized on one defect. Projector/Gram representatives are used before phase-sensitive transport whenever they retain the protected information; connection, Real, and marked-loop data are added when they carry independent orientation or holonomy content. The exterior summands and weak-direct maps fix protected species labels and the chiral Dirac support; a physical spectral state is identified only after the corresponding microscopic response develops the channel kernel described in Section 4.3. Borel-Weil-Dirac, theta, tubular, elliptic-surface, and filtered-unification constructions remain comparison or compatibility readings rather than additional dynamical assumptions.

4. Alena-Codazzi Realization of the Canonical Protected Geometry

Section 3 fixes ${\mathfrak{P}}_{\mathsf{\Gamma }}$ before microscopic dynamics is used. The realization layer is organized around one constructive class of the current-Codazzi action: a primary packet determines the effective metric-density, rotational, gauge, and current readouts, while spectral and family objects are attached only after a stationary background and its constrained Jacobi operator have been identified. Relations built into this normal form are not counted again as independent realization gates. The finite fermionic operator remains a conditional spectral descendant of the constrained microscopic package rather than an additional Parent term, and exact spectral incidences define $\mathfrak{C}\left({\mathcal{L}}_{\mathrm{Parent}}\right)$ through (8) before the reduced ledger is introduced.

4.1. Current-Codazzi Collar and Resolved Source

The local Parent is kept in the single-density Alena form

$$S_A\left[\mathsf{\Phi }\right]={\sigma }_A{\int }_X{p}_{\mathsf{\Lambda }}\left[1-{\epsilon }_A({\rho }_{\zeta },R_{\omega })\right]d{\mathrm{vol}}_{\eta },{\sigma }_A=\pm 1,$$

(57)

with primary packet

$$\mathsf{\Phi }=(\mathsf{\Psi },{\rho }_{\zeta },\vartheta ,{\mathcal{A}}_{\mathrm{g}}),\mathsf{\Psi }⟼(J,U,{\kappa }_{\mathrm{base}},R_{\mathrm{vort}}),$$

(58)

where ${\mathcal{A}}_{\mathrm{g}}$ has curvature $\mathcal{F}$, $J^2=\mathbf{1}$, $\mathrm{tr}J=0$, $J_{\eta }^{†}=J$, $P_{E}U=U$, $\eta (U,U)=1$, and $P_{E,F}=(\mathbf{1}\mp J)/2$. Euler and Codazzi equations are imposed after variation. For $p_0>0$ and a positive invariant form $G_0$ on the reconstructed gauge algebra,

$${\kappa }^2={\kappa }_{\mathrm{base}}^2-{\displaystyle \frac{1}{4p_0}}{G}_0({\mathcal{F}}_{\mu \nu },{\mathcal{F}}^{\mu \nu }),p_{\mathsf{\Lambda }}=p_0{\kappa }^2,{\eta }^{-1}k=P_E+\left({\displaystyle \frac{\kappa }{2}}-1\right)P_F,$$

(59)

on the branch $\kappa >4$. Thus $d_k=\sqrt{\left|k\right|/\left|\eta \right|}=\kappa /2-1$ and $u_{\mathrm{AC}}:=(d_k-1)/(d_k+1)=1-4/\kappa \in (0,1)$. This Parent anisotropy is not identified with the reduced coordinate u without a separate incidence. The gauge metric is chosen on the ray of (38).

On the regular scalar branch take ${\epsilon }_A(\rho ,R)=V\left(\rho \right)+\mu \left(\rho \right)R$, $\mu >0$, so that ${\zeta }^2=V\left(\rho \right)$ and $q_{\mathrm{rot}}=\mu \left(\rho \right)R$. With ${\nu }_{\vartheta }=U^{\mu }{\partial }_{\mu }\vartheta$,

$$R_{\omega }=R_{\mathrm{vort}}\left[\mathsf{\Psi }\right]-d_k{\displaystyle \frac{V\left({\rho }_{\zeta }\right)}{\mu \left({\rho }_{\zeta }\right)}}{\nu }_{\vartheta }.$$

(60)

The phase is cyclic and variation gives

$$J_{\mathrm{tr}}^{\mu }=p_{\mathsf{\Lambda }}{\zeta }^2{U}^{\mu },{\nabla }_{\mu }^{\left(k\right)}{J}_{\mathrm{tr}}^{\mu }=0.$$

(61)

Its pure Hessian vanishes with the remaining fields frozen, so this transport direction is eliminated or constrained before protected zero modes are counted. The stress interpretation follows [66], the current/vorticity slots [67], and the continuum and branch inputs [68,69,70].

For $\mathsf{Y}:=p_{\mathsf{\Lambda }}(4{\eta }^{-1}k/\kappa -\mathbf{1})$ the normal form gives $\mathsf{Y}=p_{\mathsf{\Lambda }}{u}_{\mathrm{AC}}J$ and hence the non-null Rainich square. On the reference phase slice density stationarity is $V^{\prime }+R{\mu }^{\prime }=0$. With $K_{\rho }:=V^{″}+R{\mu }^{″}>0$, the density is pointwise eliminated. On ${\mu }^{\prime }\ne 0$, using $\mathcal{V}\left(\mu \right):=V\left(\rho \right(\mu \left)\right)$ gives $R=-{\mathcal{V}}_{\mu }$ and

$${\displaystyle \frac{d\widehat{\phi }}{dR}}=-\mu ,C_R:={\displaystyle \frac{d^2\widehat{\phi }}{d{R}^2}}={\displaystyle \frac{{\left({\mu }^{\prime }\right)}^2}{K_{\rho }}}={\left({\mathcal{V}}_{\mu \mu }\right)}^{-1}>0,$$

(62)

for $\widehat{\phi }\left(R\right)=1-\mathcal{V}+\mu {\mathcal{V}}_{\mu }$. The Legendre pair $\mathsf{\Phi }=1+{\mathcal{V}}^{*}(-R)$ has

$$g_{\mathrm{sc}}={\mathcal{V}}_{\mu \mu }d{\mu }^2=C_{R}d{R}^2,{\nabla }^2(\mathsf{\Phi }\circ R)=C_{R}dR\otimes dR-\mu {\nabla }^{2}R.$$

(63)

Thus the susceptibility and phase-fixed Jacobi block come from one reduced density.

Let $B_{\mu \nu }=Y_{\mu \nu }-k_{\mu \nu }{k}^{\alpha \beta }{Y}_{\alpha \beta }/3$. For $A=\phi B$, the multiplier equation is $C_{\alpha \mu \nu }^B+{\theta }_{\alpha }{B}_{\mu \nu }-{\theta }_{\mu }{B}_{\alpha \nu }=0$, $\theta =d\log \left|\phi \right|$, with ${\theta }_{\alpha }^B=-{\beta }^{\mu \nu }{C}_{\alpha \mu \nu }^B/3$ when B is invertible.

Proposition 11

(Regular Codazzi normal form and flat one-sided reduction). On a regular multiplier patch the marked $E/F$ splitting gives $S_A=k^{-1}A={\lambda }_E{P}_E+{\lambda }_F{P}_F$, $\Delta :={\lambda }_E-{\lambda }_F\ne 0$. The standard two-eigenvalue Codazzi identities [28], with the Lorentzian extension represented by [29], imply $P_E^{*}d{\lambda }_E=P_F^{*}d{\lambda }_F=0$, $N_J=0$, umbilicity of both distributions, and

$$k={({\lambda }_E-{\lambda }_F)}^{-2}\left(h_E\left(x\right)\oplus h_F\left(y\right)\right),{\lambda }_E={\lambda }_E\left(y\right),{\lambda }_F={\lambda }_F\left(x\right).$$

(64)

Conversely, nondegenerate product data of this form define a two-eigenvalue Codazzi tensor. In the retained Parent normal form $u_{\mathrm{AC}}=3({\lambda }_E+{\lambda }_F)/({\lambda }_F-{\lambda }_E)$ and ${\lambda }_E<0<{\lambda }_F$ with $-1<{\lambda }_E/{\lambda }_F<-1/2$. Flatness of the retained background forces on each connected regular product patch

$$d{\lambda }_E\equiv 0\mathrm{or}d{\lambda }_F\equiv 0.$$

(65)

The two branches are dual affine-warps. On a nonparallel patch the flat metric is locally $\eta =d{r}^2+ϵd{y}^2+r^2\overline{h}$, $K_{\overline{h}}=1$. A connected complete flat base with globally positive affine warp has no such nonparallel branch; a global defect must meet a boundary/horizon or leave the regular patch. The full tensor Codazzi residual and reconstructed flatness remain final checks. Writing $S_B=b_E{P}_E+b_F{P}_F$, the multiplier slope and mean-curvature readings are

$$\begin{array}{cc}\hfill {\theta }^B& =-P_E^{*}d\log |b_E|-P_F^{*}d\log \left|b_F\right|,\hfill \\ \hfill H_E& ={\displaystyle \frac{P_F({\left(d{b}_E\right)}^{♯}+b_E{\left({\theta }^B\right)}^{♯})}{b_E-b_F}},H_F={\displaystyle \frac{P_E({\left(d{b}_F\right)}^{♯}+b_F{\left({\theta }^B\right)}^{♯})}{b_F-b_E}}.\hfill \end{array}$$

(66)

On the multiplier locus ${\theta }^B=d\log \left|\phi \right|$. Conversely, $d{\theta }^B=0$ together with (66), integrability, and umbilicity reconstructs the multiplier/Codazzi equation. If $b_E/b_F$ is constant and differs from $0,1$, then ${\nabla }^{\left(k\right)}J={\nabla }^{\left(k\right)}A=0$; a nonparallel regular sector therefore requires spectral-shape variation.

The independently extracted Hilbert/stress form gives ${\theta }^B=a_{\mathrm{AC}}\left(R\right)dR$ and hence

$$a_{\mathrm{AC}}\left(R\right)P_E^{*}dR=-P_E^{*}d\log |b_E|,a_{\mathrm{AC}}\left(R\right)P_F^{*}dR=-P_F^{*}d\log \left|b_F\right|.$$

(67)

These projected compatibility conditions reconstruct the corresponding components of $dR$ when $a_{\mathrm{AC}}\ne 0$; the scalar stiffness and mixed Hessian entries remain independent response tests. On $q_{\mathrm{rot}}\ne 0$, $H_{\omega }^{\mathrm{AC}}=u_{\mathrm{AC}}/q_{\mathrm{rot}}$ and the retained cross-layer target is

$$u_{\mathrm{AC},*}=t_{*}:={\displaystyle \frac{91-48\sqrt{3}}{37}},q_{\mathrm{rot},*}{H}_{\omega ,*}^{\mathrm{AC}}=u_{\mathrm{AC},*}.$$

(68)

The product identity is automatic, whereas $t_{*}$ is imposed only after the independently typed Parent-to-reduced incidence. Along an integral curve $\gamma$ of U, current conservation (61) gives $p_{\mathsf{\Lambda }}V\left(\rho \left(\gamma \left(s\right)\right)\right)\exp \left({\int }_{s_0}^s{\nabla }_{\mu }^{\left(k\right)}{U}^{\mu }\left(\gamma \left(\sigma \right)\right)d\sigma \right)=\mathrm{const}$, an equivalent numerical holdout when the phase equation is omitted from the Newton system. Its principal normal symbol is the scalar map $V_1\to V_0$; the affine multiplier residual contributes no independent same-field cubic constitutive vertex.

The regular flat classification reduces the reference-phase search to the two one-sided affine-warp branches. A compact hyperbolic resolved profile is retained only as a regulated scalar/Green benchmark. With ${\Sigma }_g$, $g\ge 2$, unit-mass mollifier ${\rho }_{\epsilon }$, fixed mean ${\overline{\omega }}_s\ne 0$, and the warped-product conventions of [11], set ${\beta }_{\epsilon }={\Delta }_{\gamma ,0}^{-1}(1-{\mathrm{Area}}_{\gamma }\left({\Sigma }_g\right){\rho }_{\epsilon })$ and ${\alpha }_{\epsilon }={\overline{\omega }}_s{\beta }_{\epsilon }$. Then ${\Delta }_{\gamma }{\alpha }_{\epsilon }={\overline{\omega }}_s-{\overline{\omega }}_s{\mathrm{Area}}_{\gamma }\left({\Sigma }_g\right){\rho }_{\epsilon }$, with the additive constant fixed by the standard elliptic normalization [35]. Its coupling to $R_{\mathrm{vort}}$ remains a seed-geometry incidence.

Proposition 12

(Local Parent nonemptiness and fixed-sector density realizability). Let $\eta ,J_{*},U_{*},{\kappa }_{*}>4,R_{*},{\rho }_{*}$ be constant, ${\mathcal{F}}_{*}=0$, ${\vartheta }_{*}=\mathrm{const}$, and assume ${\mu }_{*}>0$, ${\phi }_{*}>0$, $V_{*}^{\prime }+R_{*}{\mu }_{*}^{\prime }=0$. If $D{\kappa }_{\mathrm{base}}{{|}_{*}=D{R}_{\mathrm{vort}}|}_{*}=0$ and $R_{\mathrm{vort}}\left[{\mathsf{\Psi }}_{*}\right]=R_{*}$, the constant configuration is stationary for admissible primary variations and satisfies the current and Codazzi equations. On a finite-dimensional axial fixed-sector tangent with vanishing first seed jets, $D_{\mathsf{\Psi }}^2{\mathcal{L}}_A{|}_{*}={\sigma }_A[{\phi }_{*}{p}_0{D}^2\left({\kappa }_{\mathrm{base}}^2\right){|}_{*}-p_{\mathsf{\Lambda },*}{\mu }_{*}{D}^2{R}_{\mathrm{vort}}{|}_{*}]$, apart from independent derivative slots. Hence every symmetric projected local fixed-sector density Hessian is realizable for ${\phi }_{*}\ne 0$.

This establishes local class nonemptiness on the parallel/product sublocus, not existence of a nonparallel defect.

Proposition 13

(Action-native seed jets and identifiability). After density elimination ${\mathcal{L}}_{\mathrm{red}}={\sigma }_{A}P\widehat{\phi }\left(R\right)$, $P=p_{\mathsf{\Lambda }}$, $R=R_{\mathrm{vort}}\left[\mathsf{\Psi }\right]$, and

$$\begin{array}{cc}\hfill {\sigma }_A^{-1}D{\mathcal{L}}_{\mathrm{red}}& =\widehat{\phi }DP-P\mu DR,\hfill \\ \hfill {\sigma }_A^{-1}{D}^2{\mathcal{L}}_{\mathrm{red}}& =\widehat{\phi }{D}^{2}P-\mu (DP\otimes DR+DR\otimes DP)+P(C_{R}DR\otimes DR-\mu D^{2}R).\hfill \end{array}$$

(69)

Thus the Euler/Jacobi operators require the actual first and second seed jets, and higher LS coefficients the corresponding higher jets. On the gauge-flat reference phase, action jets alone do not separately identify ${\kappa }_{\mathrm{base}}$ and $R_{\mathrm{vort}}$; their provenance is supplied by the independent metric/Codazzi, current, Hilbert, and multipole interfaces.

Theorem 4

(Conditional thin-core extraction). Let the regularized core currents have locally bounded mass and boundary mass, no interior boundary on the singular-source-free collar, primitive limiting linking charge, vanishing trace-adjusted Codazzi residual away from the limiting support, and a uniform nonzero frozen gap. If the selected component is regular, timelike, and multiplicity one, a subsequence converges locally to an integral current whose selected component is a timelike worldline Γ. Its oriented normal sphere is $S_{\mathsf{\Gamma }}^2\simeq {\mathbb{CP}}_{\mathsf{\Gamma }}^1$, and the limiting charge gives the primitive line of (11); Codazzi holds distributionally off the current.

The compactness input is [71] in the form used in geometric measure theory [72]; concentrated-core comparisons are represented by [73,74]. Ordinary higher moments vanish under thin-core rescaling, so protected dipole/quadrupole data require a separately typed multipole-bearing current or equivalent derivative-distribution export. Its non-scalar normal data are $m\in N_{\mathsf{\Gamma }}$ and $M_0\in {\mathrm{Sym}}_0^2\left(N_{\mathsf{\Gamma }}\right)$ with harmonic representative in $V_1\left[1\right]\oplus V_2\left[2\right]$.

Proposition 14

(Null-Codazzi origin of the Veronese and mixed Jordan line). For $\zeta =\tau U^{♭}+\xi$, the principal Codazzi symbol on trace-free symmetric tensors has a nonzero kernel exactly for null ζ, generated by $\zeta \otimes \zeta$. Its spatial trace-free projection is the Veronese class $Q_e$ of (14); the scalar, mixed-vector, and spatial trace-free components have one common amplitude, with mixed equation $\tau \pi ={(\xi \odot q)}_0$. This supplies the multiplicity-one Jordan tensor type entering the protected product jet. Application to the actual Parent still requires the trace-free/frozen-scalar incidence and an independent normalization.

The corresponding Gauss-local charges are the current flux in $V_1$ and trace-free Codazzi charge in $V_2$:

$$\begin{array}{cc}\hfill Q_1\left(r\right)& ={\int }_{S_r^2}m{j}_{\mathrm{tr}}^i{n}_{i}dS,\hfill \\ \hfill \langle Q_2\left(r\right),M\rangle & ={\int }_{S_r^2}\left(u{\partial }_n{H}_M-H_M{\partial }_{n}u\right)dS.\hfill \end{array}$$

(70)

They are radius-independent on a source-free principal annulus and extend to the curved collar by the corresponding current and Green pairings. For nondegenerate $B_0\in V_2$ and $0\ne \xi \in N_{\mathsf{\Gamma }}^{*}$, the principal multiplier source has

$$0⟶\mathbb{R}⟶V_2\oplus V_1\stackrel{{\mathsf{\Phi }}_{\xi ,B_0}}{\to }{\mathcal{S}}_{\xi ,B_0}⟶0,0⟶\mathbb{R}{B}_0⟶V_2⟶{\mathcal{S}}_{\xi ,B_0}/\mathrm{Im}{\mathcal{M}}_{B_0}⟶0,$$

(71)

with lost/surviving/ambient ranks $(1,4,5)$; on $B_0=\beta Q_e$ the quotient splits into axial weights one and two.

Theorem 5

(Veronese Frenet-Gauss source jets). For unit-speed $e\left(t\right)\subset S^2$ with signed geodesic curvature ${\kappa }_g$, write $Q\left(e\left(t\right)\right)=Q_e+tA+t^{2}B+O\left(t^3\right)$ and $B_H=B+3Q_e/2$, ${\chi }_H=\sqrt{3}\parallel B_H{\parallel /\parallel A\parallel }^2$, ${\chi }_R=\sqrt{3}\parallel B\parallel /{\parallel A\parallel }^2$. The standard Veronese geometry gives

$${\chi }_H^2={\displaystyle \frac{3}{8}}(1+{\kappa }_g^2),{\chi }_R^2={\displaystyle \frac{3}{8}}(4+{\kappa }_g^2),{\chi }_R^2-{\chi }_H^2={\displaystyle \frac{9}{8}}.$$

(72)

and

$$Q^{\left(3\right)}=-(4+{\kappa }_g^2)A+3{\kappa }_g(v\otimes n+n\otimes v)+{\dot{\kappa }}_g(e\otimes n+n\otimes e).$$

(73)

Thus $B_H$ is the Gauss-corrected second jet rather than an independent source tensor; at ${\kappa }_g=0$, $Q^{\left(3\right)}=-4A$ and no independent cubic source direction remains.

4.2. Canonical Local Response Closure and Integral Incidence

The normal form above fixes the metric-density, Rainich, current, and scalar-Hodge bookkeeping of the admissible Parent class. The remaining local realization data are kept as one typed response packet ${\mathcal{P}}_{\mathrm{loc}}=(\Sigma ,Z_{\mathrm{H}},G,T_{\mathrm{VH}},{\nabla }_{\mathrm{ax}})$, comprising the principal symbol, constitutive product jet, positive response metric, source-Hodge graph, and marked axial transport. These entries occupy different Taylor or transport slots and are compared only after their provenance has been established. Their common target is the primitive multiplication geometry of Section 3.2.

For the marked axial-transport entry, let S denote the regular family cycle of (45), let $A_{\mathrm{mic}}^{†}=-A_{\mathrm{mic}}$ be the centered projected connection jet, and put ${\mathsf{\Pi }}_r^S\left(T\right)=3^{-1}{\sum }_{j=0}^2{\omega }^{rj}{S}^{j}T{S}^{-j}$. With $R_1={\mathsf{\Pi }}_1^S\left(A_{\mathrm{mic}}\right)$, $a_1^2={\parallel R_1\parallel }_{\mathrm{HS}}^2/3$, ${\Delta }_{\mathrm{conn}}=a_1^6-{\left|\det R_1\right|}^2$, ${\delta }_H={\Delta }_{\mathrm{conn}}/a_1^6$, and ${ϵ}_U^2={\parallel R_1^{†}{R}_1-a_1^2\mathbf{1}\parallel }_{\mathrm{HS}}^2/\left(6a_1^4\right)$, the retained readings are $\parallel {\mathsf{\Pi }}_0^S\left(A_{\mathrm{mic}}\right){\parallel }_{\mathrm{HS}}$ and ${ϵ}_U^2\le {\delta }_H\le 4{ϵ}_U^2$, with $|\det R_1/a_1^3{|}^2=1-{\delta }_H$. The determinant phase is assigned only after the marked-loop attachment. The parent angular Casimir obeys ${\mathcal{C}}_{\partial }=6{\mathcal{L}}_K$; after the common normalization the direct ray is $G_{4,t}^{\mathrm{dir}}=2t^2{P}_{V_1}+P_{V_2}$, which lies on the ${\mathcal{L}}_K$ ray at $t^2=1/6$, while its inverse lies on the $G_K$ ray of (27). The same direct/compliance pair follows from $H_{\mathrm{ARC}}=2{\varkappa }_W{\mu }_{\mathsf{\Gamma }}{P}_{V_1}+6{\varkappa }_C{P}_{V_2}$ when ${\varkappa }_C={\varkappa }_W{\mu }_{\mathsf{\Gamma }}$. These are compatibility readings of the G entry of ${\mathcal{P}}_{\mathrm{loc}}$; microscopic provenance remains separately typed.

For real normal tensors below, $u\odot v:=(u\otimes v+v\otimes u)/2$ and ${(·)}_0$ denotes trace-free projection; $V_1$ and $V_2$ carry the induced Euclidean and Frobenius metrics.

Proposition 15

(Principal Clifford incidence). Use the normal metric to identify $V_1\simeq \mathfrak{so}\left(N_{\mathsf{\Gamma }}\right)$ by $u\mapsto {\left[u\right]}_{\times }$ and $V_2={\mathrm{Sym}}_0^2\left(N_{\mathsf{\Gamma }}\right)$. For $0\ne \xi \in N_{\mathsf{\Gamma }}$, put $A_{\xi }b={\left[\xi \right]}_{\times }b$. Since $V_1\otimes V_1\simeq V_0\oplus V_1\oplus V_2$ with multiplicity one, every equivariant tensor completion has the form $C_{\xi }u=a{[\xi \times u]}_{\times }+c{(\xi \odot u)}_0$; after removal of common nonzero scale and sign its type is the single point $[a:c]\in {\mathbb{RP}}^1$. For $G_{h,\beta }=h{P}_{V_1}+\beta P_{V_2}$, isotropy of the principal Codazzi Gram is equivalent to $h=3\beta$. On this metric define

$${\mathcal{R}}_{\Sigma ,\mathrm{proj}}:={\displaystyle \frac{|6a+c|}{\sqrt{37}\sqrt{a^2+c^2}}},{\mathcal{R}}_{\Sigma ,\mathrm{proj}}=0⟺[a:c]=[1:-6].$$

(74)

Multiplicity-one orthogonality makes $(a,c)$ the Hilbert-Schmidt tomography coordinates of the microscopic principal symbol. Since ${\mathrm{Hom}}_{SO\left(3\right)}(V_0,V_1\oplus V_2)=0$, the scalar current/Ward datum cannot select a nonzero point of this projective line; the required coefficients belong to the Hilbert/stress principal variation. The zero-residual line gives orthogonality of $C_{\xi }\left(V_1\right)$ and $A_{\xi }\left(V_2\right)$; equality of the Clifford constant fixes $c^2=3$ only after the common metric normalization has been attached. Up to the common sign,

$$C_{\xi }\left(u\right)=\pm \sqrt{3}\left[{(\xi \odot u)}_0-{\displaystyle \frac{1}{6}}{[\xi \times u]}_{\times }\right],{\Sigma }_{\xi }(u,b)=C_{\xi }\left(u\right)+A_{\xi }b,$$

(75)

and

$${\Sigma }_{\xi }^{†}{G}_{3\beta ,\beta }{\Sigma }_{\eta }+{\Sigma }_{\eta }^{†}{G}_{3\beta ,\beta }{\Sigma }_{\xi }=2\beta \langle \xi ,\eta \rangle \mathbf{1}.$$

(76)

The principal condition number is $\sqrt{\max \{h/3,\beta \}/\min \{h/3,\beta \}}$, so $h=3\beta$ is the unique isotropic optimum. If the source norm and target Gram are retained by exact recovery, matching the common scale to (27) makes the fixed $A_{\xi }$ leg isometric and reduces full-symbol recovery to (76). Its associative left/right reading is the marked ratio $[1:2]$; a scalar-current residual affects only the longitudinal slice. The microscopic gate remains unless Σ is independently typed as a protected observation.

Lemma 3

(Self-adjoint constitutive first-jet normal form). For $m\in V_1$ and $M\in V_2$, put $S_{m}u={(m\odot u)}_0$, $B_{M}u=Mu$, $C_{M}b={(Mb+bM)}_0$, and $R_{M}u=[M,{\left[u\right]}_{\times }]$. These four multiplicity-one tensor lines are the Jordan and Lie projections of the ordinary associative product in the split ${\mathfrak{sl}}_3$ envelope of Theorem 1. Every $SO\left(3\right)$-equivariant formally self-adjoint zero/first frozen-moment endomorphism of $V_1\oplus V_2$ is uniquely of the form

$$Z_{\mathrm{H}}(m,M)=\left(\begin{array}{cc}{z}_1\mathbf{1}+b_{11}{B}_M& a{S}_m^{†}+b_{12}{R}_M^{†}\\ a{S}_m+b_{12}{R}_M& z_2\mathbf{1}+b_{22}{C}_M\end{array}\right),$$

(77)

with six real coefficients $z_1,z_2,a,b_{11},b_{12},b_{22}$.

For an $SO\left(3\right)$-invariant same-field cubic $F_3=\gamma \langle M,{(m\otimes m)}_0\rangle +\delta \mathrm{Tr}\left(M^3\right)$, the Hessian first jet has $a=b_{11}=2\gamma$, $b_{12}=0$, and $b_{22}=3\delta$; hence a scalar same-field cubic cannot generate the required Lie line. The mixed multiplier variation supplies that line because the trace-free part of ${\left[\eta \right]}_{\times }b$ is $-R_b\eta /2$. Adding only this Lie line to the same-field cubic leaves $b_{11}=a$ and hence $r_{\mathrm{prod}}=\sqrt{3}$, so the protected product locus still requires the independent relative Jordan response. A Schur-eliminated $V_1$ mediator may alternatively generate $b_{12}^{\mathrm{eff}}=-g_{\mathrm{med}}{h}_{\mathrm{med}}/\Delta$, but this level retains only the product $g_{\mathrm{med}}{h}_{\mathrm{med}}/\Delta$.

On the protected nonzero-product branch write $a={\sigma }_a{g}_P$, $b_{11}=h_P$, $b_{12}={\sigma }_K{g}_P/\sqrt{2}$, and $b_{22}=-h_P$, with ${\sigma }_a,{\sigma }_K=\pm 1$. The single product-jet condition on the action class is $g_P=\sqrt{3}{h}_P$, so

$$(a,b_{11},b_{12},b_{22})=h_P\left({\sigma }_a\sqrt{3},1,{\sigma }_K\sqrt{{\displaystyle \frac{3}{2}}},-1\right).$$

(78)

It constrains derivatives of (57) and introduces no additional action term. For $m=re\ne 0$ and $M=\nu Q_e$, cross-block isotropy is equivalent to $|\nu /r|=1/\sqrt{6}$. More generally the oriented axial stabilizer gives ${\dim }_{\mathbb{R}}{\mathrm{Hom}}_{SO\left(2\right)}(V_1,V_2)=3$; in adapted orthonormal bases every equivariant cross block is

$$T_{\mathrm{ax}}(\alpha ,\beta ,\gamma )=\left(\begin{array}{ccc}0& 0& \alpha \\ \beta & -\gamma & 0\\ \gamma & \beta & 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),T_{\mathrm{ax}}^{†}{T}_{\mathrm{ax}}=({\beta }^2+{\gamma }^2)P_{\perp }+{\alpha }^2{P}_{\Vert }.$$

(79)

The tangent coefficient $\beta +i\gamma$ is the unique local phase channel; an $O\left(2\right)$ reflection removes $\gamma$. At the canonical projective magnitudes, $9{\alpha }^2=16({\beta }^2+{\gamma }^2)$ and ${\beta }^2=3{\gamma }^2$, so $\left|\arg \right(\beta +i\gamma \left)\right|=\pi /6$ modulo $\pi$. On the marked positive-tangent branch this gives $U_{\mathrm{pol}}=R_{{\sigma }_G\pi /6}$ on $e^{\perp }$, extended by the identity on $\mathbb{R}e$; an overall tangent sign remains separately typed. Positive axial spectra determine ${\alpha }^2$ and ${\beta }^2+{\gamma }^2$ but not the oriented phase. The square of $U_{\mathrm{pol}}$ is a local order-six rotation; identification with the global Eisenstein ${\mu }_6$ remains part of the later marked holonomy incidence.

For an independently exported second variation, let the product be read in the left/right standard form of (28). For $G=g_1{P}_{V_1}+g_2{P}_{V_2}$ put $r_{\mathrm{prod}}=g_1/\left(3g_2\right)$ and parametrize $b_{11}=-c_J/3$, $b_{12}=c_K/\sqrt{6}$. Whitening (77) gives $|c_K/c_J|=r_{\mathrm{prod}}^{-1/2}$ and, on the factorized locus, $2|b_{12}{|}^2=\sqrt{3}\left|a{b}_{11}\right|$ with $r_{\mathrm{prod}}=\sqrt{3}|b_{11}|/|a|=3b_{11}^2/\left(2b_{12}^2\right)=2b_{12}^2/a^2$.

Proposition 16

(Common protected metric correspondence). After removal of the common positive scale, the observer/Rees form, principal-symbol isotropy, the protected product condition $r_{\mathrm{prod}}=1$, and $r_{\mathrm{stiff}}:={\varkappa }_C/\left({\varkappa }_W{\mu }_{\mathsf{\Gamma }}\right)=1$ select the same direct/compliance pair $({\mathcal{L}}_K,G_K)$. Isotropic cross-block recovery then gives $c_{\mathrm{shell}}=1/\sqrt{6}$, and for Jordan/Lie normalizations $|{\kappa }_L/{\kappa }_J|=\sqrt{2}$. These are compatibility readings of one positive target metric; microscopic provenance remains a typed Parent test.

For a nonzero independently computed second variation, the projective product defect may be read as ${\mathcal{R}}_{\mathrm{prod}}^2=\left(\right|a|/(\sqrt{3}|b_{11}{\left|\right)-1)}^2+\left(\right|b_{12}/b_{11}{|-\sqrt{3/2})}^2$. On the same-field-plus-Lie locus $a=b_{11}$ it has the fixed positive lower bound $(4-2\sqrt{3})/3$, so that locus is excluded before any mediator interpretation. The stiffness is read directly as $r_{\mathrm{stiff}}=\mathrm{Tr}\left(P_{V_2}{H}_{\mathrm{ARC}}\right)/\left[5\mathrm{Tr}\left(P_{V_1}{H}_{\mathrm{ARC}}\right)\right]$. If $Z_{\mathrm{H}}$ and $H_{\mathrm{ARC}}$ are successive Taylor levels of one Parent Hessian, exact stiffness together with (78) gives $(z_1,z_2;a,b_{11},b_{12},b_{22})=(z_1,3z_1;{\sigma }_a\sqrt{3}\mu ,\mu ,{\sigma }_K\sqrt{3/2}\mu ,-\mu )$. Relative signs remain marking-dependent.

Theorem 6

(Canonical Veronese-Hodge tangent/normal response). On the axial branch let $e=m/\left|m\right|$ and $q\left(x\right)={(x\otimes x)}_0$. Put $D_e=D{q}_e$ and normalize the source-Hodge compensator from (71) by $H_e=-{\mathcal{A}}_e^{+}{\mathcal{M}}_{Q_e}:V_1\to V_2$. Both maps have image $U_e=\mathbb{R}{Q}_e\oplus \{e\otimes v+v\otimes e:v\perp e\}$. For $v\perp e$,

$$D_{e}e=2Q_e,D_{e}v=e\otimes v+v\otimes e,H_{e}e=-Q_e,H_{e}v={\displaystyle \frac{2}{3}}(e\otimes v+v\otimes e),R_e:=H_e^{-1}{D}_e=-2P_{\Vert }+{\displaystyle \frac{3}{2}}{P}_{\perp }.$$

(80)

Let $H_e=U_e^{\mathrm{H}}\left|H_e\right|$ be the polar decomposition. Since $G_K{|}_{V_1}=3\mathbf{1}$, the Green-whitened relative-polar map

$$T_e^{\mathrm{VH}}:=U_e^{\mathrm{H}}|R_e|G_K^{-1/2}={\displaystyle \frac{1}{\sqrt{3}}}{U}_e^{\mathrm{H}}\left|R_e\right|,G_3:={\left(T_e^{\mathrm{VH}}\right)}^{†}{T}_e^{\mathrm{VH}}={\displaystyle \frac{4}{3}}{P}_{\Vert }+{\displaystyle \frac{3}{4}}{P}_{\perp }$$

(81)

has graph projector $P_3^{\mathrm{VH}}$ with compression spectrum $\{3/7,4/7,4/7\}$. Let $V_2^{\left(2\right)}$ be the axial weight-two plane and let $P_{\mathrm{bx}}$ denote its orthogonal projector. Using $\mathcal{V}$ from (14), put $\mathsf{\Phi }(x,\widehat{e})=(1/2-2x/\sqrt{3})\mathcal{V}\left(\widehat{e}\right)$. At $(0,e)$ its first fundamental form and the discriminant Hessian of (15) obey

$$d{\mathsf{\Phi }}_{(0,e)}^{†}d{\mathsf{\Phi }}_{(0,e)}=G_3,{\nabla }^2{\Delta }_{A_2}{{|}_{s{Q}_e}=4s^4{P}_{\mathrm{bx}},\ker {\nabla }^2{\Delta }_{A_2}|}_{s{Q}_e}=U_e.$$

(82)

Thus the phase-free source-Hodge image is the tangent space of the nonzero Veronese/discriminant cone, while the biaxial doublet is its normal space. The projector $P_{\mathrm{bx}}$ is geometric; its physical stiffness coefficient remains a Parent Hessian datum. If $A_P=P_{V_1}{P}_3{P}_{V_1}{|}_{V_1}$, define

$${\mathcal{R}}_{\mathrm{CM}}\left(P_3\right):=49A_P^2-49A_P+12\mathbf{1},{\mathcal{R}}_{\mathrm{CM}}\left(P_3^{\mathrm{VH}}\right)=0,\mathrm{Tr}{A}_{P_3^{\mathrm{VH}}}={\displaystyle \frac{11}{7}}.$$

(83)

The longitudinal squared gain of $|R_e|$ is 4, while the transverse Green-whitened coefficient on the geodesic branch agrees with ${\chi }_H=\sqrt{3/8}$ in (72); the denominator-seven packet is the graph normalization of these two singular gains.

The null-characteristic differential of Proposition 14, the product phase in (79), and the phase-free tangent geometry of Theorem 6 occupy different typed slots of one Parent packet. Positive Gram data determine the two singular gains but not the oriented tangent phase. On the tangent plane the protected product jet carries the oriented complex coefficient of (79), whereas the same-level real source-Hodge residual Gram with kernel $\mathrm{graph}{T}_e^{\mathrm{VH}}$ has a phase-free transverse cross block. These readings therefore cannot be identified with one Taylor-level real Hessian coefficient. A common Parent requires typed derivative and residual slots rather than one Hessian coefficient serving all roles.

On a branch satisfying the typed target (68), common typing gives ${\mathcal{J}}_{\mathrm{br}}={(4G_3/3)}^{-3/4}$, ${\mathcal{A}}_e=(3\sqrt{6}/8){(4G_3/3)}^{3/4}$, and $H_e{\mathcal{A}}_e=T_e^{\mathrm{VH}}$. Thus the Hodge anisotropy is a spectral function of the same positive source-Hodge Gram rather than an independent susceptibility. In the source-Hodge frame the phase-free scaling is $S_{\mathrm{VH}}=-2P_{\Vert }/\sqrt{3}+\sqrt{3}{P}_{\perp }/2$; the product-selected frame differs by $U_{\mathrm{pol}}^{-1}$ and is parallel under the common marked axial connection. Native provenance therefore separates into the graph test below and an output-unitary invariant normalization test.

Proposition 17

(Native Parent provenance of the Veronese-Hodge graph). Let $D{R}_{\mathrm{con}}=\left(J_1{J}_2\right):V_1\oplus V_2\to {\mathcal{E}}_{\mathrm{con}}$ be the derivative of an independently defined constitutive residual with metric $G>0$. If $J_2^{†}G{J}_2>0$ and $\mathrm{rank}D{R}_{\mathrm{con}}=5$, its kernel is the graph of $T_{\mathrm{phys}}=-{\left(J_2^{†}G{J}_2\right)}^{-1}{J}_2^{†}G{J}_1$, and

$$P_3^{\mathrm{phys}}=P_3^{\mathrm{VH}}⟺J_1+J_2{T}_e^{\mathrm{VH}}=0,{\mathcal{R}}_{\mathrm{VH}}^{\mathrm{jet}}:={\parallel J_1+J_2{T}_e^{\mathrm{VH}}\parallel }_{G,\mathrm{HS}}.$$

(84)

With $D=J_2^{†}G{J}_2$, the residual is equivalent to the graph mismatch through $\sqrt{{\lambda }_{\min }\left(D\right)}\parallel T_e^{\mathrm{VH}}-T_{\mathrm{phys}}{\parallel }_{\mathrm{HS}}\le {\mathcal{R}}_{\mathrm{VH}}^{\mathrm{jet}}\le \sqrt{{\lambda }_{\max }\left(D\right)}{\parallel T_e^{\mathrm{VH}}-T_{\mathrm{phys}}\parallel }_{\mathrm{HS}}$. At zero residual the native Gram factors through the canonical graph residual, so (84) is the decisive microscopic graph test.

On the passed graph locus, assume in addition that $D:=J_2^{†}G{J}_2$ preserves the marked axial decomposition. Writing $D=d_0{P}_0+d_1{P}_1+d_2{P}_2$ on the output sectors of multiplicities $1,2,2$, output-unitary canonicalization gives $\tilde{J}=D^{1/2}(-T_e^{\mathrm{VH}}{\mathbf{1}}_{V_2})$. If ${\mu }_0,{\mu }_1,{\mu }_2$ denote the corresponding nonzero squared singular values, then

$${\mathrm{Spec}}_{+}\left({\tilde{J}}^{†}\tilde{J}\right)=\left\{{\displaystyle \frac{7}{3}}{d}_0,{\left({\displaystyle \frac{7}{4}}{d}_1\right)}^{\left(2\right)},d_2^{\left(2\right)}\right\},d_0=d_1=d_2⟺{\displaystyle \frac{3}{7}}{\mu }_0={\displaystyle \frac{4}{7}}{\mu }_1={\mu }_2.$$

(85)

Thus graph provenance and axial normalization are independently testable without choosing an output frame; the absolute common scale and marked orientation remain Parent data. The principal-recovery and Veronese-Hodge $3|5$ splittings remain inequivalent typed decompositions; in the common whitened metric their squared principal-angle cosines are $\{4/7,13/28,13/28\}$ and their squared Hilbert-Schmidt distance is 3, while the product polar phase is a separate transported datum. Writing $\delta :=\parallel {\mathcal{R}}_{\mathrm{CM}}\left(P_3\right){\parallel }_{\mathrm{op}}$, $\delta <1/4$ confines each compressed eigenvalue to within $(1-\sqrt{1-4\delta })/14$ of $\{3/7,4/7\}$; across an isolated gap g, an off-diagonal perturbation of norm $\epsilon <g/2$ moves the corresponding projector by at most $\epsilon /(g-\epsilon )$.

Corollary 4

(Veronese-Hodge packet and integral completion). Put $P_3=P_3^{\mathrm{VH}}$, $P_5=\mathbf{1}-P_3$, ${\mathsf{\Gamma }}_{\mathrm{sel}}=P_5-P_3$, ${\mathsf{\Gamma }}_{\mathrm{gr}}=P_{V_1}-P_{V_2}$, and ${\delta }_{\mathrm{ov}}:=|\mathrm{Tr}\left(P_3{P}_{V_1}\right)-\mathrm{Tr}\left(P_5{P}_{V_1}\right)|$. Then

$${\delta }_{\mathrm{ov}}={\displaystyle \frac{1}{7}},\left|\mathrm{Tr}\left({\mathsf{\Gamma }}_{\mathrm{sel}}{\mathsf{\Gamma }}_{\mathrm{gr}}\right)\right|={\displaystyle \frac{16}{7}},\mathrm{Tr}\left(P_5{P}_{V_2}\right)={\displaystyle \frac{25}{7}}.$$

(86)

Hence the ordered rational packet is $(1,4,5;7)$. If ${\mathcal{R}}_{\mathrm{VH}}^{\mathrm{jet}}=0$ and this packet is retained as an integral reading of the same ordered $A_2$ datum as the equianharmonic torsor, the selected orientation gives $\xi =(1-4\omega )/(1-\omega )=2-\omega$, while the opposite orientation gives $\overline{\xi }=3+\omega$. The relative grading product supplies an independent CM-field checksum with nontrivial eigenvalue real parts $\pm 1/7$; it is not counted as a second selector.

The inequivalence is already witnessed by the recovery intersection $\mathbb{R}{(\xi \odot \xi )}_0\subset V_2$ and its $V_1$ compression spectrum $\{0,1/4,1/4\}$, whereas $P_3^{\mathrm{VH}}$ is a graph over all of $V_1$. On the exact VH fixed-gain leaf the remaining transverse phase is transported by the same marked connection.

If ${\mathcal{R}}_{\mathrm{VH}}^{\mathrm{jet}}=0$ and the constitutive and equianharmonic packets are retained integral readings of the same ordered protected $A_2$ correspondence, exact PLSR identifies their marked lattice class; the microscopic projector incidence remains a separate test.

The axial source-Hodge transport supplies this existence test. Before discrete reduction, the twisted-line tangent and the active weak plane have axial weights $-2$ and $+1$, so ${\mathrm{Hom}}_{SO\left(2\right)}(W_{-2},W_{+1})=0$. Their restrictions carry the same nontrivial $C_3$ character, which is why a nonzero linear bridge requires the physical $SO\left(2\right)\to C_3$ reduction. Real-compatible integral incidence then has the form

$$M=p{U}_d,p\in \mathbb{Z},U_d\in \mathbb{Z}{\left[\omega \right]}^{\times },N_{\mathrm{inc}}=p^2,\mathrm{coker}M\simeq E_{\omega }\left[\right|p\left|\right].$$

(87)

If the incidence factors through the parity coboundary $1-\tau =2$ and the pre-parity map lies on the scalar/unit class $\mathsf{\Phi }=kV$ with $V\in \mathbb{Z}{\left[\omega \right]}^{\times }$, then

$$M=U_d(1-\tau )\mathsf{\Phi }=2k{U}_{d}V,p=2k,\left|p\right|=2⟺\left|\det \mathsf{\Phi }\right|=1⟺{\mathsf{\Phi }}^T{G}_{A_2}\mathsf{\Phi }=G_{A_2}.$$

(88)

The equivalence is restricted to this Eisenstein scalar/unit class, and the unimodularity test is applied before the parity factor. Independently, $E_{\omega }\left[n\right]$ contains exactly one free $C_3$ orbit only for $n=2$, giving a second upstream selector of the same incidence modulus. For the actual neutral linear response $R_{\nu }$, let ${\rho }_1$ and ${\rho }_Y$ denote the retained axial actions on the source and target, and let ${ϵ}_{\mathrm{eq}}:={\sup }_g\parallel {\rho }_Y\left(g\right)R_{\nu }-R_{\nu }{\rho }_1\left(g\right)\parallel$ denote its equivariance defect. Reynolds projection gives

$${\mathcal{P}}_{\mathrm{eq}}{R}_{\nu }=a_{\nu }{T}_{\nu },\parallel R_{\nu }-{\mathcal{P}}_{\mathrm{eq}}{R}_{\nu }\parallel \le {ϵ}_{\mathrm{eq}}.$$

(89)

Equivariance fixes the retained line but not its positive amplitude. After an actual first response $T_1:V_1\to {\mathcal{A}}_1$ is known, let $Q\left(m\right)$ be the retained finite product of $T_{1}m$ with itself and put $q_{\mathrm{src}}\left(m\right)={(m\odot m)}_0$; let $T_2^{*}$ denote the trace-free polarized restriction of this quadratic response. The generated square is tested by

$$b_{\square }:={\displaystyle \sum _{i=1}^3}Q\left(e_i\right),{\mathcal{G}}_{\square }:=\underset{\left|m\right|=1}{\sup }\parallel Q\left(m\right)-T_2^{*}{q}_{\mathrm{src}}\left(m\right)\parallel ={\displaystyle \frac{\parallel b_{\square }\parallel }{3}},{\mathcal{G}}_{\square }=0.$$

(90)

When this residual vanishes, the weight-two connection and normalization are inherited from the same product; if ${\eta }_1,{\eta }_2$ denote the corresponding difference-connection one-forms, covariant differentiation of the retained square gives ${\eta }_2=2{\eta }_1$ and $F_{E_2}-F_{W_2}=2(F_{E_1}-F_{W_1})$. The full statement, including the axial bundle and Real marking, is recorded as Proposition 18.

Proposition 18

(Axial source-Hodge transport). On the aligned branch, the weight-one and weight-two source planes are associated to the marked axial bundle. After restriction to the physical $C_3$ marking, their integral incidence is (87); parity factorization is (88); and the generated square is certified by (90). For a nonzero retained line map $T_{\nu }$, the difference connection may be written ${\nabla }^{\nu }{T}_{\nu }-T_{\nu }{\nabla }^{\left(1\right)}=i{\eta }_{\nu }{T}_{\nu }$. Existence requires equality of the first Chern classes; parallel attachment requires equal curvature and trivial marked difference holonomy, with equal curvature sufficient on $S^2$. Once this line and connection are matched, two parallel antiunitary Real structures differ only by the marked sign and introduce no additional continuous phase. Thus the remaining local data are the integral incidence, the positive response gain, and the marked connection comparison.

Proposition 19

(Primitive 2-3-5 connection packet). Let $\mathcal{A}$ and $\mathcal{B}$ be the typed weight-five and weight-three Hermitian lines with unitary connections on the resolved sphere. The local curvature gate is

$$3F_{\mathcal{A}}=5F_{\mathcal{B}}.$$

(91)

Then ${\mathcal{B}}^{\otimes 5}\otimes {\mathcal{A}}^{-\otimes 3}$ is parallel-trivial, and $L:={\mathcal{B}}^{\otimes 2}\otimes {\mathcal{A}}^{-1}$ is the unique common connection root with $L^{\otimes 3}{\simeq }_{\nabla }\mathcal{B}$ and $L^{\otimes 5}{\simeq }_{\nabla }\mathcal{A}$. If the generated weight-two line is typed as $L^{\otimes 2}$, the holonomies arise from one $h\in U\left(1\right)$ exactly when

$$H_3^2=H_2^3,H_5=H_2{H}_3,h=H_3{H}_2^{-1}.$$

(92)

Once a typed $H_2$ is evaluated on the retained norm-seven target, its two conjugate values select the arithmetic orientation branch. Lifting a fixed $H_2$ to the primitive root leaves only $h\mapsto -h$; an independently typed odd-power datum $H_3$ resolves this ${\mathbb{Z}}_2$ root parity while testing the first relation in (92). No additional continuous phase is introduced by this lift. If $\mathcal{A}=L_{\mathsf{\Gamma }}^5$ is additionally typed with the fifth power of the structural connection, uniqueness identifies $L{\simeq }_{\nabla }{L}_{\mathsf{\Gamma }}$ and $\mathcal{B}{\simeq }_{\nabla }{L}_{\mathsf{\Gamma }}^3$. The curvature gate fixes the local common root but does not identify the physical marked loop. For a marked loop sweep with lifted holonomy $e^{i\varphi \left(s\right)}$, put $\tau \left(s\right)={\varphi }^{\prime }\left(s\right)$; curvature transgression gives ${\int }_0^1\tau \left(s\right)ds=2\pi d$. On the primitive branch $d=\pm 1$, ${\mathcal{R}}_{\mathrm{TV}}:={\left(2\pi \right)}^{-1}{\int }_0^1\left|\tau \left(s\right)\right|ds-1$ is nonnegative and vanishes exactly when $d\tau \left(s\right)\ge 0$; without a flat interval the selected target has a unique crossing, simple when $\tau \left(s_{*}\right)\ne 0$. A nontrivial phase on the resolved sphere is attached through the marked sweep rather than intrinsic flat monodromy.

On the two conjugate retained norm-seven targets the marked weight-two holonomy satisfies $H_2^3=e^{i{\sigma }_7^{\mathrm{ar}}{\delta }_7}$, where ${\sigma }_7^{\mathrm{ar}}=\pm 1$ is the arithmetic orientation and $e^{i{\delta }_7}$ is the coefficient phase used in (220). After the physical marked-loop incidence identifies the realized conjugate branch, the reduced CKM phase is the cubic reading of the same $H_2$ rather than an additional continuous phase.

Corollary 5

(Marked holonomy error budget). For a discrete marked loop of rank-one projectors put $Z=\mathrm{Tr}(P_0\cdots P_{N-1})$ and $H_2=Z/\left|Z\right|$. For perturbed projectors put $\tilde{Z}=\mathrm{Tr}({\tilde{P}}_0\cdots {\tilde{P}}_{N-1})$ and, when $\tilde{Z}\ne 0$, ${\tilde{H}}_2=\tilde{Z}/\left|\tilde{Z}\right|$. If $\parallel {\tilde{P}}_j-P_j\parallel \le {\epsilon }_j$ and $\Delta :=2{\sum }_j{\epsilon }_j<\left|Z\right|$, then

$$|\tilde{Z}-Z|\le \Delta ,\left|\arg \left({\tilde{H}}_2{H}_2^{-1}\right)\right|\le \arcsin \left({\displaystyle \frac{\Delta }{\left|Z\right|}}\right),$$

(93)

where the principal argument is used. Thus the existing Riesz/projector perturbation bounds propagate directly to the discrete Bargmann weight-two phase. On the norm-seven branch the cubic phase error is at most $3\arcsin (\Delta /|Z\left|\right)$; the conjugate branch is preserved whenever $\Delta /\left|Z\right|<3\sqrt{3}/14$. The lower bound on $\left|Z\right|$ is part of the certificate; near $Z=0$ the phase is ill-conditioned. Identification with a continuous marked-loop holonomy additionally requires mesh control.

The self-dual Lorentz carrier does not supply ${\sigma }_7^{\mathrm{phys}}$ by itself: for $g\in S{L}_2\left(\mathbb{C}\right)$ one has $\det {\mathrm{Sym}}^2\left(g\right)=1$. A nontrivial norm-seven determinant phase therefore requires the separately typed line/determinant connection and marked loop already used in Proposition 19.

For a finite parent functional $\mathcal{E}(B,x)$ with stationary branch $B_{*}\left(x\right)$, put $t=Tx$, $g_{\mathrm{sh}}={\nabla }_B\mathcal{E}(t,x)$, and $\Delta E_A=\mathcal{E}(t,x)-\mathcal{E}(B_{*},x)$. If $\lambda \mathbf{1}⪯{\nabla }_B^2\mathcal{E}⪯\mathsf{\Lambda }\mathbf{1}$ on the relevant convex neighborhood, strong convexity gives

$${\displaystyle \frac{\parallel g_{\mathrm{sh}}{\parallel }^2}{2\mathsf{\Lambda }}}\le \Delta E_A\le {\displaystyle \frac{\parallel g_{\mathrm{sh}}{\parallel }^2}{2\lambda }},g_{\mathrm{sh}}=0⟺\Delta E_A=0⟺B_{*}=t,{\kappa }_{\mathrm{gg}}:={\displaystyle \frac{\parallel g_{\mathrm{sh}}{\parallel }^2}{2\Delta E_A}}\in [\lambda ,\mathsf{\Lambda }](\Delta E_A>0).$$

(94)

On a multiplicity-one line this directly bounds $|r_A-1|$ and is kept as the microscopic K2 certificate.

Proposition 20

(Axial diagnostics and constrained VH selector). On an oriented axial branch, $V_1=V_1^{\left(0\right)}\oplus V_1^{\left(1\right)}$ and $V_2=V_2^{\left(0\right)}\oplus V_2^{\left(1\right)}\oplus V_2^{\left(2\right)}$. Before ensemble restriction every $SO\left(2\right)$-equivariant self-adjoint Hessian has the real blocks $H_0=\left(\begin{array}{cc}{\alpha }_0& x_0\\ x_0& {\beta }_0\end{array}\right)$, $H_1=\left(\begin{array}{cc}{\alpha }_1& z\\ \overline{z}& {\beta }_1\end{array}\right)$, and $k_2{\mathbf{1}}_2$, with

$$\det H={\Delta }_0{\Delta }_1^2{k}_2^2,{\Delta }_0={\alpha }_0{\beta }_0-x_0^2,{\Delta }_1={\alpha }_1{\beta }_1-{\left|z\right|}^2.$$

(95)

Thus a generic ${\Delta }_0$ zero is simple, whereas ${\Delta }_1$ or $k_2$ zeros are double-soft; these remain unreduced diagnostics. For the physical fixed-charge problem let Q be the conserved charge associated with (61), and decompose the augmented Hessian and charge differential into retained and hard variables as ${\mathcal{H}}_{\mathrm{aug}}=\left(\begin{array}{cc}{H}_s& B\\ B^{†}& C\end{array}\right)$, $C\succ 0$, and $g=DQ={(g_s,g_h)}^T$. Put $S=H_s-B{C}^{-1}{B}^{†}$ and assume it remains invertible on the local regular branch. If $g_h\ne 0$, define $\tilde{g}=g_s-B{C}^{-1}{g}_h$ and $h=g_h^{†}{C}^{-1}{g}_h$; exact hard minimization at fixed charge gives

$$K_{Q,\mathrm{eff}}=S+{\displaystyle \frac{\tilde{g}{\tilde{g}}^{†}}{h}},{\chi }_Q=g^{†}{\mathcal{H}}_{\mathrm{aug}}^{-1}g={\tilde{g}}^{†}{S}^{-1}\tilde{g}+h.$$

(96)

For $g_h=0$ the fixed-charge operator is the restriction of S to $\ker g_s^{†}$. Hence fixed-charge restriction and hard Schur elimination commute only on this latter branch. On the passed native Veronese-Hodge graph ${\mathcal{G}}_{\mathrm{VH}}={\mathcal{G}}_0\oplus {\mathcal{G}}_1$, with $\dim {\mathcal{G}}_0=1$ and $\dim {\mathcal{G}}_1=2$. Once the charge-VH ensemble incidence identifies the physical graph Hessian with the corresponding restriction of $K_{Q,\mathrm{eff}}$, axial equivariance gives

$$S_G=a_0{P}_0+a_1{P}_1.$$

(97)

With the complementary gaps open, a genuine simple stability loss can therefore occur only through

$$a_0\left(s_{*}\right)=0,a_1\left(s_{*}\right)>0,{\dot{a}}_0\left(s_{*}\right)\ne 0.$$

(98)

The weight-one mixing phase remains separately typed by (79).

For a one-coordinate Parent branch with physical coordinate x, unresolved nuisance/completion variables $\nu$, and exact mandatory residual vector $\mathcal{R}(x,\nu )$, put $r_{\mathrm{sel}}:=\mathrm{rank}{D}_{(x,\nu )}\mathcal{R}-\mathrm{rank}{D}_{\nu }\mathcal{R}\in \{0,1\}$. At locally constant rank, $r_{\mathrm{sel}}=1$ removes the physical branch coordinate and $r_{\mathrm{sel}}=0$ fixes only nuisance data. Selector rank is evaluated before restriction to the zero set under test. At a simple master zero, any smooth scalar branch residual vanishing at the same point factors locally through that master residual; it is therefore a dependent checksum unless its provenance supplies an independent condition. After the structural incidences have passed, the charge discriminant below supplies the single singular selector, while (98) is its VH reading.

Proposition 21

(Noether-Fredholm lock and charge discriminant). For the phase-relaxed scalar block let $R:=R_{\mathrm{vort}}$, $G\xi :=D{R}_{\mathrm{vort}}\left[\xi \right]$, $a_{\mathrm{ph}}:=d_{k}R$, $T\theta :=a_{\mathrm{ph}}{U}^{\mu }{\nabla }_{\mu }\theta$, $w:=p_{\mathsf{\Lambda }}{C}_R$, $B_{\mathrm{ph}}:=\sqrt{w}T$, and $j:=p_{\mathsf{\Lambda }}{\zeta }^2$ as in (61). On $a_{\mathrm{ph}}\sqrt{w}\ne 0$, current conservation gives ${\psi }_{\mathrm{tr}}:=j/\left(a_{\mathrm{ph}}\sqrt{w}\right)\in \ker B_{\mathrm{ph}}^{*}$. If $\ker B_{\mathrm{ph}}^{*}=\mathbb{C}{\psi }_{\mathrm{tr}}$ and $\langle {\widehat{\psi }}_{\mathrm{tr}},\sqrt{w}G\xi \rangle$ is proportional to $DQ\left[\xi \right]$, phase relaxation gives $Q_C^{\mathrm{rel}}\left(\xi \right)=|c_Q{|}^2{\left|DQ\left[\xi \right]\right|}^2$; the phase Gram term is therefore charge-normal and vanishes on the fixed-Q tangent.

Let Ω parametrize regular relative equilibria of ${\mathcal{L}}_{\mathsf{\Omega }}:=S_A-\mathsf{\Omega }Q$, with $H_{\mathsf{\Omega }}:=D^2{\mathcal{L}}_{\mathsf{\Omega }}$ and $q_{\mathsf{\Omega }}:=DQ$. Wherever $H_{\mathsf{\Omega }}$ is invertible,

$$H_{\mathsf{\Omega }}{\partial }_{\mathsf{\Omega }}\mathsf{\Phi }=q_{\mathsf{\Omega }},z:=H_{\mathsf{\Omega }}^{-1}{q}_{\mathsf{\Omega }},{\chi }_Q:=\langle q_{\mathsf{\Omega }},z\rangle ={\displaystyle \frac{dQ}{d\mathsf{\Omega }}}.$$

(99)

Thus ${\chi }_Q=0$ is the charge turning point. At such a point $z_{*}\in \ker q_{*}$ and, for one scalar constraint, $\mathrm{rad}\left(H_{*}{|}_{\ker q_{*}}\right)=\mathbb{R}{z}_{*}$. In a finite chart, if $U_Q$ is an orthonormal frame of $\ker q_{\mathsf{\Omega }}^{†}$ and $K_Q:=U_Q^{†}{H}_{\mathsf{\Omega }}{U}_Q$, then $\det K_Q=(\det H_{\mathsf{\Omega }}/\parallel q_{\mathsf{\Omega }}{\parallel }^2){\chi }_Q$; the Fredholm analogue is a determinant-line statement. A simple fold is therefore ${\chi }_Q\left({\mathsf{\Omega }}_{*}\right)=0$, $Q_{*}^{\prime \prime }\ne 0$.

On the $g_h\ne 0$ branch of (96), $z_s=S^{-1}\tilde{g}$ and $K_{Q,\mathrm{eff}}{z}_s=({\chi }_Q/h)\tilde{g}$. After the charge-VH lock,

$${\dot{a}}_0\left({\mathsf{\Omega }}_{*}\right)=-{\displaystyle \frac{Q_{*}^{\prime \prime }}{\parallel z_{s,*}{\parallel }^2}},{\beta }_Q=-{\displaystyle \frac{Q_{*}^{\prime \prime }}{\parallel z_{*}{\parallel }^3}}.$$

(100)

Moreover $n_{-}\left(K_{Q,\mathrm{eff}}\right)=n_{-}\left(S\right)-n_{-}\left({\chi }_Q\right)$ and, for $C\succ 0$, $n_{-}\left(H_{\mathsf{\Omega }}\right)=n_{-}\left(S\right)$, so a simple sign change of ${\chi }_Q$ changes the constrained Morse index by one while the full index remains fixed.

For ${\chi }_Q\ne 0$, the fixed-charge KKT inverse is

$$G_Q=H_{\mathsf{\Omega }}^{-1}-{\displaystyle \frac{z\otimes z}{{\chi }_Q}},\underset{\mathsf{\Omega }\to {\mathsf{\Omega }}_{*}}{\lim }{\chi }_Q{G}_Q=-z_{*}\otimes z_{*}.$$

(101)

For scalar readouts $F_a$ of the same Parent solution, with ${\dot{F}}_a=D{F}_a\left[z\right]$,

$${\mathsf{\Xi }}_{ab}^{\left(Q\right)}=\langle D{F}_a,H_{\mathsf{\Omega }}^{-1}D{F}_b\rangle -{\displaystyle \frac{{\dot{F}}_a{\dot{F}}_b}{{\chi }_Q}},-{\chi }_Q{\mathsf{\Xi }}^{\left(Q\right)}⟶{\dot{\mathbf{F}}}_{*}{\dot{\mathbf{F}}}_{*}^T.$$

(102)

A soft-blind readout need not diverge. In the finite chart the oriented product of hard constrained eigenvalues satisfies $D_{\mathrm{hard},*}=-\det H_{*}\parallel z_{*}{\parallel }^2/{\parallel q_{*}\parallel }^2$. The constrained Schur zero, VH soft mode, determinant-line zero, KKT pole, intrinsic charge turning, and cubic crossing are dependent readings of the same discriminant.

Theorem 7

(Fredholm-Feshbach reduction of the charge selector). Let the gauge-fixed Hessian act on a common form/graph domain split by complementary retained and tail projections as

$$H=\left(\begin{array}{cc}A& B^{†}\\ B& C\end{array}\right),q=\left({\displaystyle \genfrac{}{}{0pt}{}{p}{r}}\right),\parallel C^{-1}\parallel \le {\gamma }^{-1}.$$

(103)

The corresponding bordered operator is $\mathbb{K}=\left(\begin{array}{ccc}A& B^{†}& p\\ B& C& r\\ p^{†}& r^{†}& 0\end{array}\right)$. Assume that C is invertible on the tail and define

$$S=A-B^{†}{C}^{-1}B,p_{\mathrm{eff}}=p-B^{†}{C}^{-1}r,\tau =r^{†}{C}^{-1}r.$$

(104)

Tail elimination gives

$${\mathbb{K}}_{\mathrm{eff}}=\left(\begin{array}{cc}S& p_{\mathrm{eff}}\\ p_{\mathrm{eff}}^{†}& -\tau \end{array}\right),\ker \mathbb{K}\simeq \ker {\mathbb{K}}_{\mathrm{eff}}.$$

(105)

If $n_{-}\left(C\right)<\infty$, Schur congruence gives $n_{-}\left(\mathbb{K}\right)=n_{-}\left(C\right)+n_{-}\left({\mathbb{K}}_{\mathrm{eff}}\right)$ and $n_{-}\left(H\right)=n_{-}\left(C\right)+n_{-}\left(S\right)$; for $C\succ 0$ the tail terms vanish. Wherever S is invertible,

$$z_P=S^{-1}{p}_{\mathrm{eff}},z_Q=C^{-1}(r-B{z}_P),{\chi }_Q=\tau +p_{\mathrm{eff}}^{†}{S}^{-1}{p}_{\mathrm{eff}}.$$

(106)

If ${\mathbb{K}}_P=\left(\begin{array}{cc}A& p\\ p^{†}& 0\end{array}\right)$ and $D=(B,r)$, then ${\mathbb{K}}_{\mathrm{eff}}={\mathbb{K}}_P-D^{†}{C}^{-1}D$ and

$$\mathrm{dist}(0,\mathrm{spec}{\mathbb{K}}_P)>{\displaystyle \frac{{\parallel D\parallel }^2}{\gamma }}⟹0\notin \mathrm{spec}{\mathbb{K}}_{\mathrm{eff}}.$$

(107)

At every point or validated box where this inequality holds, the tail cannot hide a full KKT wall behind the retained block; segmentwise exclusion requires the bound along the certified chain.

For a posteriori comparison with the naive retained tangent $A^{-1}p$, put $a_A:={\parallel A^{-1}\parallel }^{-1}$, $b:=\parallel B\parallel$, $p_0:=\parallel p\parallel$, $r_0:=\parallel r\parallel$, $e=b^2/\gamma$, $d=b{r}_0/\gamma$, and $\sigma =a_A-e>0$. Then

$${\epsilon }_P:={\displaystyle \frac{e{p}_0/a_A+d}{\sigma }},{\epsilon }_Q:={\displaystyle \frac{r_0+b(p_0+d)/\sigma }{\gamma }},{\epsilon }_z:=\sqrt{{\epsilon }_P^2+{\epsilon }_Q^2}$$

(108)

bounds the retained and full tangent errors, and

$$\left|{\chi }_Q-\langle p,A^{-1}p\rangle \right|\le {\epsilon }_{\chi }:=p_0{\epsilon }_P+r_0{\epsilon }_Q.$$

(109)

These quantities are solver/error certificates rather than additional selector conditions.

Theorem 8

(Protected seed-to-fold certificate). Let a regular gauge-fixed branch start at a fixed-Q stable seed ${\mathsf{\Omega }}_0$, with $K_{Q,0}\succ 0$, $H_0$ invertible, $h_0:={\parallel H_0^{-1}\parallel }^{-1}>0$, $q_0:=\parallel q_{{\mathsf{\Omega }}_0}\parallel >0$, $r_0:=\parallel H_0^{-1}{q}_{{\mathsf{\Omega }}_0}\parallel$, and ${\chi }_0<0$. Fix $0<\vartheta <1$, $\Delta >0$, and $\overline{h}=(1-\vartheta )h_0$. Assume on the maximal local continuation in the declared chart that $\parallel D^{2}Q\parallel \le C_2$, $\parallel D^3{\mathcal{L}}_{\mathsf{\Omega }}\parallel \le C_3$, and along the actual stationary tangent

$$D^{2}Q[z,z]\ge {\mu }_{\Vert }{\parallel z\parallel }^2,{\mu }_{\Vert }>0.$$

(110)

For each bounded competing structural operator $A_j$, put $g_{j,0}:={\sigma }_{\min }\left(A_j\left({\mathsf{\Omega }}_0\right)\right)>0$ and assume $\parallel {\left({\partial }_{\mathsf{\Omega }}{A}_j\right)}_{\mathsf{\Phi }}\parallel \le {\alpha }_j$, $\parallel D_{\mathsf{\Phi }}{A}_j\parallel \le {\beta }_j$. With $a=2C_2/\overline{h}$, $b=C_3/\overline{h}$, define

$$Y\left(\tau \right)={\displaystyle \frac{r_0{e}^{a\tau }}{1-(b{r}_0/a)(e^{a\tau }-1)}},I_Y(\Delta )=-{\displaystyle \frac{1}{b}}\log \left[1-{\displaystyle \frac{b{r}_0}{a}}(e^{a\Delta }-1)\right],$$

(111)

with continuous $a=0$ or $b=0$ limits. Assume that the closed state-space ball of radius $I_Y(\Delta )$ remains in the admissible chart and put

$${\overline{B}}_H=C_2\Delta +C_3{I}_Y,{\overline{B}}_q=C_2{I}_Y,{\overline{B}}_j={\alpha }_j\Delta +{\beta }_j{I}_Y.$$

(112)

Set $q_{-}^{\mathrm{boot}}=q_0-{\overline{B}}_q$, $h_{+}^{\mathrm{boot}}=\parallel H_0\parallel +{\overline{B}}_H$, ${\Delta }_{3,\Vert }^{\mathrm{boot}}=3{\mu }_{\Vert }-C_{3}Y(\Delta )$, and $m_{\mathrm{boot}}={(q_{-}^{\mathrm{boot}}/h_{+}^{\mathrm{boot}})}^2{\Delta }_{3,\Vert }^{\mathrm{boot}}$. If

$$1-{\displaystyle \frac{b{r}_0}{a}}(e^{a\Delta }-1)>0,{\overline{B}}_H<\vartheta h_0,{\overline{B}}_q<q_0,{\overline{B}}_j<g_{j,0},{\Delta }_{3,\Vert }^{\mathrm{boot}}>0,m_{\mathrm{boot}}\Delta >-{\chi }_0,$$

(113)

then the Hessian, charge covector, and listed structural gaps remain regular on the certified continuation, while ${\chi }_Q^{\prime }=Q^{\prime \prime }\ge m_{\mathrm{boot}}>0$. Hence ${\chi }_Q$ has exactly one zero in $({\mathsf{\Omega }}_0,{\mathsf{\Omega }}_0+\Delta )$, which is the only constrained stability wall on that segment. The same conclusion holds if the directional coercivity route is replaced by an independent certified bound ${\chi }_Q^{\prime }\ge m>0$ together with the same regularity budgets.

Proof. 

The provisional bound $\parallel H^{-1}{\parallel }^{-1}\ge \overline{h}$ gives $d\parallel z\parallel /d\mathsf{\Omega }\le (2C_2/\overline{h})\parallel z\parallel +(C_3/\overline{h}){\parallel z\parallel }^2$, so Bihari comparison gives (111); its integral controls the state displacement. The budgets (112) then close the Hessian bootstrap and preserve the charge and competing gaps. Since ${\chi }_Q^{\prime }=3D^{2}Q[z,z]-D^3{\mathcal{L}}_{\mathsf{\Omega }}[z,z,z]$, (110) and $\parallel z\parallel \ge q_{-}^{\mathrm{boot}}/h_{+}^{\mathrm{boot}}$ give the stated lower bound and monotone crossing. □

Failure of the sufficient bounds above does not falsify a protected fold; sharper directional or fourth-jet control, or subdivision into certified segments, may close the same branch without introducing another selector.

Corollary 6

(Validated segment continuation). Suppose a finite chain of overlapping certified boxes carries locally unique full Parent solutions in one fixed gauge/graph-norm convention, the solutions agree on each overlap, the Feshbach tail gate (107) and all competing structural gaps remain open, and ${\chi }_Q^{\prime }\ge m_j>0$ on box j. Then the boxes glue to one regular Parent branch with ${\chi }_Q^{\prime }\ge m_{\mathrm{chain}}:={\min }_j{m}_j$. If certified endpoint enclosures have opposite signs, the chain contains exactly one zero of ${\chi }_Q$; when it starts at the physical fixed-Q stable anchor, this zero is the first protected wall on the covered branch. Sign tests based on retained numerics use the enclosure (109).

Corollary 7

(Intrinsic charge curvature at the fold). Let $r=\parallel z\parallel$, $\widehat{z}=z/r$, and $d{\ell }_{\mathsf{\Phi }}/d\mathsf{\Omega }=r$. Then

$${\displaystyle \frac{dQ}{d{\ell }_{\mathsf{\Phi }}}}={\displaystyle \frac{{\chi }_Q}{r}},{\left.{\displaystyle \frac{d^{2}Q}{d{\ell }_{\mathsf{\Phi }}^2}}\right|}_{*}={\displaystyle \frac{Q_{*}^{\prime \prime }}{r_{*}^2}}=3D^{2}Q[{\widehat{z}}_{*},{\widehat{z}}_{*}]-r_{*}{D}^3{\mathcal{L}}_{{\mathsf{\Omega }}_{*}}[{\widehat{z}}_{*},{\widehat{z}}_{*},{\widehat{z}}_{*}].$$

(114)

Hence $Q_{*}^{\prime \prime }>0$ gives an intrinsic local minimum of Q on the stationary curve.

Theorem 9

(Exact-fold critical residue from stable-side response). For a finite panel of regular scalar readouts set $\mathbf{f}={\left(D{F}_a\left[z\right]\right)}_a$ and ${\mathsf{A}}_{ab}:=\langle D{F}_a,H_{\mathsf{\Omega }}^{-1}D{F}_b\rangle$. On the certified stable side define the desingularized response

$$\mathsf{M}\left({\chi }_Q\right):=-{\chi }_Q{\mathsf{\Xi }}^{\left(Q\right)}=\mathbf{f}{\mathbf{f}}^T-{\chi }_Q\mathsf{A}.$$

(115)

Assume $\mathsf{M}$ is $C^2$ as a function of ${\chi }_Q$ on the terminal certified segment and put $B_2^{\chi }:=\sup {\parallel {\partial }_{{\chi }_Q}^2\mathsf{M}\parallel }_{\mathrm{op}}$. Then the exact-fold residue is

$${\mathsf{R}}_{*}:=\mathsf{M}\left(0\right)={\mathbf{f}}_{*}{\mathbf{f}}_{*}^T.$$

(116)

For two distinct stable-side nodes ${\chi }_1,{\chi }_2<0$,

$${\mathsf{R}}_{12}^{\left[1\right]}:={\displaystyle \frac{{\chi }_2\mathsf{M}\left({\chi }_1\right)-{\chi }_1\mathsf{M}\left({\chi }_2\right)}{{\chi }_2-{\chi }_1}},\parallel {\mathsf{R}}_{12}^{\left[1\right]}-{\mathsf{R}}_{*}{\parallel }_{\mathrm{op}}\le {\displaystyle \frac{B_2^{\chi }}{2}}\left|{\chi }_1{\chi }_2\right|.$$

(117)

If numerical matrices satisfy $\parallel {\widehat{\mathsf{M}}}_i-\mathsf{M}\left({\chi }_i\right){\parallel }_{\mathrm{op}}\le {\epsilon }_i$, then

$$E_{12}:={\displaystyle \frac{|{\chi }_2|{\epsilon }_1+\left|{\chi }_1\right|{\epsilon }_2}{|{\chi }_2-{\chi }_1|}}+{\displaystyle \frac{B_2^{\chi }}{2}}\left|{\chi }_1{\chi }_2\right|.$$

(118)

Then (118) bounds $\parallel {\widehat{\mathsf{R}}}_{12}^{\left[1\right]}-{\mathsf{R}}_{*}{\parallel }_{\mathrm{op}}$. For interval-enclosed nodes the two intervals must lie in $(-\infty ,0)$ and have positive separation; the interpolation weights and $E_{12}$ are then evaluated by interval arithmetic.

Under the simple single-soft KKT hypothesis, ${\mathsf{R}}_{*}$ has rank one. Hence

$${\sigma }_2\left({\widehat{\mathsf{R}}}_{12}^{\left[1\right]}\right)\le E_{12}$$

(119)

is necessary after the Feshbach, typing, and smoothness certificates have passed. If $s_{*}:={\parallel {\mathbf{f}}_{*}\parallel }^2\ge s_{-}>E_{12}$, the dominant projector obeys

$$\parallel {\widehat{\mathsf{\Pi }}}_1-{\mathsf{\Pi }}_{*}{\parallel }_{\mathrm{op}}\le {\displaystyle \frac{E_{12}}{s_{-}{E}_{12}}},|{\widehat{\lambda }}_1-s_{*}|\le E_{12}.$$

(120)

After one oriented reference channel fixes the global sign and ${\widehat{\lambda }}_1>E_{12}$, the corresponding coefficient vector satisfies

$$\parallel {\widehat{\mathbf{f}}}_{*}-{\mathbf{f}}_{*}\parallel \le {\displaystyle \frac{E_{12}}{\sqrt{{\widehat{\lambda }}_1}+\sqrt{{\widehat{\lambda }}_1-E_{12}}}}+\sqrt{2({\widehat{\lambda }}_1+E_{12})}{\displaystyle \frac{E_{12}}{s_{-}{E}_{12}}}.$$

(121)

Proof. 

Equation (115) follows from (102). Linear interpolation of a Banach-valued $C^2$ map at ${\chi }_1,{\chi }_2$ gives the remainder in (117); numerical errors propagate through the same weights. Weyl and one-dimensional spectral-subspace perturbation estimates give (119)-(121). □

On a segment with $\mathcal{T}:=d{\chi }_Q/d\mathsf{\Omega }\ge m>0$,

$${\partial }_{{\chi }_Q}^2\mathsf{M}={\displaystyle \frac{\ddot{\mathsf{M}}}{{\mathcal{T}}^2}}-{\displaystyle \frac{\dot{\mathsf{M}}\dot{\mathcal{T}}}{{\mathcal{T}}^3}},B_2^{\chi }\le {\displaystyle \frac{M_2^{\mathsf{\Omega }}}{m^2}}+{\displaystyle \frac{M_1^{\mathsf{\Omega }}{B}_{\mathcal{T}}}{m^3}}.$$

(122)

A sufficient implementation uses bounds on $\mathbf{f},\dot{\mathbf{f}},\ddot{\mathbf{f}}$ and $\mathsf{A},\dot{\mathsf{A}},\ddot{\mathsf{A}}$; componentwise

$${\ddot{f}}_a=D^3{F}_a[z,z,z]+3D^2{F}_a[w,z]+D{F}_a\left[j\right],w=\dot{z},j=\dot{w}.$$

(123)

Thus the $C^2\left({\chi }_Q\right)$ hypothesis can be certified from finite Parent/observable jets without making those jets additional physical assumptions.

Corollary 8

(Frame-complete critical tangent and causal holdout). Let a typed finite target $V_X$ be observed by a frame $L_X:V_X\to {\mathbb{R}}^{m_X}$ with $A_X{\parallel v\parallel }^2\le \parallel L_X{v\parallel }^2\le B_X{\parallel v\parallel }^2$. For $v_X:=DX\left[z_{*}\right]$ and the corresponding residue subvector ${\mathbf{f}}_X=L_X{v}_X$,

$$v_X=L_X^{†}{\mathbf{f}}_X,L_X^{†}={\left(L_X^{*}{L}_X\right)}^{-1}{L}_X^{*},\parallel {\widehat{v}}_X-v_X\parallel \le {\displaystyle \frac{{\epsilon }_{f_X}}{\sqrt{A_X}}}.$$

(124)

Hence probe-complete panels reconstruct, within their declared target spaces, typed critical tangents such as $D{P}_Q\left[z_{*}\right]$, $D{H}_R\left[z_{*}\right]$, and $D{\ell }_{\mathrm{leak}}\left[z_{*}\right]$; this does not reconstruct the full Parent tangent space. With three or more certified boxes, independently extrapolated pairs must satisfy

$$\parallel {\widehat{\mathsf{R}}}_{ij}^{\left[1\right]}-{\widehat{\mathsf{R}}}_{kl}^{\left[1\right]}{\parallel }_{\mathrm{op}}\le E_{ij}+E_{kl}.$$

(125)

For a proposed strong mediation $Y=F\left(X\right)$, $J=DF\left(X_{*}\right)$, reconstructed tangents satisfy $v_Y=J{v}_X$. If their errors are $e_X,e_Y$ and $\parallel \widehat{J}-J\parallel \le e_J$, then

$$\parallel {\widehat{v}}_Y-\widehat{J}{\widehat{v}}_X\parallel >e_Y+\parallel \widehat{J}\parallel e_X+e_J{V}_X^{+}$$

(126)

falsifies that directed mediation whenever $\parallel v_X\parallel \le V_X^{+}$. Failure of this stronger relation does not remove the weaker common-germ interpretation.

For a nonzero response vector $\mathbf{f}\left(\mathsf{\Omega }\right)$, put ${\nu }_f=\mathbf{f}/\parallel \mathbf{f}\parallel$ and ${\mathsf{\Pi }}_f={\nu }_f{\nu }_f^T$. Its projective turning rate is

$${\kappa }_f={\displaystyle \frac{\parallel P_{{\mathbf{f}}^{\perp }}\dot{\mathbf{f}}\parallel }{\parallel \mathbf{f}\parallel }},{\parallel {\dot{\mathsf{\Pi }}}_f\parallel }_{\mathrm{op}}={\kappa }_f.$$

(127)

The integrated turning is an independent no-switching checksum between certified boxes; it is not required by the two-box recovery itself.

Higher-order N-node extrapolation is available when it reduces the certified total error: Lagrange weights $w_i^{\left(N\right)}$ cancel powers through degree $N-1$, with remainder bounded by $(B_N^{\chi }/N!){\prod }_i\left|{\chi }_i\right|$ plus ${\sum }_i\left|w_i^{\left(N\right)}\right|{\epsilon }_i$. The two-box form is the default because it requires only $C^2$ control.

The physical selectable line is fixed by the charge discriminant together with the charge-VH incidence. The dual-Hessian construction below remains a local intrinsic chart of the same scalar event.

Proposition 22

(Dual-Hessian susceptibility and oriented first loss). On a nonhomogeneous fixed-phase tangent block with $D{p}_{\mathsf{\Lambda }}=0$, put $R:=R_{\mathrm{vort}}$, $g={\left(dR\right)}^{♯}$, $q=\parallel g\parallel >0$, $e=g/q$, and $S=D^{2}R$. Relative to $\mathbb{R}e\oplus \ker dR$, write $S=\left(\begin{array}{cc}{s}_{00}& b^{†}\\ b& S_{\perp }\end{array}\right)$. By (63), the phase-fixed scalar block is

$$A\left(C_R\right)=C_R{q}^2{P}_e-\mu S,C_{\mathrm{crit}}:={\displaystyle \frac{\mu }{q^2}}\left(s_{00}-b^{†}{S}_{\perp }^{-1}b\right).$$

(128)

If $S_{\perp }\prec 0$ and the remaining projected terms are absent or independently controlled, $C_R=C_{\mathrm{crit}}$ is a simple kernel with $v_{*}=e-S_{\perp }^{-1}b$. For the retained positive Jacobi sign, the signed Schur determinant margin is $\det A/\left[q^2\det (-\mu S_{\perp })\right]=C_R-C_{\mathrm{crit}}$, so the scalar inertia changes by one across the simple crossing. The transnormal residual ${\mathcal{R}}_{\mathrm{tn}}:={\parallel [P_e,S]\parallel }_{\mathrm{HS}}/\sqrt{2}$ vanishes exactly when this soft vector is radial. Moreover $[P_e,S]=0$ is equivalent to $P_{e^{\perp }}\nabla \left(q^2\right)=0$; locally $q^2$ is then a function of R, and hence of Φ. On this locus Φ is a local transnormal coordinate and $\mathcal{B}\left(\mathsf{\Phi }\right):={\parallel \nabla \mathsf{\Phi }\parallel }^2$ satisfies

$${\lambda }_{\Vert }\left(\mathsf{\Phi }\right)={\nabla }^2\mathsf{\Phi }\left({\displaystyle \frac{\nabla \mathsf{\Phi }}{\parallel \nabla \mathsf{\Phi }\parallel }},{\displaystyle \frac{\nabla \mathsf{\Phi }}{\parallel \nabla \mathsf{\Phi }\parallel }}\right)={\displaystyle \frac{1}{2}}{\displaystyle \frac{d\mathcal{B}}{d\mathsf{\Phi }}}.$$

(129)

For a branch leaving a stable anchor with orientation ${\sigma }_{\mathsf{\Phi }}:=\mathrm{sgn}\dot{\mathsf{\Phi }}$, an isolated scalar first-loss terminal therefore obeys

$${\mathcal{B}}_{\mathsf{\Phi }}\left({\mathsf{\Phi }}_{*}\right)=0,{\sigma }_{\mathsf{\Phi }}{\mathcal{B}}_{\mathsf{\Phi }\mathsf{\Phi }}\left({\mathsf{\Phi }}_{*}\right)<0.$$

(130)

The locus and transversality are invariant under $\mathsf{\Phi }\mapsto g\mathsf{\Phi }+c$, $g>0$. If ${\sigma }_{\mathsf{\Phi }}{\mathcal{B}}_{\mathsf{\Phi }\mathsf{\Phi }}\le -m_B<0$ on the oriented interval and the endpoint signs bracket zero, the first-loss terminal is unique and an approximate root satisfies $|\widehat{\mathsf{\Phi }}-{\mathsf{\Phi }}_{*}|\le |{\mathcal{B}}_{\mathsf{\Phi }}\left(\widehat{\mathsf{\Phi }}\right)|/m_B$.

Under identification of the dual-Hessian and constrained VH lines, (130) is a local chart of the fold selected by Theorem 8; phase relaxation is handled upstream and no dual-Hessian zero is counted as an additional selector.

Proposition 23

(Simple-soft branch and local closure). Suppose a $C^4$ gauge-fixed Parent energy has a stationary branch parametrized by x, with a self-adjoint Fredholm Hessian H of index zero, $\ker H_{*}=\mathbb{R}{v}_{*}$, $\parallel v_{*}\parallel =1$, a gapped complement, and $\alpha :=\langle v_{*},{\left({\partial }_{x}H\right)}_{*}{v}_{*}\rangle \ne 0$. The standard simple-eigenvalue theorem [75] gives one local nontrivial stationary branch. Put $P_{\perp }=\mathbf{1}-|v_{*}\rangle \langle v_{*}|$, $L_{\perp }=P_{\perp }{H}_{*}{P}_{\perp }{|}_{\mathrm{Ran}{P}_{\perp }}$, and $q_3:=P_{\perp }{\left(D^3\mathcal{E}[v_{*},v_{*},·]\right)}^{♯}$. With $\tau =\alpha (x-x_{*})$, Lyapunov-Schmidt reduction gives ${\mathcal{E}}_{\mathrm{red}}-{\mathcal{E}}_0=\tau a^2/2+{\beta }_{\mathrm{LS}}{a}^3/6+{\gamma }_{\mathrm{LS}}{a}^4/24+\mathrm{higher}\mathrm{order}$, where ${\beta }_{\mathrm{LS}}=D^3\mathcal{E}[v_{*},v_{*},v_{*}]$ and ${\gamma }_{\mathrm{LS}}=D^4\mathcal{E}[v_{*},v_{*},v_{*},v_{*}]-3\langle q_3,L_{\perp }^{-1}{q}_3\rangle$. For the physical fixed-Q fold of Proposition 21, the third-jet ensemble incidence identifies ${\beta }_{\mathrm{LS}}={\beta }_Q$ in (100); hence $Q_{*}^{\prime \prime }\ne 0$ forces the generic cubic class. An exact-even branch with ${\beta }_{\mathrm{LS}}=0$ remains a valid alternative local event but cannot represent the same simple charge fold. For ${\gamma }_{\mathrm{LS}}>0$ and ${\beta }_{\mathrm{LS}}\ne 0$, the quartic approximation has ${\tau }_J=0$, ${\tau }_{\mathrm{co}}^{\left(4\right)}={\beta }_{\mathrm{LS}}^2/\left(3{\gamma }_{\mathrm{LS}}\right)$, ${\tau }_{\mathrm{sp}}^{\left(4\right)}=3{\beta }_{\mathrm{LS}}^2/\left(8{\gamma }_{\mathrm{LS}}\right)$, and coexistence barrier ${\beta }_{\mathrm{LS}}^4/\left(24{\gamma }_{\mathrm{LS}}^3\right)$, within the controlled Lyapunov-Schmidt radius.

If the same branch breaks the protected dark port with ${\mathsf{\Gamma }}_{\mathrm{dark}}={\kappa }_n{\left|a\right|}^{2n}+{o\left(\right|a|}^{2n})$, put $\Delta \mathcal{E}={\mathcal{E}}_{\mathrm{red}}(a_{\sec },x)-{\mathcal{E}}_0\left(x\right)$ and $K=|{\partial }_a^2{\mathcal{E}}_{\mathrm{red}}|$ on the secondary branch. The leading slope-free laws are

$$\begin{array}{cccc}\hfill {\beta }_{\mathrm{LS}}\ne 0:|\Delta \mathcal{E}|& ={\displaystyle \frac{2}{3{\beta }_{\mathrm{LS}}^2}}{K}^3+o\left(K^3\right),\hfill & \hfill {\mathsf{\Gamma }}_{\mathrm{dark}}& ={\kappa }_n{\left({\displaystyle \frac{2}{|{\beta }_{\mathrm{LS}}|}}\right)}^{2n}{K}^{2n}+o\left(K^{2n}\right),\hfill \\ \hfill {\beta }_{\mathrm{LS}}=0,{\gamma }_{\mathrm{LS}}>0:|\Delta \mathcal{E}|& ={\displaystyle \frac{3}{8{\gamma }_{\mathrm{LS}}}}{K}^2+o\left(K^2\right),\hfill & \hfill {\mathsf{\Gamma }}_{\mathrm{dark}}& ={\kappa }_n{\left({\displaystyle \frac{3}{{\gamma }_{\mathrm{LS}}}}\right)}^n{K}^n+o\left(K^n\right).\hfill \end{array}$$

(131)

Consequently

$${\mathcal{U}}_n:={\displaystyle \frac{{\mathsf{\Gamma }}_{\mathrm{dark}}{K}^n}{{\kappa }_n{|\Delta \mathcal{E}|}^n}}⟶\left\{\begin{array}{cc}{6}^n,\hfill & {\beta }_{\mathrm{LS}}\ne 0,\hfill \\ 8^n,\hfill & {\beta }_{\mathrm{LS}}=0\mathrm{by}\mathrm{exact}\mathrm{symmetry},{\gamma }_{\mathrm{LS}}>0.\hfill \end{array}\right.$$

(132)

The coefficient ${\kappa }_n$ is extracted from the outgoing absorptive operator, so this remains a same-branch closure test. Write $\mathcal{G}\left(Q\right)$ for the dual fixed-Q generating functional and $d_{\mathsf{\Phi }}:=\parallel {\mathsf{\Phi }}_u-{\mathsf{\Phi }}_s\parallel$ for the field-space separation of its two local sheets. For the same simple charge fold,

$$\Delta {\mathcal{G}}_{\mathrm{fold}}={\displaystyle \frac{4\sqrt{2}}{3\sqrt{|Q_{*}^{\prime \prime }|}}}|Q-Q_{*}{|}^{3/2}+O\left(\right|Q-Q_{*}{|}^2)={\displaystyle \frac{|{\beta }_Q|}{12}}{d}_{\mathsf{\Phi }}^3+O\left(d_{\mathsf{\Phi }}^4\right),$$

(133)

with the stable sheet lower when $\mathcal{G}$ is the physical fixed-Q energy/free-energy functional. If the same fold is an exact same-arrow dark point, the dark projector is differentiated along the same multiplier tangent, leakage first appears at order n, and the spectral root is simple with ${dE/d\mathsf{\Omega }|}_{*}\ne 0$, then

$${\mathsf{\Gamma }}_{\mathrm{dark}}\sim |Q-Q_{*}{|}^n\sim {|E-E_{*}|}^{2n}\sim \Delta {\mathcal{G}}_{\mathrm{fold}}^{2n/3}.$$

(134)

These are exponent relations; the absolute width coefficient still requires the outgoing spectral density.

Theorem 10

(Conditional protected one-coordinate reduction). Assume on one realized current-Codazzi branch: exact axial $SO\left(2\right)$ invariance; the native source-Hodge provenance test (84); the Noether-Fredholm charge lock and charge-VH ensemble incidence of Proposition 21; a protected seed-to-fold segment satisfying Theorem 8, with the weight-one, hard, Callias, AIII, port, and Riesz gaps included among its protected margins; and Parent stationarity or constraint equations for every remaining eliminated variable with invertible linearization. Then the axial fixed line is an invariant branch and all gapped tomographic variables are locally either zero by symmetry or smooth functions of one longitudinal coordinate x. The protected reduced data are ${\mathcal{P}}_{\mathrm{AC},\mathrm{red}}=\{V_{\mathrm{eff}}\left(x\right),L_{\mathrm{eff}}\left(x\right),H_{\mathrm{jet}}\left(x\right),{\nabla }_{\mathrm{root}}\left(x\right)\}$. The unique zero of ${\chi }_Q$ is the simple terminal on that protected segment; when the certified coverage starts at the physical stable anchor it is the first protected terminal. Its VH image is (98). Weight-one or biaxial terminals retain even real multiplicity. The number of gapped microscopic probe directions therefore need not equal the number of retained physical slow moduli.

Proof. 

Axial equivariance preserves the fixed-point set and the implicit-function theorem solves the gapped stationary variables locally as functions of x. The seed-to-fold certificate keeps the competing margins open and gives one discriminant zero; exact constrained Schur elimination and the charge-VH incidence identify its retained Hessian reading with (97). □

Along the one-coordinate branch let $a_{\mathrm{cen}}\left(x\right)$ denote the coefficient of the centered response line fixed by (25), let $b_{\mathrm{tr}}\left(x\right)$ denote the scalar trace readout entering the reduced flavor denominator, and let $t_{\mathrm{tr}}\ne 0$ be its marked configuration-space tangent. On a nonhomogeneous branch with $d{R}_{\mathrm{vort}}\ne 0$, the rank-one scalar Gram contribution selects ${\ell }_R=\mathrm{span}\left\{{\left(d{R}_{\mathrm{vort}}\right)}^{♯}\right\}$; its incidence with the protected trace line is tested by

$${\mathcal{R}}_{\mathrm{align}}:=1-{\displaystyle \frac{|d{R}_{\mathrm{vort}}\left(t_{\mathrm{tr}}\right){|}^2}{\parallel d{R}_{\mathrm{vort}}{\parallel }^2{\parallel t_{\mathrm{tr}}\parallel }^2}}=0.$$

(135)

This condition is distinct from ${\mathcal{R}}_{\mathrm{tn}}$ of Proposition 22: (135) identifies the scalar response line with the flavor-trace tangent, while ${\mathcal{R}}_{\mathrm{tn}}=0$ keeps the critical soft vector on that line. The operator-native family normalization and the reduced slope are imposed only after the actual family block has been constructed in Section 4.3. The constant collar rescaling $(\phi ,B)\mapsto (c\phi ,B/c)$ leaves the Codazzi product and $d\log \left|\phi \right|$ unchanged, while the Veronese-Hodge selector fixes projective directions and singular-gain ratios but leaves the physical stiffness normalization Parent-dependent. For a nonzero two-control export with susceptibility $J=D_{\xi }(a_{\mathrm{cen}},b_{\mathrm{tr}})$, let $C_{\mathrm{ctrl}}\succ 0$, put $r_{*}=-80/147$, ${\widehat{c}}_{*}={(1,r_{*})}^T/\sqrt{1+r_{*}^2}$, and ${\widehat{n}}_{*}={(-r_{*},1)}^T/\sqrt{1+r_{*}^2}$. The invariant diagnostics are $q_{\det }:=\sqrt{\det \left(J{C}_{\mathrm{ctrl}}{J}^{†}\right)}/\mathrm{Tr}\left(J{C}_{\mathrm{ctrl}}{J}^{†}\right)$ and ${ϵ}_{*}^2:={\widehat{n}}_{*}^{†}J{C}_{\mathrm{ctrl}}{J}^{†}{\widehat{n}}_{*}/\mathrm{Tr}\left(J{C}_{\mathrm{ctrl}}{J}^{†}\right)$. Under an invertible control reparametrization ${\xi }^{\prime }=S\xi$, $J^{\prime }=J{S}^{-1}$ and $C_{\mathrm{ctrl}}^{\prime }=S{C}_{\mathrm{ctrl}}{S}^{†}$, so $J{C}_{\mathrm{ctrl}}{J}^{†}$ is unchanged. For $C_{\mathrm{ctrl}}=\mathbf{1}$, $q_{\det }=\left|\det J\right|/{\parallel J\parallel }_F^2$; for ${ϵ}_{*}<1$, ${\sigma }_2\left(J{C}_{\mathrm{ctrl}}^{1/2}\right)/{\sigma }_1\left(J{C}_{\mathrm{ctrl}}^{1/2}\right)\le {ϵ}_{*}/\sqrt{1-{ϵ}_{*}^2}$. These remain executable range diagnostics, while the primary cross-layer normalization gate is (175).

The isolated Riesz projector supplies the response metric (138). Its arclength is typed to the response path and is distinct from the stationary-state arclength of Corollary 7; the reduced coordinate is fixed only after the centered same-response morphism of Proposition 35. If only one Schur stiffness softens while complementary modes remain gapped, the singular inverse and the divergent control-space QGT are rank one. A second visible divergent response direction falsifies this branch, while collinear readout of distinct microscopic soft modes still requires an independent Hessian/QGT gap for exclusion.

4.3. Microscopic Spectral Attachment

The pole of (101) is a zero-frequency constrained response on the stationary Parent branch. Spectral states and outgoing resonances below are instead defined in the independent spectral coordinate E; identifying the two requires a typed same-object channel map and is not part of the fold certificate. Let $L_b$ be the microscopic normal operator, $N_{+}=L_b^{†}{G}_{\mathrm{mic}}{L}_b$, and $P_b$ the Riesz projector onto an isolated low cluster. On an exact constant-rank kernel branch, with $P=P_{\ker L_0}$, $Q=\mathbf{1}-P$, $A=L_{0}Q$ and graph jets $b\mathcal{B}+b^2\mathcal{C}/2+O\left(b^3\right)$, kernel persistence gives

$$A\mathcal{B}=-{\dot{L}}_{0}P,A\mathcal{C}=-({\ddot{L}}_{0}P+2{\dot{L}}_0\mathcal{B}).$$

(136)

For a near-zero cluster the usual Riesz derivative and common gap are retained; standard perturbation theory is used in the form of [8]. For a rank-one parent field $\mathsf{\Psi }=\sqrt{\rho }u{v}^{†}$, the anti-Hermitian Moore-Penrose current reduces to ${\mathcal{A}}_R=[D{P}_u,P_u]/2+\vartheta P_u$, with $D{P}_u=2[{\mathcal{A}}_R,P_u]$ and $\parallel {\mathcal{A}}_R{\parallel }_{\mathrm{HS}}^2=\parallel D{P}_u{\parallel }_{\mathrm{HS}}^2/4+{\left|\vartheta \right|}^2$; its projective part is the minimum-Hilbert-Schmidt lift modulo the stabilizer.

At the base point put $P=P_b$, $Q=\mathbf{1}-P$, $M=L_{0}Q$, and $G_H=M^{†}M$ on the active high subspace. For a tangent direction X, set $A_X={\left({\partial }_{X}L\right)}_{0}P$, $M^{+}=G_H^{-1}{M}^{†}$, ${\mathsf{\Pi }}_{\mathrm{cok}}=\mathbf{1}-M{M}^{+}$, ${\mathcal{J}}_X={\mathsf{\Pi }}_{\mathrm{cok}}{A}_X$, and ${\mathcal{B}}_X=-M^{+}{A}_X=Q\left({\partial }_{X}P\right)P$.

Proposition 24

(Microscopic Moore-Penrose-Riesz splitting). If M is injective on the active high subspace, then

$$A_X^{†}{A}_Y={\mathcal{J}}_X^{†}{\mathcal{J}}_Y+{\mathcal{B}}_X^{†}{G}_H{\mathcal{B}}_Y.$$

(137)

The two terms measure stationary obstruction and projected-bundle motion. For a one-parameter branch, ${\partial }_{x}P={\mathcal{B}}_x+{\mathcal{B}}_x^{†}$ and therefore

$$g_R\left(x\right):={\displaystyle \frac{1}{2}}\mathrm{Tr}\left[{\left({\partial }_{x}P\right)}^2\right]=\parallel {\mathcal{B}}_x{\parallel }_{\mathrm{HS}}^2={\parallel M^{+}{A}_x\parallel }_{\mathrm{HS}}^2.$$

(138)

If $g_H\mathbf{1}⪯G_H⪯G_H^{\max }\mathbf{1}$ and ${\Delta }_x:=\parallel A_x{\parallel }_{\mathrm{HS}}^2-{\parallel {\mathcal{J}}_x\parallel }_{\mathrm{HS}}^2$, then

$${\displaystyle \frac{{\Delta }_x}{G_H^{\max }}}\le g_R\left(x\right)\le {\displaystyle \frac{{\Delta }_x}{g_H}}.$$

(139)

Thus the intrinsic Riesz length is obtained directly from the Parent graph jet without numerical differentiation of P.

Along a one-parameter branch $L\left(t\right)=L_0+t{L}_1+t^2{L}_2/2+O\left(t^3\right)$, put $K_1={\mathsf{\Pi }}_{\mathrm{cok}}{L}_{1}P$; on the first-order unobstructed branch $K_1=0$, the second pre-Gram jet is

$$K_2={\mathsf{\Pi }}_{\mathrm{cok}}\left(L_{2}P-2L_1{M}^{+}{L}_{1}P\right),P{N}_{+}^{\prime }\left(0\right)P=0,P{N}_{+}^{″}\left(0\right)P=2{\left(L_{1}P\right)}^{†}\left(L_{1}P\right).$$

(140)

Thus the Gram family retains the Hermitian quadratic combination of the first pre-Gram jet but not its oriented complex representative. After the active high block is Schur-eliminated on $K_1=0$, the same second obstruction gives

$$L_{\mathrm{eff}}\left(t\right)={\displaystyle \frac{1}{2}}{t}^2{K}_2+O\left(t^3\right),N_{\mathrm{eff}}\left(t\right)={\displaystyle \frac{1}{4}}{t}^4{K}_2^{†}{K}_2+O\left(t^5\right).$$

(141)

Thus the reduced small Gram spectrum opens at fourth order; the $t^2$ term in $P\left(L^{†}L\right)P$ is a pre-Schur contribution. Put $Q_{\mathrm{mic}}^{\mathrm{raw}}:=P{N}_{+}^{″}\left(0\right)P/2$ on the first-order unobstructed branch and whiten it by the same retained metric used for the local protected response packet. When $\mathrm{Tr}{Q}_{\mathrm{mic}}^{\mathrm{raw}}>0$, put ${\widehat{Q}}_{\mathrm{mic}}:=3Q_{\mathrm{mic}}^{\mathrm{raw}}/\left(2\mathrm{Tr}{Q}_{\mathrm{mic}}^{\mathrm{raw}}\right)$. The raw trace is retained as a coupling-strength datum only when the common Gram scale is independently protected. For an observable reduced by P, Riesz motion alone cannot create a first response from a block-diagonal baseline: if P reduces $O_0$, then $D{O}_{\mathrm{low}}{|}_0=P{\left(DO\right)}_{0}P$. On an exact persistent-kernel branch the graph jets are already fixed by (136).

Proposition 25

(Spin-one polynomial checksum of the axial response packet). On the axial branch complexify $V_1$ and put $J_e=-i{\left[e\right]}_{\times }$. Let $A_{\mathrm{SH}}=P_{V_1}{P}_3^{\mathrm{phys}}{P}_{V_1}{|}_{V_1}$ and use $Q_e$ from (14). For independently exported $J_e,Q_e,A_{\mathrm{SH}},{\widehat{Q}}_{\mathrm{mic}}$ define

$$\begin{array}{cc}\hfill {\mathcal{R}}_{\mathrm{ax}}^2=& \parallel [{\widehat{Q}}_{\mathrm{mic}},J_e]{\parallel }_{\mathrm{HS}}^2+{∥J_e^2-\left({\displaystyle \frac{2}{3}}\mathbf{1}-Q_e\right)∥}_{\mathrm{HS}}^2\hfill \\ \hfill & +\parallel J_e^2-(7A_{\mathrm{SH}}-3\mathbf{1}){\parallel }_{\mathrm{HS}}^2+{\parallel {(2{\widehat{Q}}_{\mathrm{mic}}-\mathbf{1})}^2-J_e^2\parallel }_{\mathrm{HS}}^2.\hfill \end{array}$$

(142)

Then ${\mathcal{R}}_{\mathrm{ax}}=0$ implies ${\widehat{Q}}_{\mathrm{mic}}=(\mathbf{1}\pm J_e)/2$. Consequently the Veronese axis, the source-Hodge graph compression, and the normalized primitive first Gram defect are complementary polynomial readings of one spin-one generator. The sign is not fixed by the even graph data and remains part of the orientation/Real/global attachment.

This checksum concerns the common axial generator and source axis. The full principal Clifford symbol, the product/stiffness tomography, and the native residual provenance remain typed tests of the same Parent packet; the polynomial residual does not replace them. Its graph-Gram part gives a coordinate-free common-axis test before any separate axis fit. The local minimal square root is read on a Hopf lift, consistently with Proposition 3; a global fixed two-frame is not an admissible target.

A marking-preserving unitary residual-frame change conjugates $N_{+}$, $P_b$, $G_H$, ${\mathcal{J}}_X$, and ${\mathcal{B}}_X$ by the marked unitaries; rank, spectrum, gap, projected connection, trace invariants, and determinant holonomy therefore descend independently of the residual frame.

Proposition 26

(Marked four-moment primitive-port certificate). Let $S_0={\mathbf{1}}_C,S_1,\dots ,S_4$ be a normalized Hermitian three-port moment packet and put

$$K=S_2-S_1^2,R=S_3-S_1{S}_2-S_2{S}_1+S_1^3,S_4^{\mathrm{pred}}=S_1{S}_3+K{S}_2+R{S}_1+R{K}^{+}R.$$

(143)

where $K^{+}$ is the Moore-Penrose inverse. At a regular point with $K⪰0$, $\mathrm{rank}K=2$, unit $d\in \ker K$, and $Rd=0$, put ${\mathsf{\Omega }}_4:=S_4-S_4^{\mathrm{pred}}$. The packet has a finite self-adjoint realization through order four exactly when ${\mathsf{\Omega }}_4⪰0$ and ${\mathsf{\Omega }}_{4}d=0$; its minimal cyclic dimension is $5+\mathrm{rank}{\mathsf{\Omega }}_4\in \{5,6,7\}$. Hence ${\mathsf{\Omega }}_4=0$ is the exact rank-five stopping certificate. On this stratum the generic residual normal coordinates consist of one simple-null condition for K, five real components of $Rd$, and the nine Hermitian components of ${\mathsf{\Omega }}_4$.

If the port is independently marked as $C={\mathrm{Sym}}^{2}W$ in the orthonormal basis $(s_1^2,\sqrt{2}{s}_1{s}_2,s_2^2)$, the first coupling is primitive exactly when $K_{02}=0$, $K_{01}=K_{12}$, and $2K_{11}=K_{00}+K_{22}$. These are five additional real linear residuals. On the nonzero rank-two branch, $\alpha =K_{00}+K_{22}$ and the projective spinor $s=a{s}_1+b{s}_2$ is reconstructed by ${\left|a\right|}^2=K_{00}/\alpha$, ${\left|b\right|}^2=K_{22}/\alpha$, and $a\overline{b}=\sqrt{2}{K}_{01}/\alpha$. Thus the exact-five generic and primitive residual maps have local codimensions 15 and 20, respectively, at regular rank-two points. Singular strata, including rank changes and gap closures, require separate treatment. If the full normalized port response is independently known to have minimal Hermitian realization degree five, the same certificate identifies its first coupling; on the primitive branch the realization is unitarily equivalent to $C\oplus W$ up to the hidden $U\left(2\right)$ frame.

Proof. 

Choose X with $X{X}^{†}=K$. Since $Rd=0$ and R is Hermitian, $R=XE{X}^{†}$ for a Hermitian hidden block E, unique up to the hidden $U\left(2\right)$ frame. Necessity follows from the Gram decomposition ${\mathsf{\Omega }}_4={\left(H^{2}V\right)}^{†}(\mathbf{1}-P_1)H^{2}V$, which gives ${\mathsf{\Omega }}_4⪰0$ and ${\mathsf{\Omega }}_{4}d=0$. Conversely these conditions place ${\mathsf{\Omega }}_4$ on $\mathrm{Ran}K$, so ${\mathsf{\Omega }}_4=XT{T}^{†}{X}^{†}$ for a matrix T of rank $\mathrm{rank}{\mathsf{\Omega }}_4$; adjoining the corresponding hidden directions gives a Hermitian realization, while no smaller cyclic extension can reproduce this Gram rank. The marked symmetric-power conditions reconstruct $K=\alpha j_s{j}_s^{†}$ as before. □

A nonzero admissible ${\mathsf{\Omega }}_4$ opens additional finite spectral support but does not by itself determine a continuum pole or lifetime. On the exact-five stratum the reconstructed self-adjoint realization $H_5$ and the marked inclusion $J:C\hookrightarrow C\oplus W$ already generate the finite Weyl function $m_5\left(z\right)=J^{†}{(H_5-z\mathbf{1})}^{-1}J$; its large-z expansion is determined by the retained moments. The additional microscopic gate is therefore provenance: the actual channel response must be this marked finite response, or its affine passive Schur/continuum extension, rather than an independently chosen denominator. The standard positive spectral-measure framework is the following.

Proposition 27

(Passive Schur-Herglotz response and minimal active realization). For $\mathbb{K}\left(z\right)=\left(\begin{array}{cc}A-z{Z}_L& B\\ B^{†}& D-z{Z}_H\end{array}\right)$ self-adjoint on the real axis, with $Z_L,Z_H\succ 0$, put $\mathsf{K}=Z_H^{-1/2}D{Z}_H^{-1/2}$ and $\mathsf{C}=B{Z}_H^{-1/2}$. The negative low Schur complement

$$\mathcal{W}\left(z\right)=-A+z{Z}_L+\mathsf{C}{(\mathsf{K}-z\mathbf{1})}^{-1}{\mathsf{C}}^{†}$$

(144)

is matrix Herglotz-Nevanlinna [76]; on the finite active subspace

$$\mathcal{W}\left(z\right)=-A+z{Z}_L+\sum _a{\displaystyle \frac{R_a}{{\lambda }_a-z}},R_a=\mathsf{C}{P}_a{\mathsf{C}}^{†}⪰0.$$

(145)

For $\mathsf{K}\succ 0$ and $\alpha \ge 0$,

$$M_n\left(\alpha \right)={\displaystyle \frac{{(-1)}^n}{n!}}{\displaystyle \frac{d^n}{d{\alpha }^n}}\left[\mathcal{W}(-\alpha )+A+\alpha Z_L\right]=\mathsf{C}{(\mathsf{K}+\alpha \mathbf{1})}^{-(n+1)}{\mathsf{C}}^{†}⪰0.$$

(146)

The block-Hankel rank is the reachable cyclic dimension, hence the minimal finite realization degree; its reconstruction is the finite positive moment problem [77]. Invisible heavy directions remain in the dark complement.

Proposition 28

(Protected channel gap states). If (144) is equivariant under the retained compact gauge action, a multiplicity-one irreducible channel has ${\mathcal{W}\left(E\right)|}_{V_{\lambda }}=w_{\lambda }\left(E\right)\mathbf{1}$. In every real gap, positivity gives $w_{\lambda }^{\prime }\left(E\right)>0$ on a nonconstant channel, so there is at most one zero per gap; for k finite visible positive-residue poles the $k+1$ simple zeros interlace them. At a zero, the full-operator kernel has the same multiplet multiplicity whenever the complementary Feshbach block is invertible. A simple gap state obeys $d{E}_i/ds=-\left({\partial }_s{w}_{\lambda }\right)/\left({\partial }_E{w}_{\lambda }\right)$ and persists while the gap, positivity, equivariance, and visibility remain open. If a visible residue r at λ tends to zero and the regular part satisfies $h\left(\lambda \right)\ne 0$, then $E_r=\lambda +r/h\left(\lambda \right)+O\left(r^2\right)$; the visible pole-zero pair disappears without changing the microscopic carrier dimension.

Proposition 29

(Finite active inverse-spectral closure). For a scalar compression $m_n=u^{†}{M}_n\left(\alpha \right)u$ with d visible nodes $x_a={({\lambda }_a+\alpha )}^{-1}$ and positive weights, let $H_0={\left(m_{i+j}\right)}_{i,j=0}^{d-1}$ and $H_1={\left(m_{i+j+1}\right)}_{i,j=0}^{d-1}$. Then $H_0\succ 0$, the generalized eigenvalues of $H_{1}v=x{H}_{0}v$ are the $x_a$, and $m_0,\dots ,m_{2d-1}$ reconstruct the visible poles and weights. If $\parallel H_j-{\tilde{H}}_j\parallel \le {\epsilon }_j$, $\tilde{h}={\lambda }_{\min }\left({\tilde{H}}_0\right)>{\epsilon }_0$, and $h=\tilde{h}-{\epsilon }_0$, then

$$|x_a-{\tilde{x}}_a|\le {\delta }_x:={\displaystyle \frac{{\epsilon }_1}{h}}+{\displaystyle \frac{\parallel {\tilde{H}}_1\parallel {\epsilon }_0}{h\tilde{h}}},{\displaystyle \frac{1}{{\tilde{x}}_a+{\delta }_x}}-\alpha \le {\lambda }_a\le {\displaystyle \frac{1}{{\tilde{x}}_a-{\delta }_x}}-\alpha ,$$

(147)

for ${\tilde{x}}_a>{\delta }_x$. The stabilized Hankel rank is the SISO realization degree; cyclicity of the chosen port is required for full active recovery. Exact vanishing of $m_n{m}_{n+2}-m_{n+1}^2$ characterizes a one-pole scalar compression.

Normalize the visible spectral mass and let $\mathfrak{V}=({\nu }_0{\nu }_2-{\nu }_1^2)/{\nu }_0^2$. If one bright node has a dark contribution of weight $w_D\le 1/2$ at distances in $[{\Delta }_{-},{\Delta }_{+}]$, then

$$w_D(1-w_D){\Delta }_{-}^2\le \mathfrak{V}\le w_D{\Delta }_{+}^2,{\displaystyle \frac{\mathfrak{V}}{{\Delta }_{+}^2}}\le w_D\le {\displaystyle \frac{2\mathfrak{V}}{{\Delta }_{-}^2}}.$$

(148)

For the same leaked component in an outgoing form ${\alpha }_{\mathrm{out}}\mathbf{1}⪯A_{\mathrm{out}}⪯{\beta }_{\mathrm{out}}\mathbf{1}$,

$${\displaystyle \frac{{\alpha }_{\mathrm{out}}}{{\Delta }_{+}^2}}\mathfrak{V}\le \mathsf{\Gamma }\le {\displaystyle \frac{2{\beta }_{\mathrm{out}}}{{\Delta }_{-}^2}}\mathfrak{V}.$$

(149)

Thus finite leakage controls a non-asymptotic width interval once same-object outgoing coercivity is supplied. The fold-dark exponent diagnostics remain those of (131); a noninteger positive inferred leakage order falsifies the single-analytic-jet mechanism.

For matrix data the block-Hankel pencil reconstructs the reachable spectrum after Krylov-rank stabilization. If $X={(\mathsf{K}+\alpha \mathbf{1})}^{-1}$ has minimal polynomial $p_X\left(t\right)=t^d+c_{d-1}{t}^{d-1}+\dots +c_0$, then every typed port pair sharing this denominator obeys

$$M_{n+d}^{ab}+c_{d-1}{M}_{n+d-1}^{ab}+\dots +c_0{M}_n^{ab}=0.$$

(150)

For two slices related by ${\mathsf{K}}_1=U({\mathsf{K}}_0+\Delta \mathbf{1})U^{†}$ and transported ports,

$${\mu }_{n,1}^{ab}={\displaystyle \sum _{j=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\Delta }^{n-j}{\mu }_{j,0}^{ab},{\mathcal{M}}_{n,1}^{ab}\left(\alpha \right)={\mathcal{M}}_{n,0}^{ab}(\alpha +\Delta ).$$

(151)

One scalar compression fixes Δ; the remaining moments are no-refit holdouts. If $\tilde{h}\le {\epsilon }_0$, the pole packet remains unresolved at the certified error level.

Proposition 30

(Projective one-pole and arithmetic-shell certificate). For a marked unit scalar channel ℓ, subtract the affine Herglotz part and put $m_{\mathrm{act}}\left(E\right):=a_0+{\ell }^{†}(\mathcal{W}\left(E\right)+A-E{Z}_L)\ell$. One finite visible active pole gives $m_{\mathrm{act}}\left(E\right)=a+r/(\lambda -E)$, $r>0$, hence zero Schwarzian, rank-one Loewner/Pick kernels, and preserved cross-ratios. Conversely these projective rank-one conditions on a non-affine rational Herglotz branch give the one-pole class up to a real Möbius post-map. For the family packet the same-object Cayley lock is $\mathcal{C}\left(m_{\mathrm{act}}\right)=\mathcal{C}\left(\sqrt{3}{q}_{\mathrm{fam}}\right)$ with $\mathcal{C}\left(x\right)=(x+i)/(x-i)$; Cayley injectivity then fixes $m_{\mathrm{act}}=\sqrt{3}{q}_{\mathrm{fam}}$.

On one trace-one arithmetic orientation, $q_m=2m+1$ gives

$$E_m=\lambda -{\displaystyle \frac{r}{\sqrt{3}(2m+1)-a}},{\displaystyle \frac{1}{\lambda -E_{m+1}}}-{\displaystyle \frac{1}{\lambda -E_m}}={\displaystyle \frac{2\sqrt{3}}{r}}.$$

(152)

Three consecutive shells reconstruct $(\lambda ,r,a)$; in particular

$$\lambda ={\displaystyle \frac{2E_{m-1}{E}_{m+1}-E_m(E_{m-1}+E_{m+1})}{E_{m-1}+E_{m+1}-2E_m}},r={\displaystyle \frac{2\sqrt{3}}{{(\lambda -E_{m+1})}^{-1}-{(\lambda -E_m)}^{-1}}},$$

(153)

with $a=\sqrt{3}(2m+1)-r/(\lambda -E_m)$. The fourth shell is then a no-refit holdout and the cross-ratio of four consecutive same-orientation values is $4/3$. For $q\left(E\right)=(a+r/(\lambda -E))/\sqrt{3}$ and $q^{\prime \prime }\ne 0$, the local inverse is $\lambda =E+2q^{\prime }/q^{\prime \prime }$, $r=4\sqrt{3}{\left(q^{\prime }\right)}^3/{\left(q^{\prime \prime }\right)}^2$, and $a=\sqrt{3}[q-2{\left(q^{\prime }\right)}^2/q^{\prime \prime }]$. Thus the global shell tower and local characteristic response reconstruct the same $(\lambda ,r,a)$ and form one redundant certificate.

Proposition 31

(Product-commutator centrality certificate). Let the finite active block carry the product action generated by the fundamental $SU\left(3\right)$ and $SU\left(2\right)$ factors together with the marked family Weyl pair $S,Z$, with $SU\left(n\right)$ generators normalized by $\mathrm{Tr}\left(T_a^{†}{T}_a\right)=1/2$. Let ${\mathbb{E}}_{\mathrm{prod}}$ be the trace-preserving twirl onto the product commutant and put ${\mathcal{C}}_{\mathrm{prod}}\left(X\right)={\sum }_a\parallel [X,T_a]{\parallel }_{\mathrm{HS}}^2+\parallel [X,S]{\parallel }_{\mathrm{HS}}^2+{\parallel [X,Z]\parallel }_{\mathrm{HS}}^2$. The factor commutator Laplacians have spectra $\{0,3\}$ on $M_3$, $\{0,2\}$ on $M_2$, and $\{0,3,6\}$ for the marked family Weyl pair; the tensor-sum spectrum therefore has smallest and largest positive values 2 and 11. Hence on the orthogonal complement of the scalar commutant, ${\mathcal{C}}_{\mathrm{prod}}\left(X\right)/11\le {\parallel X-{\mathbb{E}}_{\mathrm{prod}}X\parallel }_{\mathrm{HS}}^2\le {\mathcal{C}}_{\mathrm{prod}}\left(X\right)/2$. For $R_H\left(z\right)={(K_H-z)}^{-1}$ and $g_H\left(z\right)=\mathrm{dist}(z,\sigma \left(K_H\right))>0$,

$$\parallel R_H-{\mathbb{E}}_{\mathrm{prod}}{R}_H{\parallel }_{\mathrm{HS}}\le {\displaystyle \frac{\sqrt{{\mathcal{C}}_{\mathrm{prod}}\left(K_H\right)/2}}{g_H{\left(z\right)}^2}}.$$

(154)

A typed Schur edge multiplies the bound by at most its squared operator norm. Exact product equivariance makes the denominator scalar on the protected product factor; approximate equivariance gives a nonperturbative propagation error budget.

A finite-dimensional symmetric block cannot itself supply the nontrivial boundary space $C\simeq {\mathbb{C}}^3$. In the standard boundary-operator setting of [78,79], an enlarged closed symmetric core/collar has $n_{+}=n_{-}=3$; its boundary triple fixes no preferred positive frame normalization. If the same response is realized as a Weyl function $M\left(z\right)$, the usual boundary determinant $\det (\mathsf{\Theta }-M(z\left)\right)$ counts extension poles by the argument principle, while threshold statements require separate continuum asymptotics.

Proposition 32

(Primitive port continuum and moment protection). Let the continuum couple behind the hidden W block through the same primitive rank-two arrow $X:W\to C$. If the dynamical matrix-Weyl term factorizes as ${\Sigma }_C^{\mathrm{dyn}}\left(z\right)=XM\left(z\right)X^{†}$, then

$$L_{\mathrm{dark}}\subseteq \ker {\Sigma }_C^{\mathrm{dyn}}\left(z\right)\mathrm{for}\mathrm{every}z.$$

(155)

In particular, for an absorptive density ${\rho }_W⪰0$ with $\Im {\Sigma }_C=X{\rho }_W{X}^{†}$, one has $\mathrm{rank}\Im {\Sigma }_C\le 2$ and $L_{\mathrm{dark}}\subseteq \ker \Im {\Sigma }_C$. Let $A_{\mathrm{eff}}$ denote the Hermitian C-block of the finite response after the retained subtraction/counterterm contribution at the selected real spectral point. For a unit generator d of $L_{\mathrm{dark}}$, dynamic protection is tested by ${\mathcal{R}}_{\mathrm{dark}}:=\parallel (\mathbf{1}-d{d}^{†})A_{\mathrm{eff}}d\parallel$. If ${\mathcal{R}}_{\mathrm{dark}}=0$, so that $A_{\mathrm{eff}}d=E_{d}d$, the same-arrow identity gives $[A_{\mathrm{eff}}+{\Sigma }_C^{\mathrm{dyn}}\left(z\right)]d=E_{d}d$ for every z in the Weyl domain. Thus the continuum cannot shift or broaden this line; exact BIC status additionally requires the appropriate outgoing/domain condition. If the retained line moves, exact covariant protection requires $(\mathbf{1}-P)H_{\mathrm{dyn}}P=i\hslash (\mathbf{1}-P)\left({\nabla }_{t}P\right)P$. Bright leakage of order $O\left({\epsilon }^n\right)$ gives width $O\left({\epsilon }^{2n}\right)$ when the outgoing density is regular; the coefficient is controlled by (149) once the same leaked component and outgoing coercivity are identified. In the primitive spin-one block a generic spherical-tensor perturbation of rank $k=1$ or 2 has direct $q=1$ or $q=2$ matrix elements, so typically $n=1$; higher order requires typed symmetry or vanishing reduced matrix elements. If a new tail first occurs at graph distance L from the unchanged finite port, every closed port walk visiting it has length at least $2L$; hence $S_n^{\mathrm{new}}=S_n^{\mathrm{old}}$ for $n<2L$, with $S_{2L}$ the first moment that can change.

When the isolated Riesz module and the Borel-Weil target are marking-preserving multiplicity-one modules of type $E_3\oplus E_2$, their unitary incidence is fixed up to the marked block phases; Berry-Kato transport is then fixed by the projector path. The conditional magnetic-charge-three comparison uses the standard Nahm-Hitchin correspondence [80] in the ADHMN convention of [81]. Axial and rational-map comparisons are represented by [82,83,84]; explicit cyclic charge-three spectral data are represented by [85,86,87]. At the strongly centered axial charge-three comparison point, in the standard $(\eta ,\zeta )$ spectral-curve coordinates and fixed normalization, the admissible pure-phase tangent has $\delta q=0$, no $\eta {\zeta }^2$ variation, and spans $i\mathbb{R}\zeta ({\zeta }^4-1)$; this is retained as a microscopic tangent checksum before Dirac-Riesz reduction. The physical charge normalization is not fixed by the degree-three associated line.

Under the standard nonsingularity hypotheses of the Nahm-Hitchin correspondence, the spectral direct image satisfies ${\pi }_{*}{L}^s\left(k\right)\simeq \mathcal{O}{\left(1\right)}^{\oplus k}$ for $0<s<2$. Thus the charge-three comparison gives $L_{\mathsf{\Gamma }}\otimes {\mathbb{C}}^3$, and an independently marked $C_3$ attachment gives $L_{\mathsf{\Gamma }}\otimes \mathbb{C}\left[C_3\right]$; the rank-three residue and rank-two inverse ADHMN kernel reproduce the $E_3$ and $E_2$ comparison types. This remains a comparison: the family torsor is fixed by the protected $A_2$ coefficient geometry, not by the monopole construction.

The full exterior coefficient module is retained before parity locking. With ${\mathsf{\Gamma }}_V={(-1)}^{N_C+N_W}$,

$${\mathsf{\Gamma }}_{\mathrm{lock}}={\gamma }_{\mathrm{br}}^5\otimes {\mathsf{\Gamma }}_V,$$

(156)

and a separately typed microscopic Weyl block passes the weak-direct factorization when one section $h\in W$ and four family matrices $Y_f$ give

$$\begin{array}{cc}\hfill {\tilde{Q}}_F^{\mathrm{P}}& =T_u\left(h\right)\otimes Y_u+T_d\left(h\right)\otimes Y_d+T_e\left(h\right)\otimes Y_e+T_{\nu }\left(h\right)\otimes Y_{\nu },\hfill \\ \hfill Q_{F,c}^{\mathrm{P}}& ={\tilde{Q}}_F^{\mathrm{P}}+{\left({\tilde{Q}}_F^{\mathrm{P}}\right)}^{t_{\nu }},D_F^{\mathrm{P}}=Q_{F,c}^{\mathrm{P}}+{\left(Q_{F,c}^{\mathrm{P}}\right)}^{†}.\hfill \end{array}$$

(157)

The channel factors are multiplicity one by (33); determinant-dual completion adds no family matrix. On a smooth gapped carrier family the marked kernels form ${\mathcal{V}}^{\mathrm{P}}={\mathcal{C}}^{\mathrm{P}}\oplus {\mathcal{W}}^{\mathrm{P}}$, and the split-preserving connection induces the standard superconnection ${\nabla }^{F,\mathrm{P}}+D_F^{\mathrm{P}}$ in the sense of [15]. The collar is spin, so the corresponding spacetime Dirac operator is $i{\gamma }^{\mu }{\nabla }_{\mu }^{S\otimes F,\mathrm{P}}+{\gamma }_{\mathrm{br}}^5\otimes D_F^{\mathrm{P}}$ and anticommutes with ${\mathsf{\Gamma }}_{\mathrm{lock}}$; commutant/order-one and full Real compatibility remain separate attachment tests.

The gauge quadratic form is the $G_0$ already present in (59). Its scale-free departure from the protected trace ray is the traceless part of the associated positive operator $K_{\mathrm{P}}$; typed gauge variation gives $D_{\mu }\left(\phi K_{\mathrm{P}}{\mathcal{F}}^{\mu \nu }\right)={\mathcal{J}}^{\nu }$, while the common normalization remains a matching datum. For a parity-locked high lift, family×parity threshold matrices $M_W,M_C\in M_6\left(\mathbb{C}\right)$ are compared by their normalized singular spectra. Equality of these spectra together with $\parallel M_C{\parallel }_{\mathrm{F}}/{\parallel M_W\parallel }_{\mathrm{F}}=16$ gives $|\det M_C|/|\det M_W|={16}^6$; a determinant phase needs the marked determinant line. The structural scalar index 256 does not by itself imply this tensorial statement, and the common high scale remains completion data.

For faithful block maps, Schur functoriality transports the principal support as $V_2|V_1|V_2\mapsto C\left|W_{\sigma }\right|C$, i.e. $3\left|2\right|3$. The centered connection weights are $(a_C,a_W)=r_{\vartheta }(-2/5,3/5)$ and the relative family line has weight five. Thus the common primitive lift has weights $(-2,3,5)$; for extracted coefficients $v=({\alpha }_C,{\alpha }_W,{\alpha }_F)$ its least-squares amplitude is $a_{*}=(-2{\alpha }_C+3{\alpha }_W+5{\alpha }_F)/38$. These are the differential readings of the centered cocharacter and relative determinant line already fixed in Section 3.

Higher integer source moments do not create new first-order low channels. In particular:

Proposition 33

(Schur suppression of higher integer moments). If $P{T}_{\ell }P=0$ and $Q{K}_{0}Q$ has gap $g_{\mathrm{CSch}}$, then for $\ell \ge 3$ the effective perturbation starts at second order,

$$\delta K_{\mathrm{eff}}^{(\ell )}=-{\epsilon }^{2}P{T}_{\ell }Q{\left(Q{K}_{0}Q\right)}^{-1}Q{T}_{\ell }P+O\left({\displaystyle \frac{{\left|\epsilon \right|}^3{\parallel T_{\ell }\parallel }^3}{g_{\mathrm{CSch}}^2}}\right).$$

(158)

For the retained $V_3$ line, Clebsch-Gordan multiplicity one gives

$$\dim {\mathrm{Hom}}_{SU\left(2\right)}\left(V_3,\mathrm{Hom}(C,R_{\mathsf{\Gamma },4})\right)=1,{\mathcal{T}}_3^{\mathrm{mic}}=a_3{T}_3^{\mathrm{cat}}.$$

(159)

The scalar $a_3$ remains a microscopic line normalization.

Proposition 34

(Locked central low-sector factorization). Under exact finite-shadow covariance, parity closure, and an isolated finite-core primitive projection, the low module factorizes into the primitive channel and the retained central family factor. A central perturbation below half the isolating gap preserves the product rank. Since the retained parity factor is one-dimensional and the central family factor has dimension three, every simple primitive channel has a rank-three low cluster.

For an isolated Riesz projector P, the projected connection is the standard Berry-Kato connection

$${\nabla }^{\mathrm{R}}=Pd,F^{\mathrm{R}}=P{\left(dP\right)}^{2}P.$$

(160)

Its Abelian geometric-phase form is due to [88]; the non-Abelian extension is [89].

For approximate rank-three spectral data, let $P,Q$ be the actual and reference Riesz projectors and put ${A=PQP|}_{\mathrm{Ran}P}$ and

$${\delta }_R:=\det A={\displaystyle \frac{{\left(\mathrm{Tr}A\right)}^3-3\mathrm{Tr}A\mathrm{Tr}{A}^2+2\mathrm{Tr}{A}^3}{6}}.$$

(161)

The condition ${\delta }_R>0$ is equivalent to $\parallel P-Q\parallel <1$, with ${\parallel P-Q\parallel }^2\le 1-{\delta }_R$, and gives the standard Kato-Sz.-Nagy unitary transport. With ${\mathcal{A}}_{PQ}:=P{U}_{PQ}^{*}d{U}_{PQ}P$,

$$U_{PQ}^{*}\left(Qd\right)U_{PQ}=Pd+{\mathcal{A}}_{PQ}.$$

(162)

For a marked unit vector $T\in \mathrm{Ran}P$, put $R=U_{PQ}T$, $V=QPT$, and $c_{*}=\langle R,V\rangle$. The scalar-relay leakage is

$${\mathcal{R}}_{\mathrm{sc}}={\parallel V\parallel }^2-{\left|c_{*}\right|}^2=0,g_{\mathrm{rel}}=c_{*}\in [\sqrt{{1-\parallel P-Q\parallel }^2},1].$$

(163)

Proposition 35

(Canonical centered-Riesz recovery). Let ${\mathcal{R}}_P\left(s_R\right)$ be a Parent response path in the retained positive ambient, parametrized by its Riesz arclength, with $L_R=\mathrm{span}\left\{{\nabla }_{s_R}{\mathcal{R}}_P\right\}$, and let $a_{\mathrm{cen}}\left(s_R\right)$ be the ${\mu }_K$-projection of this same response. Along a regular Parent chart Ω put $v_R:=d{s}_R/d\mathsf{\Omega }={\parallel {\nabla }_{\mathsf{\Omega }}{\mathcal{R}}_P\parallel }_{\mathrm{prot}}$. No identification of $v_R$ with the stationary-state speed $\parallel H^{-1}DQ\parallel$ is made without an additional isometric-speed incidence. The same-response provenance generates the marking-preserving line morphism ${\mathcal{T}}_{cR}:L_R\to L_{\mathrm{cen}}$; a post-hoc normalized line isometry is only a metric checksum. In local unit frames write ${\mathcal{T}}_{cR}{e}_R=\rho e^{i\theta }{e}_{\mathrm{cen}}$ on a connected nondegenerate segment with $\rho >0$. Then $\rho ={\kappa }_{\mathrm{cen}}|d{a}_{\mathrm{cen}}/d{s}_R|$ for ${\kappa }_{\mathrm{cen}}^2:={\parallel {\mu }_K\parallel }_{\mathrm{prot}}^2$, and Cauchy-Schwarz gives $\rho \le 1$ after the common protected normalization. In that normalized ambient $\rho =|\langle e_{\mathrm{cen}},e_R\rangle |=\cos \vartheta$ is the principal-angle cosine. Common ambient, differential recovery, and non-expansivity are therefore consequences of the same-response incidence rather than independent gates. Since the segment is connected and $\rho >0$, $d{a}_{\mathrm{cen}}/d{s}_R$ never vanishes and has fixed sign. With ${\mathsf{\Theta }}_{cR}:={\mathcal{T}}_{cR}^{-1}{\nabla }^{\mathrm{Hom}}{\mathcal{T}}_{cR}$, metric compatibility gives

$$\Re {\mathsf{\Theta }}_{cR}=d\log \rho .$$

(164)

If $\rho \left(s_{R,a}\right)=1$, parallelism gives $\rho \equiv 1$; more generally, $D\left(s_R\right):={\int }_{s_{R,a}}^{s_R}|\Re {\mathsf{\Theta }}_{cR}\left({\partial }_{\sigma }\right)|d\sigma$ gives $e^{-D\left(s_R\right)}\le \rho \left(s_R\right)/\rho \left(s_{R,a}\right)\le e^{D\left(s_R\right)}$. Choose the physical orientation ${\tilde{a}}_{\mathrm{cen}}$ so that it increases with u. With $c_K:=2/\left(3{\kappa }_{\mathrm{cen}}\right)$ and the centered normalization of (193),

$$u\left(\mathsf{\Omega }\right)-u\left({\mathsf{\Omega }}_a\right)=c_K{\int }_{s_R\left({\mathsf{\Omega }}_a\right)}^{s_R\left(\mathsf{\Omega }\right)}\rho \left(s_R\right)d{s}_R={\displaystyle \frac{2}{3}}[{\tilde{a}}_{\mathrm{cen}}\left(\mathsf{\Omega }\right)-{\tilde{a}}_{\mathrm{cen}}\left({\mathsf{\Omega }}_a\right)].$$

(165)

Thus the quadrature and the centered endpoint difference are two representations of the same transfer. For the structural anchor ${\mathsf{\Omega }}_{\mathrm{str}}$ with $u=0$ define

$$\mathcal{U}\left({\mathsf{\Phi }}_{\mathsf{\Omega }}\right):={\displaystyle \frac{2}{3}}[{\tilde{a}}_{\mathrm{cen}}\left(\mathsf{\Omega }\right)-{\tilde{a}}_{\mathrm{cen}}\left({\mathsf{\Omega }}_{\mathrm{str}}\right)],u\left(\mathsf{\Omega }\right)=\mathcal{U}\left({\mathsf{\Phi }}_{\mathsf{\Omega }}\right).$$

(166)

If the protected charge discriminant selects ${\mathsf{\Phi }}_{*}$, then

$$u_{\mathrm{Parent}}=u_{*}=\mathcal{U}\left({\mathsf{\Phi }}_{*}\right),{\chi }_Q\left({\mathsf{\Phi }}_{*}\right)=0.$$

(167)

The gain, Riesz length, and line turning remain mechanism/provenance diagnostics rather than additional data required to define $u_{*}$. Exact unit gain gives $u\left(\mathsf{\Omega }\right)-u\left({\mathsf{\Omega }}_a\right)=c_K[s_R\left(\mathsf{\Omega }\right)-s_R\left({\mathsf{\Omega }}_a\right)]$. In the ordinary Hilbert-Schmidt convention $\parallel {\mu }_K{\parallel }_{\mathrm{HS}}^2=15/2$ and $c_K=\sqrt{8/135}$; this coefficient becomes a protected recovery coefficient only after the common-metric incidence has passed.

Corollary 9

(Selector-coordinate prediction). On a certified seed-to-fold segment put $\alpha =du/d\mathsf{\Omega }=(2/3)d{\tilde{a}}_{\mathrm{cen}}/d\mathsf{\Omega }$ and $\mathcal{T}=d{\chi }_Q/d\mathsf{\Omega }$. For $\alpha ,\mathcal{T}>0$,

$${\displaystyle \frac{du}{d{\chi }_Q}}={\displaystyle \frac{\alpha }{\mathcal{T}}},u_{*}-u_0={\int }_{{\chi }_0}^0{\displaystyle \frac{\alpha \left(\chi \right)}{\mathcal{T}\left(\chi \right)}}d\chi .$$

(168)

If $0<{\alpha }_{-}\le \alpha \le {\alpha }_{+}$ and $0<m\le \mathcal{T}\le M$, then $({\alpha }_{-}/M)(-{\chi }_0)\le u_{*}-u_0\le ({\alpha }_{+}/m)(-{\chi }_0)$. With $\beta =d^{2}u/d{\mathsf{\Omega }}^2$, $\dot{\mathcal{T}}=d\mathcal{T}/d\mathsf{\Omega }$, $\left|\alpha \right|\le A_1$, $\left|\beta \right|\le A_2$, $|\dot{\mathcal{T}}|\le B_{\mathcal{T}}$, and $\mathcal{T}\ge m$,

$$H_{u\chi }:={\displaystyle \frac{A_2}{m^2}}+{\displaystyle \frac{A_1{B}_{\mathcal{T}}}{m^3}},{\widehat{u}}_{*}^{\left(1\right)}:=u_0-{\chi }_0{\displaystyle \frac{{\alpha }_0}{{\mathcal{T}}_0}},|u_{*}-{\widehat{u}}_{*}^{\left(1\right)}|\le {\displaystyle \frac{1}{2}}{H}_{u\chi }{\chi }_0^2.$$

(169)

On a terminal segment with $0<du/d{\chi }_Q\le g_{+}$,

$$|u-u_{*}|\le g_{+}|{\chi }_Q|,|\tilde{u}-u_{*}|\le {\epsilon }_u+g_{+}\left(\right|\tilde{\chi }|+{\epsilon }_{\chi }).$$

(170)

Here one may take $g_{+}={\alpha }_{+}/m$; the Feshbach error (109) is an admissible contribution to ${\epsilon }_{\chi }$. For mechanism audit, $V_R={\nabla }_{\mathsf{\Omega }}{\mathcal{R}}_P$ and $A_R={\nabla }_{\mathsf{\Omega }}{V}_R$ give ${\kappa }_R=\parallel P_{V_R^{\perp }}{A}_R\parallel /\parallel V_R{\parallel }^2$.

Let ${\mathcal{K}}_{\mathrm{fam}}$ be the family bundle, $S_{\mathrm{fam}}$ its transported regular cycle, and $K_0=K-(\mathrm{Tr}K/3)\mathbf{1}$ the traceless constrained-Jacobi block. With $z=\mathrm{Tr}\left(S_{\mathrm{fam}}^{†}{K}_0\right)$, $m=2\Re z/3$, and $q_E=-\Im z/(\sqrt{3}\Re z)$, the $C_3$-twirl is $K_c=mJ\left(q_E\right)$ and

$$N_{\mathrm{eff}}\left(q_E\right)={\displaystyle \frac{1+3q_E^2}{4}},\mathrm{Spec}J\left(q_E\right)=\left\{1,{\displaystyle \frac{3q_E-1}{2}},-{\displaystyle \frac{3q_E+1}{2}}\right\}.$$

(171)

Trace-one Eisenstein attachment gives $q_E\in 2\mathbb{Z}+1$; on the nonunit branch the first shell is $q_E=\pm 3$. With the protected normalization $\mathrm{Tr}{\widehat{J}}^2=42$,

$$C_{*}\left(q\right)={\displaystyle \frac{4}{\sqrt{7}}}{\displaystyle \frac{9q^2-1}{{(1+3q^2)}^{3/2}}},C_{*}\left(q\right)={\displaystyle \frac{40}{49}}⟺\left|q\right|=3\left(\right|q|>1),$$

(172)

and the family certificate is

$$[J_{\mathrm{fam}},S_{\mathrm{fam}}]=0,\mathrm{Tr}{J}_{\mathrm{fam}}=0,\mathrm{Tr}{J}_{\mathrm{fam}}^2=42,C_{*}\left(J_{\mathrm{fam}}\right)={\displaystyle \frac{40}{49}}.$$

(173)

It fixes the simple spectrum $\{1,4,-5\}$ up to the conjugate arithmetic orientation.

The shape layer has

$${\mathcal{R}}_{B0}^{\mathrm{mic}}:=B_0^{\mathrm{car}}-B_0^{\mathrm{fam}}=0,B_0^{\mathrm{car}}={\displaystyle \frac{cw}{{\left(T^{*}\right)}^2}},B_0^{\mathrm{fam}}={\displaystyle \frac{6}{N_{\mathrm{eff}}{\left(q_E\right)}^2}},$$

(174)

together with

$${\mathcal{R}}_{JS}:=C_R-C_{*}\left(J_{\mathrm{fam}}\right)=0,$$

(175)

and, after the centered same-response incidence,

$${\mathcal{R}}_{\mathrm{tr}}:={\displaystyle \frac{d{b}_{\mathrm{tr}}}{d{a}_{\mathrm{cen}}}}+{\displaystyle \frac{80}{147}}=0.$$

(176)

The first equality is a common-origin constraint only when the carrier and family observations factor through the same protected positive anchor line.

For the projector layer let $P_D^{\mathrm{dyn}}$ be the dark rank-one Riesz projector and $P_F$ the rank-three family projector. With ${\rho }_D={\mathrm{Tr}}_2{P}_D^{\mathrm{dyn}}$ and ${\rho }_F={\mathrm{Tr}}_{{\mathbb{C}}^3}{P}_F/3$,

$$P_D^{\mathrm{dyn}}=P_Q^{\otimes 2}⟺\mathrm{Tr}{\rho }_D^2=1,P_F=P_Q\otimes {\mathbf{1}}_3⟺\mathrm{Tr}{\rho }_F^2=1.$$

(177)

On the respective exact loci the reconstructed roots are $P_Q^D={\rho }_D$ and $P_Q^F={\rho }_F$, and common-root closure also requires $P_Q^D=P_Q^F$. More generally, for ${\mathcal{R}}_D:=1-\mathrm{Tr}{\rho }_D^2$ and ${\mathcal{R}}_F:=1-\mathrm{Tr}{\rho }_F^2$, a residual $\mathcal{R}<1/2$ gives a unique nearest rank-one root projector with maximal normalized overlap $(1+\sqrt{1-2\mathcal{R}})/2$ with the corresponding factorized projector. On a smooth common-root germ parametrized by any regular branch coordinate t,

$${\mathrm{Tr}}_2{\displaystyle \frac{d^r{P}_D^{\mathrm{dyn}}}{d{t}^r}}={\displaystyle \frac{1}{3}}{\mathrm{Tr}}_{{\mathbb{C}}^3}{\displaystyle \frac{d^r{P}_F}{d{t}^r}}={\displaystyle \frac{d^r{P}_Q}{d{t}^r}},r\ge 0,$$

(178)

and for an ordered finite loop,

$$\mathrm{Tr}\left({\displaystyle \prod _{j=1}^N}{P}_D^{\mathrm{dyn}}\left(t_j\right)\right)={\left[{\displaystyle \frac{1}{3}}\mathrm{Tr}\left({\displaystyle \prod _{j=1}^N}{P}_F\left(t_j\right)\right)\right]}^2.$$

(179)

The finite-loop identity is phase-free and does not replace connection provenance. Since $L_D^{\mathrm{dyn}}\simeq Q_W^{\otimes 2}$ and $\det {\mathcal{K}}_{\mathrm{fam}}\simeq Q_W^{\otimes 3}$ on this locus,

$$Q_{\mathrm{Bez}}:=\left(\det {\mathcal{K}}_{\mathrm{fam}}\right)\otimes {\left(L_D^{\mathrm{dyn}}\right)}^{-1}\simeq Q_W.$$

(180)

This is a dependent line-bundle reconstruction after common-root provenance, not an additional root selector.

The differential and two-point checks

$$3g_D=2g_F^{\mathrm{Tr}},3{\mathsf{\Omega }}_D=2{\mathsf{\Omega }}_F^{\mathrm{Tr}},\mathrm{Tr}\left(P_D^{\mathrm{dyn}}\left(s\right)P_D^{\mathrm{dyn}}\left(t\right)\right)={\left[{\displaystyle \frac{1}{3}}\mathrm{Tr}\left(P_F\left(s\right)P_F\left(t\right)\right)\right]}^2,$$

(181)

are independent holdouts before smooth common-root provenance is established and dependent consequences of (178)-(179) afterwards. The phase-sensitive connection test is

$${\mathfrak{R}}_{\mathrm{conn}}:=\left({\nabla }_F{J}_{\mathrm{fam}},{\mathcal{F}}_F-{\displaystyle \frac{1}{2}}{\mathsf{\Omega }}_D{\mathbf{1}}_3\right).$$

(182)

After the marked dark-port incidence, ${\mathfrak{R}}_{\mathrm{conn}}=0$ identifies the induced physical root connection with the structural $Q_W$. For the induced root line, an alternative segmentwise certificate is one seed connection identification plus equality of mixed curvatures on a certified $I\times S^2$; the identification then propagates because $H^1(I\times S^2;\mathbb{R})=0$, while the weight-one line remains an independent holdout. The open-neighborhood degree checksum is

$${\displaystyle \sum _{r=0}^2}{\parallel F_r-F_{\mathsf{\Gamma }}\parallel }_{L^2}^2<{\displaystyle \frac{8{\pi }^2}{A_{\mathsf{\Gamma }}}}⟹(d_0,d_1,d_2)=(1,1,1).$$

(183)

The determinant-family interpretation is represented by [13].

The Callias index is used in its standard form [9,10]. For asymptotic mass ${\mathsf{\Phi }}_{\infty }=v(T_3+\nu \mathbf{1})$, exact target degree q forces $\left|k\right|=1$, and a common scalar shift for distinct integer target dimensions exists only for adjacent dimensions. Together with protected total rank five this fixes $\left\{q\right\}=\{3,2\}$, common chamber $-1<\nu <-1/2$, and maximal asymptotic gap $v/4$ at $\nu =-3/4$. With common self-adjoint domains and adjoint coercivity, the index gives

$$K_{\mathrm{bulk}}:=\ker {\mathcal{D}}_3^{\mathrm{P}}\oplus \ker {\mathcal{D}}_2^{\mathrm{P}},\dim \ker {\mathcal{D}}_3^{\mathrm{P}}=3,\dim \ker {\mathcal{D}}_2^{\mathrm{P}}=2.$$

(184)

If $K_{\mathrm{bulk}}$ and $C\oplus W$ lie in one marked Hilbert space with projectors $P_K,P_E$ and $s=\parallel P_K-P_E\parallel <1$, the canonical incidence is

$$U_{K\to E}:=P_E{P}_K{\left(P_K{P}_E{P}_K\right)}^{-1/2}:\mathrm{Ran}{P}_K\stackrel{\sim }{\to }\mathrm{Ran}{P}_E,$$

(185)

with singular margin at least $\sqrt{1-s^2}$. Common $SU\left(2\right)$ and Real markings then transport the exterior/CAR structure. The family rank-three Callias block remains separately typed: identifying its kernel with ${\mathcal{H}}_{\mathrm{cen}}$ is a separate marked incidence, and its degree-three class does not distinguish electric weight three on primitive flux from unit weight on magnetic flux three. A sufficient comparison-gap condition is

$$\parallel {\mathcal{A}}_{PQ}{\parallel }_{L^{\infty }}+{\parallel V_{\mathrm{act}}-V_{\mathrm{ref}}\parallel }_{L^{\infty }}<{\delta }_{\mathrm{ref}}.$$

(186)

The terminal relay has three typed attachments: the integral incidence (87), positive response (94), and common-ambient scalarity (163). For $n=\left|p\right|$,

$${\mathcal{R}}_{\mathrm{term}}^{\left(n\right)}={∥r_{A}V-{\displaystyle \frac{2}{n}}R∥}^2=r_A^2{\mathcal{R}}_{\mathrm{sc}}+{\left(r_A{c}_{*}-{\displaystyle \frac{2}{n}}\right)}^2,$$

(187)

and

$$|q_{23}|={\displaystyle \frac{r_{A}n{g}_{\mathrm{rel}}}{2\sqrt{2}}},{\widehat{\mathsf{\Xi }}}_{23}={\displaystyle \frac{\sqrt{3}{r}_{A}n{g}_{\mathrm{rel}}}{4\sqrt{2}}},{\mathcal{R}}_{\mathrm{term}}^{\left(n\right)}=0⟺\left|q_{23}\right|={\displaystyle \frac{1}{\sqrt{2}}}.$$

(188)

Winding, positive gain, and common-ambient overlap remain recovered data rather than symmetry outputs.

Theorem 11

(PLSR elementary attachment and terminal closure). Assume the integral incidence, finite-parent regularity, and marked common ambient of this section. If K1-K3 recover their corresponding protected data by exact PLSR, then $\left|p\right|=2$, selecting the two-torsion cokernel in (87), while $r_A=1$, ${\mathcal{R}}_{\mathrm{sc}}=0$, and $g_{\mathrm{rel}}=1$. Consequently (187) vanishes and (188) gives $|q_{23}|=1/\sqrt{2}$ and ${\widehat{\mathsf{\Xi }}}_{23}^2=3/8$. On the scalar-relay branch, $\left||q_{23}|-1/\sqrt{2}\right|\le \sqrt{{\mathcal{R}}_{\mathrm{term}}^{\left(2\right)}/2}$.

Corollary 10

(Terminal-Hodge incidence). For the marked source-Hodge folding, let ${\delta }_q$ be the normalized common-to-relative third-jet defect. The source geometry gives $q_{*}=4+{\kappa }_g^2$, ${\delta }_q=9{\kappa }_g^2+{\dot{\kappa }}_g^2$, and $q_{*}{r}_{\mathrm{src}}^2=8/3$. On the exact terminal branch define ${\mathcal{R}}_{\mathrm{TH}}:={\chi }_H^2-{\widehat{\mathsf{\Xi }}}_{23}^2$. Using Theorem 5 and Theorem 11,

$${\mathcal{R}}_{\mathrm{TH}}={\displaystyle \frac{3}{8}}{\kappa }_g^2,q_{*}-4={\displaystyle \frac{8}{3}}{\mathcal{R}}_{\mathrm{TH}},r_{\mathrm{src}}^2={\displaystyle \frac{2}{3+2{\mathcal{R}}_{\mathrm{TH}}}},{\delta }_q=24{\mathcal{R}}_{\mathrm{TH}}+{\dot{\kappa }}_g^2.$$

(189)

Thus ${\mathcal{R}}_{\mathrm{TH}}=0$ is the single terminal-Hodge incidence that fixes the geodesic source point and $q_{*}=4$; ${\delta }_q=0$ remains an independent pointwise covariant-prolongation test. Along one smooth branch on which ${\mathcal{R}}_{\mathrm{TH}}$ vanishes identically, the same relation forces the prolongation defect to vanish identically. Terminal closure alone does not imply this incidence.

4.4. Realization Criterion and Retained Low Channels

The realization certificate is ordered structural-before-singular. The Parent first realizes the one-sided Codazzi geometry of Proposition 11, the current and seed-jet interfaces, the primitive-port/Source-Hodge normalization, and the Callias carrier incidences. On the resulting regular branch the single selector is (99). Theorem 7 transfers the finite retained reading to the full Fredholm/KKT problem with an explicit tail gate, while Theorem 8 and Corollary 6 certify the unique protected zero on the branch covered from the physical stable anchor. The VH zero (98), KKT pole (101), dual chart (130), and cubic coefficient (100) remain dependent readings after their typed incidences.

The selected fold exports two independent kinds of information. The same-response centered observable gives the reduced coordinate through (167); the gain, Riesz length, and turning remain provenance diagnostics. Independently, Theorem 9 reconstructs the desingularized critical residue from stable-side response boxes, and Corollary 8 converts probe-complete residue components into typed tangents within their declared finite target spaces. Pair consistency, projective transport, and the causal residual are holdouts of the common-germ and stronger mediated interpretations rather than additional selection conditions.

Physical spectral states require separately that the microscopic/boundary response realize the equivariant passive channel structure of Proposition 28; widths additionally require the same primitive arrow of Proposition 32. Family shape, common-root, orientation, and connection provenance remain independently typed. Once the common-root germ has passed, (178) and (179) organize its descendants, while the Bézout reconstruction (180) reduces the weight-$2,3,5$ connection forcing rank without replacing the marked connection/holonomy test.

After the carrier incidence, exterior calculus is functorial. The gauge-Dirac attachment supplies the block-preserving connection and weak-direct factorization (157); microscopic Real/order-one compatibility and identification of $\left[h\right]$ with the reduced scalar line remain separate. The family characteristic assignment is (173), the reduced trace coordinate additionally requires (135), and the operator (157) with its common Riesz gap and Proposition 34 supplies the low/high splitting used below. Numerical promotion requires certified continuum or tail enclosures propagated through these typed interfaces.

5. Exact Schur Transfer and Low-Energy Tests

The structural packet is fixed in Section 3, and Section 4 supplies its Parent, selector, spectral, and transport certificates. Schur-Feshbach reduction is applied only on this passed image. The reduced outputs form one map $Y:(u,{\sigma }_7^{\mathrm{ar}})\mapsto Y(u,{\sigma }_7^{\mathrm{ar}})$; affine, flavor, and mass relations are coordinate readings of this map, while normalization, poles, and running are completion fibers. After mandatory gates leave one Parent branch, the physical stability selector is the unique charge-discriminant zero on the certified protected path from the physical stable anchor, obtained by Theorem 8 and segment continuation when needed; (98) and (130) are its VH and intrinsic scalar readings. Its transfer to u is the centered endpoint pushforward (167); Proposition 35 supplies the same-response provenance and equivalent Riesz decomposition. Spectral-energy equations are then solved on this selected background and do not add Parent-branch codimension. The exterior incidence is fixed by (33), and the $V_3$ low-high direction by (159).

Lemma 4

(Exact Schur-homological transfer). For an invertible complementary Hessian block,

$$H_i^{\mathrm{red}}=H_{xx}-H_{xh}{H}_{hh}^{-1}{H}_{hx},$$

(190)

and for an isolated splitting $P+Q=\mathbf{1}$,

$${\mathfrak{F}}_P\left(K\right)=PKP-PKQ{\left(QKQ\right)}^{-1}QKP.$$

(191)

The Neumann/homological-perturbation expansion of ${\left(QKQ\right)}^{-1}$ is the standard expansion of this same Schur complement [90]. For the marked zero-diagonal Eisenstein companion splitting its scalar reading is $N_E\left(\xi \right)/\mathrm{Tr}\xi$; an orthonormal real presentation gives twice this value, so the splitting remains part of the protected datum.

Lemma 5

(Filtered-graph Schur calculus). Let $H\left(t\right)=D+V\left(t\right)$ be a finite Hermitian matrix with invertible diagonal zeroth-order block D, and let the nonzero off-diagonal entries of V define a filtered graph with edge valuations ${\nu }_e>0$. The Neumann expansion of $H{\left(t\right)}^{-1}$ is a signed sum over walks. The valuation of an $(i,j)$ entry is at least the minimum total valuation of walks connecting i to j; the coefficient at that order is the sum over all minimizing walks, and equality holds when this sum does not cancel. After vertex rephasings, independent edge phases are classified by the cycle space of the underlying graph, with $b_1=\left|E\right|-\left|V\right|+c$ for c connected components.

Proposition 36

(Gapped Schur stability). Let $K=\left(\begin{array}{cc}A& B\\ B^{†}& D\end{array}\right)$, $S=A-B{D}^{-1}{B}^{†}$, and assume ${\sigma }_{\min }\left(D\right)\ge g>0$. Put $\tilde{S}=(A+\delta A)-(B+\delta B){(D+\delta D)}^{-1}{(B+\delta B)}^{†}$, and let all norms be operator norms. If $\parallel \delta A\parallel ,\parallel \delta B\parallel ,\parallel \delta D\parallel \le \epsilon <g$ and $M=\parallel B\parallel$, then

$$\parallel \tilde{S}-S\parallel \le \epsilon \left[1+{\displaystyle \frac{M+\epsilon }{g-\epsilon }}+{\displaystyle \frac{M(M+\epsilon )}{g(g-\epsilon )}}+{\displaystyle \frac{M}{g}}\right].$$

(192)

Consequently an independently established operator-readout bound $\epsilon \le c\sqrt{{\delta }_{\mathrm{fid}}}$ gives $O\left(\sqrt{{\delta }_{\mathrm{fid}}}\right)$ stability of the reduced operator while the heavy gap remains open.

For the scalar-denominator transfer $F_g\left(A\right):=A{(\mathbf{1}+gA)}^{-1}$, rational functional calculus gives $F_g\left(F_h\left(A\right)\right)=F_{g+h}\left(A\right)$ and ${\partial }_g{F}_g\left(A\right)=-F_g{\left(A\right)}^2$. A nonzero self-adjoint A is therefore not a fixed point for $g\ne 0$; the Schur parameter is treated as a Riccati-Möbius flow coordinate, separately from the isometric recovery maps of PLSR.

The active inverse-spectral and Hankel closure has already been established in Proposition 29; the present section uses only its reconstructed reduced data and no-refit holdouts.

5.1. Reduced Low Operator and Common Affine Cell

When the gauge-Dirac attachment is realized, the fermionic low block is the Riesz/Schur compression of $D_F^{\mathrm{P}}$ in (157) on the isolated module of Proposition 34; its four weak-direct family coefficients are the $Y_f$. The common affine cell below is the independent reduced Hessian/trace response of the finite Parent certificate. Identifying the weak-direct line $\left[h\right]$ with its scalar line, or the family-shape operators $H_f$ with a fixed function of $Y_f^{†}{Y}_f$, requires separate microscopic Schur incidences. The Berry/Riesz connection (160) remains a spectral attachment and is identified with the bulk block-gauge connection only when the corresponding connection incidence passes. Differential-index transgression is standard [14]; a real or parallel trivialization of the generated determinant line remains an additional global condition.

The neutral trace reservoir has angular support $C\oplus W_{\sigma }\oplus W_{-\sigma }$ and radial support $C\oplus W_{\sigma }$. The canonical theta/Borel-Weil intertwiners are one-dimensional in each marked block, so PLSR descent fixes ${\mathfrak{I}}_{\mathrm{neu}}=(P_C^{\angle },P_C^{\angle }\otimes P_{W_{\sigma }}^{\angle },P_C^{\angle }\otimes P_C^r)$. With $(c,w)=(3,2)$ and $T^{*}=c+2w=7$, the normalized traces are $A_0=3/7$, $B_0=6/49$, and $C_0=9/35$. The constitutive packet $\mathrm{rank}{P}_3=3$, $\mathrm{mult}(4/7)=2$, $\mathrm{rank}{P}_5=5$, with spectral denominator seven, gives the same weights as an independent spectral checksum. The centered tangent fixed by (25) gives

$$A=A_0-{\displaystyle \frac{3}{2}}u,B=B_0,C=C_0+u.$$

(193)

At an unspecified positive scale $v_{\mathrm{EW}}^{\mathrm{prim}}$, the tree-level reading is $m_W^2={\left(v_{\mathrm{EW}}^{\mathrm{prim}}\right)}^{2}A/4$, $m_Z^2={\left(v_{\mathrm{EW}}^{\mathrm{prim}}\right)}^2(A+B)/4$, and $m_h^2={\left(v_{\mathrm{EW}}^{\mathrm{prim}}\right)}^{2}C$. Eliminating the common scale gives

$$15{\left(m_h^{\mathrm{prim}}\right)}^2=266{\left(m_Z^{\mathrm{prim}}\right)}^2-306{\left(m_W^{\mathrm{prim}}\right)}^2.$$

(194)

These are reduced/primitive Hessian readings. For a self-adjoint $C^2$ completion channel, let $\lambda \left(s\right)$ be an isolated simple real eigenbranch on $0\le s\le M$ with $\lambda \left(M\right)=0$ and $\lambda \left(0\right)>0$. The exact back-transport coefficient and pole slope are

$$\overline{z}:={\displaystyle \frac{\lambda \left(0\right)}{M}}={\displaystyle \frac{1}{M}}{\int }_0^M[-{\lambda }^{\prime }\left(s\right)]ds,z^{\mathrm{pole}}:=-{\lambda }^{\prime }\left(M\right).$$

(195)

Let $P=|v\rangle \langle v|$, $Q=\mathbf{1}-P$, $H=Q(F-\lambda \mathbf{1})Q$, $q=Q{F}^{\prime }v$, and assume $g\mathbf{1}⪯H⪯G\mathbf{1}$ and $\parallel F^{″}\left(s\right)\parallel \le {\kappa }_2$ along the tracked branch. With $g_R^{\left(s\right)}=\mathrm{Tr}\left[{\left(P^{\prime }\right)}^2\right]/2$ and ${\mathcal{E}}_R^{\mathrm{pole}}=2M^{-1}{\int }_0^{M}s{g}_R^{\left(s\right)}ds$, standard eigenbranch perturbation gives

$$g{\mathcal{E}}_R^{\mathrm{pole}}-{\displaystyle \frac{M{\kappa }_2}{2}}\le z^{\mathrm{pole}}-\overline{z}\le G{\mathcal{E}}_R^{\mathrm{pole}}+{\displaystyle \frac{M{\kappa }_2}{2}}.$$

(196)

The affine matrix pencil is the case ${\kappa }_2=0$; sampled Riesz projectors may be used to bound the mixing term without differentiating P. Thus direct operator curvature and Riesz-line rotation are separated before any pole value is compared with the reduced target.

The neutral cell (193) is the centered response. For the reduced flavor chart, the structural anchor $B_{\mathrm{fl}}\left(0\right)=B_0$ and the family/scalar characteristic lock (175), with the reduced coordinate supplied by the centered endpoint functional (166), give

$$B_{\mathrm{fl}}\left(u\right)=B_0+{\displaystyle \frac{40}{49}}u,{\lambda }_{\mathrm{cen}}\left(u\right):={\displaystyle \frac{B_{\mathrm{fl}}\left(u\right)}{A\left(u\right)+B_{\mathrm{fl}}\left(u\right)}}.$$

(197)

Thus the coefficient $40/49$ is the primitive characteristic value of the actual family block after the scalar normalization incidence, while (176) is the equivalent reduced slope checksum. With the scalar lift omitted, the centered-only readout is $B_0/(A+B_0)=1-r_{WZ}$; the difference isolates the remaining flavor-provenance datum. The standard electroweak convention is used only for this matching reading [91]; the analogous spectral-action relation provides a comparison [92]. For an independently selected $u_{*}$, set $a_W=4\sqrt{2}{G}_F{m}_W^2$, $a_Z=4\sqrt{2}{G}_F{m}_Z^2$, and $a_h=G_F{m}_h^2/\sqrt{2}$. The required back-transport means are

$${\overline{z}}_W^{\mathrm{req}}={\displaystyle \frac{3/7-3u_{*}/2}{a_W}},{\overline{z}}_Z^{\mathrm{req}}={\displaystyle \frac{27/49-3u_{*}/2}{a_Z}},{\overline{z}}_h^{\mathrm{req}}={\displaystyle \frac{9/70+u_{*}/2}{a_h}}.$$

(198)

If $|u_{*}-\widehat{u}|\le E_u$, these define target intervals $T_i^{\mathrm{req}}=[{\overline{z}}_i^{\mathrm{req}}\left(\widehat{u}\right)-s_i{E}_u,{\overline{z}}_i^{\mathrm{req}}\left(\widehat{u}\right)+s_i{E}_u]$, with $s_W=3/\left(2a_W\right)$, $s_Z=3/\left(2a_Z\right)$, and $s_h=1/\left(2a_h\right)$. If an independent pole solver gives $z_i^{\mathrm{pole}}\in [p_i^{-},p_i^{+}]$ and (196) gives $z_i^{\mathrm{pole}}-{\overline{z}}_i\in [L_i,U_i]$, then

$$B_i^{\mathrm{pole}}:=[p_i^{-}-U_i,p_i^{+}-L_i],B_i^{\mathrm{pole}}\cap T_i^{\mathrm{req}}\ne ⌀$$

(199)

is necessary for exact same-defect pole closure. The signed separation ${\mathcal{X}}_i=\max \{T_{i,-}^{\mathrm{req}}-B_{i,+}^{\mathrm{pole}},B_{i,-}^{\mathrm{pole}}-T_{i,+}^{\mathrm{req}}\}$ is a direct exclusion score: ${\mathcal{X}}_i>0$ falsifies the completion, whereas ${\mathcal{X}}_i\le 0$ is only non-exclusion.

Put $r_{WZ}:=m_W^2/m_Z^2$ and ${\eta }_h:={\left(m_h^{\mathrm{prim}}\right)}^2/\left(4m_Z^2\right)$. On $0\le u<2/7$, $r_{WZ}$ is strictly decreasing, while ${\eta }_h$ and the trace-completed ${\lambda }_{\mathrm{cen}}\left(u\right)=4(20u+3)/(54-67u)$ are strictly increasing. After (197) they are global charts of the same reduced one-u packet, with

$$u={\displaystyle \frac{2(7-9r_{WZ})}{49(1-r_{WZ})}}={\displaystyle \frac{6(9{\lambda }_{\mathrm{cen}}-2)}{67{\lambda }_{\mathrm{cen}}+80}}={\displaystyle \frac{18(15{\eta }_h-7)}{245(3{\eta }_h+2)}}.$$

(200)

Eliminating u gives

$${\lambda }_{\mathrm{cen}}={\displaystyle \frac{427-507r_{WZ}}{427-360r_{WZ}}},{\displaystyle \frac{{\left(m_h^{\mathrm{prim}}\right)}^2}{{\left(m_Z^{\mathrm{prim}}\right)}^2}}={\displaystyle \frac{14(831{\lambda }_{\mathrm{cen}}+100)}{15(169-120{\lambda }_{\mathrm{cen}})}}.$$

(201)

The structural origin $u=0$ is the unique interior coincidence of the central Rees and neutral determinant charts; it gives $r_{WZ}=7/9$, ${\lambda }_{\mathrm{cen}}=2/9$, and ${\left(m_h^{\mathrm{prim}}\right)}^2/m_Z^2=28/15$. It is used as the distinguished correspondence anchor, while the physical point may lie at nonzero u. The algebraic reduced maps are defined on ${\mathcal{M}}_{\mathrm{alg}}=[0,2/7)$, whereas preservation of the labels $m_1<m_2<m_3$ in the neutral mass-shape formulas requires $A\left(u\right)>B_0$ and hence ${\mathcal{M}}_{\mathrm{NO}}=[0,10/49)$.

For the restricted normal-ordered neutral mass-shape branch put $x_{\nu }=\sqrt{m_2/m_1}$ and $y_{\nu }=\sqrt{m_3/m_1}$. Define $G_1:={\lambda }_{\mathrm{cen}}(427-360r_{WZ})-(427-507r_{WZ})$ and $G_2:=30{\eta }_h-133+153r_{WZ}$. An independent relation basis is completed by

$$G_3=(1-r_{WZ})x_{\nu }-r_{WZ}=0,G_4=6(1-r_{WZ})y_{\nu }-(273-337r_{WZ})=0.$$

(202)

The Jacobian of $(G_1,G_2,G_3,G_4)$ with respect to $(r_{WZ},{\lambda }_{\mathrm{cen}},{\eta }_h,x_{\nu },y_{\nu })$ has rank four on the interior branch. Thus this five-observable packet is one-dimensional with four independent local relations; further eliminations, including (224), do not increase its local codimension. Microscopic realization requires the characteristic lock (175) and the centered endpoint recovery (167); when first loss is the active selector, Proposition 22 supplies its stability and transversality tests.

5.2. Fixed-Q Family Shells and Flavor Transfer

By (35) and $Q=T_3+Y$, the weak factor splits as $W=W_0\oplus W_{+}$. The ordered direct-sector projectors have surviving ranks $(5,4,3)$; with normalized trace ${\rho }_k=(5-k)/5$, direct Hodge gives

$$Q_f={\displaystyle \frac{10-k_f}{3(5-k_f)}},$$

(203)

for $k_f=0,1,2$, hence $Q_{\ell }=2/3$, $Q_d=3/4$, and $Q_u=8/9$. For normalized amplitudes $r_{f,i}=\sqrt{m_{f,i}}/{\sum }_j\sqrt{m_{f,j}}$, all three sectors lie on the single fixed-Q surface ${\mathcal{S}}_Q:=\{r_i>0:{\sum }_i{r}_i=1,{\sum }_i{r}_i^2=Q\}$. The cubic, Cartan-angle, determinant, and conic formulas below are charts of this surface. The same $(1,4,5)$ ranks occur in the source sequence (71) as an incidence checksum; the canonical Veronese-Hodge packet is fixed by Corollary 4, while physical equality with the constitutive projector remains the Parent test (84).

Proposition 37

(Fixed-Q$A_2$ shell, determinant completion, and Rees flow). Let $r_i$ be the eigenvalues of $H_f/\mathrm{Tr}{H}_f$. For $1/2<Q_f<1$, the constraints ${\sum }_i{r}_i=1$, ${\sum }_i{r}_i^2=Q_f$ define the intersection of the affine $A_2$ plane with a circle. Up to the $D_3$ action it is parametrized by

$$r_k={\displaystyle \frac{1}{3}}+A_Q\cos \left(\theta +{\displaystyle \frac{2\pi k}{3}}\right),A_Q=\sqrt{{\displaystyle \frac{2}{3}}\left(Q_f-{\displaystyle \frac{1}{3}}\right)},k=0,1,2.$$

(204)

Positivity and ordering select the physical arc. The symmetric determinant ${\pi }_f=r_1{r}_2{r}_3$ reconstructs the unordered amplitudes through

$$z^3-z^2+{\displaystyle \frac{1-Q_f}{2}}z-{\pi }_f=0,$$

(205)

with three distinct positive roots for $0<{\pi }_f<{\pi }_{+}\left(Q_f\right)$, ${\pi }_{+}\left(Q\right)=[5-9Q+\sqrt{2}{(3Q-1)}^{3/2}]/54$. Thus ${\pi }_f$, $\mathrm{Tr}{X}_f^3$, the Cartan angle, and the ordered conic

$$(1-Q_f)(1+a^2+b^2)=2Q_f(a+b+ab),$$

(206)

are charts of the same one-dimensional orbit, where $a=\sqrt{m_1/m_3}$ and $b=\sqrt{m_2/m_3}$. Equal-amplitude collisions are the labeled chart singularities, while ${\pi }_f=0$ is the distinct rank-loss boundary. Projecting $H_R=(3/2)H_F-(1/2)K_F$ to the shell tangent gives

$${\displaystyle \frac{d\theta }{dt}}={\displaystyle \frac{3\cos \theta -\sin \theta /\sqrt{3}}{\sqrt{2(Q_f-1/3)}}}.$$

(207)

A linear $S_3$-equivariant endomorphism preserving the affine plane and $Q_f$ restricts to $\pm \mathbf{1}$ on the centered plane; nontrivial shell motion therefore requires a nonlinear Parent response or a separately retained sector gain. A universal identical raw gain is a Parent test rather than a consequence of the Rees grading.

Proposition 38

(Determinant-fiber degree and jet hierarchy). At fixed $I_2=\mathrm{Tr}{X}_f^2$, Cayley-Hamilton reduces every conjugation-invariant polynomial Parent selector to a polynomial in $I_3=\mathrm{Tr}{X}_f^3$. Degree $d\le 5$ gives no generic strict interior well, $6\le d\le 8$ at most one quadratic well, $d\ge 9$ first permits an interior barrier, and $d\ge 12$ first permits two strict interior minima; in general M strict interior minima require $d\ge 6M$. Through degree seven the fiber curvature is common after the locations are fixed; $I_2{I}_3^2$ at degree eight first releases this stiffness and degree nine first permits a cubic fiber jet. Writing the same dependence as $h\left(Q_f\right)I_3$ does not introduce a separately typed sector marker.

For the Hermitian source reading $X_{\mathrm{src}}=B-i{\left[a\right]}_{\times }$, one has $I_{2,\mathrm{src}}=\mathrm{Tr}{B}^2+2{\left|a\right|}^2$ and $I_{3,\mathrm{src}}=\mathrm{Tr}{B}^3-3a^{T}Ba$. Under $(a,B)\mapsto (\alpha a,\beta B)$, a universal normalized-cubic bridge requires $\left|\alpha \right|=\left|\beta \right|$; otherwise source composition remains visible. This is a Parent-transfer compression test, not a family prediction.

A conditional family-parity Schur candidate supplies the quadratic attenuation $1/2$. Used as the relative magnitude in the minimal noncommuting family kernel below, it places positive isolated roots of all three Direct-Hodge shells in one $C_3\times$conjugation orbit at fixed ${\lambda }_{\mathrm{cen}}$. Its operator identification with the microscopic cross response remains to be derived, and blind mass-ratio/direct-mixing holdouts exclude its identification with the complete charged spectra or CKM source. The coefficient-side candidate $z_F=(5+4\omega )/\sqrt{21}$ satisfies $z_F{j}_7=i$ and $z_F^6=\overline{e^{i{\delta }_7}}$, with Cayley ratio $|1+z_F|/|1-z_F|=\sqrt{(7+\sqrt{21})/(7-\sqrt{21})}$; these identities do not fix the microscopic cross phase.

Proposition 39

(Scalar finite quark endpoint). Let ${\widehat{H}}_f=H_f/\mathrm{Tr}{H}_f$, $f=u,d$, have simple ordered eigenvalues $r_{f,i}$ fixed by (205). Put $P_i^f$ for the spectral projectors, $B_{ij}=\mathrm{Tr}\left(P_i^u{P}_j^d\right)$, $M_{ab}=\mathrm{Tr}\left({\widehat{H}}_u^a{\widehat{H}}_d^b\right)$, $M={\left(M_{ab}\right)}_{a,b=0}^2$, and $V_f={\left(r_{f,j}^a\right)}_{a=0,1,2;j=1,2,3}$. The entries $M_{a0}$ and $M_{0b}$ are fixed by the two spectra. Then

$$M=V_{u}B{V}_d^{\mathsf{T}},B=V_u^{-1}M{V}_d^{-\mathsf{T}}.$$

(208)

so the four new scalars $M_{11},M_{12},M_{21},M_{22}$ reconstruct the finite overlap matrix B; for ordered eigenframes $U_f$ and $V=U_u^{†}{U}_d$, one has $B_{ij}={\left|V_{ij}\right|}^2$. The fixed one-sector moments $M_{a0}$ and $M_{0b}$ enforce unit row and column sums. Their Jacobian on the doubly-stochastic interior has determinant ${\Delta }_u^2{\Delta }_d^2\ne 0$, with ${\Delta }_f={\prod }_{i<j}(r_{f,i}-r_{f,j})$, hence four continuous mixed scalars are generically locally minimal. The reconstructed doubly-stochastic matrix B admits a unitary realization exactly when $B_{ij}\ge 0$ and

$$\mathcal{U}\left(B\right):=4B_{11}{B}_{12}{B}_{21}{B}_{22}-{\left(B_{13}{B}_{23}-B_{11}{B}_{21}-B_{12}{B}_{22}\right)}^2\ge 0,\left|J_q\right|={\displaystyle \frac{1}{2}}\sqrt{\mathcal{U}\left(B\right)}.$$

(209)

For $\mathcal{U}>0$ only the two conjugate orientations remain; at $\mathcal{U}=0$ the CP sign is not an independent datum.

The finite magnitude/rephasing endpoint therefore requires no eigenvector-phase export; for $\mathcal{U}>0$ the remaining conjugation choice is the global orientation bit. The same overlap can be reconstructed rowwise from any two real response functions $f,g$ whose node-value vectors are independent modulo the singlet: normalization and the two feature moments determine the three weights, while the inverse fails exactly when the centered feature area collapses. Probability-simplex rank loss is a separate degeneracy. Polynomial moments and known broadened kernels are equivalent charts of this row measure; unknown family-dependent kernels are not. Row data give only the capacity bound $4J_q^2\le {\prod }_j{B}_{ij}$, whereas the four mixed moments of Proposition 39 remain the minimal static doubly-stochastic reconstruction. A local Sylvester or Hodge-Eisenstein realization is a stronger dynamical incidence and still requires the marked operator path.

Let $X=Z$ and $J=\omega Z^{2}S$ be the selected grade-one and grade-two Weyl directions. Exact positive-scaling equivariance and Weyl-central denominator give $A\left(t\right)=txX+t^{2}yJ/2$, $G=g\mathbf{1}$, and Lemma 4 resums the transfer as

$$\mathcal{S}\left(t\right)={\displaystyle \frac{A\left(t\right)-gA{\left(t\right)}^2+g^2\left(a{\left(t\right)}^3+b{\left(t\right)}^3\right)\mathbf{1}}{1+g^3\left(a{\left(t\right)}^3+b{\left(t\right)}^3\right)}},$$

(210)

where $a\left(t\right)=tx$, $b\left(t\right)=t^{2}y/2$ and $A{\left(t\right)}^3=(a{\left(t\right)}^3+b{\left(t\right)}^3)\mathbf{1}$. Hence the exact transfer lies in ${\mathcal{A}}_{\mathrm{allow}}=\mathrm{span}\{\mathbf{1},S,Z,Z{S}^2,Z^2,Z^{2}S\}$ and excludes cyclically $S^2$, $ZS$, $Z^2{S}^2$. If

$${\displaystyle \frac{1}{1+g^3{x}^3{t}^3+g^3{y}^3{t}^6/8}}=\sum _{k\ge 0}{d}_k{t}^{3k},d_{-1}=0,d_0=1,$$

(211)

then $d_k=-g^3{x}^3{d}_{k-1}-g^3{y}^3{d}_{k-2}/8$, so all allowed Taylor coefficients obey one second-order recurrence. In the marked clock-shift frame the forbidden coefficients are $F_{S^2}=Y_{31}+Y_{12}+Y_{23}$, $F_{ZS}={\omega }^2{Y}_{21}+\omega Y_{32}+Y_{13}$, and $F_{Z^2{S}^2}={\omega }^2{Y}_{31}+Y_{12}+\omega Y_{23}$. Admissible presentation changes preserving both ordered MASAs have projective common normalizer $G_{\mathrm{pol}}\simeq {\mathbb{Z}}_3^2⋊{\mathbb{Z}}_2$, so the basis-robust falsifier is

$${\mathcal{R}}_{\mathrm{forb},\mathrm{pol}}^2\left(Y\right):=\underset{U\in G_{\mathrm{pol}}}{\min }{\displaystyle \frac{{\mathrm{dist}}_{\mathrm{HS}}^2(UY{U}^{†},{\mathcal{A}}_{\mathrm{allow}})}{{\parallel Y\parallel }_{\mathrm{HS}}^2}}.$$

(212)

The support theorem is conditional on the exact graded numerator and central denominator; finite local jets alone do not exclude additional nonlinear Parent vertices.

The intrinsic Rees-Picard response is the non-generated grade-two component. With $R_1=a_{1}X$, unitary $X\in {\mathcal{A}}_1$, $Y\in {\mathcal{A}}_2$, and ${\mathsf{P}}_r^X=3^{-1}{\sum }_{j=0}^2{\omega }^{-rj}{\mathrm{Ad}}_X^j$, put $Y_{\perp }=({\mathsf{P}}_1^X+{\mathsf{P}}_2^X)Y=[X^{†},[X,Y]]/3$ and ${\rho }_{\mathrm{RP}}={\parallel [X,Y]\parallel }_{\mathrm{HS}}/{\parallel R_1\parallel }_{\mathrm{HS}}^2$. On $Y_{\perp }\ne 0$, the involution ${\mathcal{C}}_X=({\mathrm{Ad}}_X-{\mathrm{Ad}}_X^{-1})/\left(i\sqrt{3}\right)$ gives $p={\displaystyle \frac{1}{2}}(1+\langle Y_{\perp },{\mathcal{C}}_X{Y}_{\perp }\rangle /\parallel Y_{\perp }{\parallel }^2)$ and ${\Delta }_{\mathrm{pol}}:=3p(1-p)$. The multiplicative gate is ${\mathcal{G}}_2:=\parallel {\mathsf{P}}_0^{X}Y-R_1^2{\parallel }_{\mathrm{HS}}/{\parallel R_1^2\parallel }_{\mathrm{HS}}$, with exact descent ${\mathcal{G}}_2=0$; on the marked $X=Z$ branch this is ${\mathcal{G}}_2=|{\alpha }_0-1|$, ${\alpha }_0:=\mathrm{Tr}\left(Z^{-2}Y\right)/\left(3a_1^2\right)$. For ${\sigma }_{23}=c_2{\sigma }_{12}^2$ and ${\sigma }_{13}=c_3{\sigma }_{12}^3$, the Rees relations are $c_2^2=(1+{\rho }_{\mathrm{RP}}^2)/3$ and $c_3={\rho }_{\mathrm{RP}}/3$; eliminating ${\rho }_{\mathrm{RP}}$ gives ${({\sigma }_{23}/{\sigma }_{12}^2)}^2-3{({\sigma }_{13}/{\sigma }_{12}^3)}^2=1/3$.

Corollary 11

(Primitive contraction fixed point and Jacobi response). Assume that the source/fuzzy contraction, source-Hodge map, and family Jacobi response are typed as three readouts of the same normalized $V_1$ contraction. If $\gamma \in [0,1]$ is its normalized defect eigenvalue, survival/defect complementarity gives $c_2^2=1-\gamma$ and $c_3=\gamma$. Together with the Rees relations this forces

$$\gamma ={\displaystyle \frac{1}{3}},{\rho }_{\mathrm{RP}}=1,c_2^2={\displaystyle \frac{2}{3}},c_3={\displaystyle \frac{1}{3}}.$$

(213)

Compatibility with the adjacent fuzzy tower then selects $p=2$; on the typed source-Hodge branch the same relay gives $q_{*}=4$, ${\kappa }_g=0$, $r_{\mathrm{src}}^2=2/3$ and ${\chi }_H^2=3/8$. The identity $r_{\mathrm{src}}^{-2}-{\chi }_H^2=9/8$ is retained as a cross-reading. Polarization is not fixed by complementarity; after the selected orientation also gives ${\Delta }_{\mathrm{pol}}=0$, the primitive complement satisfies $\mathrm{Tr}{Y}_{\perp }=\mathrm{Tr}\left(Y_{\perp }^2\right)=0$, $Y_{\perp }^3=\left(\det Y_{\perp }\right)\mathbf{1}$, and ${[X,Y_{\perp }]}^{†}[X,Y_{\perp }]=3a_1^4\mathbf{1}$. Complex conjugation gives the opposite orientation.

Thus ${\rho }_{\mathrm{RP}}=1$, the source-Hodge geodesic normalization and the primitive fuzzy level are dependent consequences after the same-object contraction relay has passed; their operator provenance, rather than equality of the final scalars, is the remaining gate. On the weaker marked multiplicative source transfer, before imposing the terminal geodesic incidence, the primitive family value gives $r_{\mathrm{src}}=\sqrt{{\displaystyle \frac{8}{3(4+{\kappa }_g^2)}}}\le \sqrt{2/3}$ and $r_{\mathrm{src}}^{-2}-{\chi }_H^2=9/8$, providing an off-geodesic falsifier. On the stronger exact same-response specialization of (210), exact Jacobi and square attachment give $y=x^2$ and $g=-1/2$. The denominator becomes $1-z-z^2$ with $z=x^3{t}^3/8$; its Taylor coefficients are Fibonacci with exact convergence domain $\left|xt\right|<2{\varphi }^{-1/3}$, $\varphi =(1+\sqrt{5})/2$, and at either denominator root the numerator has characteristic polynomial ${\lambda }^3-8$. The finite-transfer pole is therefore non-removable. A physical resonance still requires spectral identification and an outgoing-sheet prescription.

Proposition 40

($H_R$ bigrading, geodesic filtration, and Parent valuations). Assume a marked identification of the family $A_2$ frame carrying $H_R$ in (50) with the positive response $A_2$ frame. In the marked order $h_1=4/3$, $h_2=1/3$, $h_3=-5/3$ give metric degrees ${\nu }_{12}=1$, ${\nu }_{23}=2$, and ${\nu }_{13}=3={\nu }_{12}+{\nu }_{23}$. Relative to the common projective $C_3$ stabilizer, $(E_{12},E_{23},E_{13})$ have characters $(\omega ,\omega ,{\omega }^2)$, so the product $E_{12}{E}_{23}=E_{13}$ preserves the bigrading $(1,1)+(2,1)=(3,2)$ in ${\mathbb{Z}}_{\ge 0}\times {\mathbb{Z}}_3$. Hence $F_p{M}_3:=\mathrm{span}\{E_{ij}:{\nu }_{ij}\le p\}$ is a multiplicative metric filtration and the degree-three 13 channel is generated at the graded level. The remaining dynamical family-frame gate is the actual Parent valuation

$$\mathrm{ord}(12,23,13)=(1,2,3)$$

(214)

in this same marked $H_R$ frame. Let the leading positive Hermitian germ be $H\left(t\right)=D+t(a_{12}{E}_{12}+{\overline{a}}_{12}{E}_{21})+t^2(a_{23}{E}_{23}+{\overline{a}}_{23}{E}_{32})+t^3(a_{13}{E}_{13}+{\overline{a}}_{13}{E}_{31})+\dots$, with $D>0$ diagonal. The filtration fixes the order of the 13 channel but not its direct coefficient: $a_{13}$ remains independent of the composite $a_{12}{a}_{23}$. The inverse has the same lower valuations by Lemma 5. With all three leading edges present, the first oriented cycle has weighted degree six; a scalar Hermitian determinant sees its real part but cannot fix its orientation. On a tree stratum the cycle phase disappears while the order-three hierarchy remains. This is Froggatt-Nielsen-type power counting [93] with geodesic Rees degree replacing an additional flavor charge.

The marked frame identification in Proposition 40 is a Parent incidence; abstract isomorphism of the two $A_2$ organizers is not used as an identification. In the same marked frame $J_{\mathrm{ax}}=\mathrm{diag}(1,0,-1)$ has adjoint weights $(1,1,2)$ on $(E_{12},E_{23},E_{13})$, so the geodesic $H_R$ filtration is a distinct typed grading. Exact $S_3$ symmetry would retain a degenerate doublet, while the physical orientation still requires an oriented channel/Jarlskog or marked-holonomy datum rather than the scalar pole set.

Proposition 41

(Direct/composite Wilson quotient and leading family packet). On the all-edge-nonzero germ of Proposition 40, vertex rephasing leaves one phase-bearing coordinate ${\mathfrak{W}}_{▵}:=a_{12}{a}_{23}\overline{a_{13}}/{\left|a_{12}{a}_{23}\right|}^2=\overline{a_{13}}/\overline{a_{12}{a}_{23}}$, with ${\mathcal{I}}_{▵}={\left|a_{12}{a}_{23}\right|}^2\Im {\mathfrak{W}}_{▵}$. The quotient is invariant under vertex rephasing and regular rescaling of the Rees coordinate. If $a_{13}=\kappa a_{12}{a}_{23}$ with real κ, then ${\mathcal{I}}_{▵}=0$; for $a_{13}=\kappa a_{12}{a}_{23}+a_{13}^{\mathrm{dir}}$, only the direct/composite interference contributes to the imaginary invariant. Thus the Rees composition fixes the hierarchy order but does not generate CP by itself. For the same leading relative generator, the same-object amplitude incidence is $|a_{23}|/|a_{12}{|}^2=c_2$ and $|a_{13}|/|a_{12}{|}^3=c_3$, so $|{\mathfrak{W}}_{▵}|=c_3/c_2$. After the primitive contraction fixed point (213) and the physical norm-seven holonomy have both passed, the leading complex target is

$$\left(\gamma ,{\mathfrak{W}}_{▵}\right)=\left({\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{\sqrt{6}}}{e}^{i{\sigma }_7^{\mathrm{phys}}{\delta }_7}\right),\underset{\lambda \to 0}{\lim }{\displaystyle \frac{J_q}{{\lambda }^6}}=(1-\gamma )\Im {\mathfrak{W}}_{▵}.$$

(215)

Hence $c_2^2=2/3$, $c_3=1/3$, the ratio $1/\sqrt{6}$, the CP orientation and the leading Jarlskog coefficient are dependent readings of one passed contraction-holonomy packet rather than independent fits.

The first relative quark basis response is the standard Sylvester response: for a simple Hermitian baseline $H_f$ with eigenvalues $h_{f,i}$ and perturbation $X_f$, ${\left({\mathsf{\Theta }}_f\right)}_{ij}={\left(X_f\right)}_{ij}/(h_{f,j}-h_{f,i})$ for $i\ne j$. Under $(H_f,X_f)\mapsto (c^p{H}_f,c^q{X}_f)$, ${\mathsf{\Theta }}_f\mapsto c^{q-p}{\mathsf{\Theta }}_f$, so a common scale cancels when baseline and numerator have the same parent homogeneity. Proposition 39 already fixes the finite invariant endpoint; the normalized $(H_f,X_f)$ path is needed only for its local dynamics, and the neutral terminal gain of Theorem 11 does not normalize the quark numerator. CKM conventions follow [94,95,96]. Common co-rotation cancels, while a shared non-circulant numerator can carry CP when the two Sylvester responses differ, with invariant normalization as in [97]. Put $J_{\mathrm{cen}}:=\Im \mathrm{Tr}\left({[{\mathsf{\Gamma }}_u,{\mathsf{\Gamma }}_d]}^3\right)$ after removal of the common circulant baseline; on the canonical clock-shift representative $J_{\mathrm{cen}}^2=(9/16)\mathrm{Disc}{\chi }_{{\Delta }_{\theta }}$, with orientation fixing the sign. Let $\mathfrak{q}$ be the microscopic common-to-relative family-jet response, tested by (218), and let $L_{ij}=E_{ij}-E_{ji}$. The Hodge-Eisenstein completion is

$$\begin{array}{cc}\hfill A_u\left(ϵ\right)& =(\mathfrak{q}-\frac{1}{2})ϵL_{12}+(\mathfrak{t}-\frac{c_2}{2}){ϵ}^2{L}_{23}-\frac{\mathfrak{h}}{2}{ϵ}^3{L}_{13},\hfill \\ \hfill A_d\left(ϵ\right)& =(\mathfrak{q}+\frac{1}{2})ϵL_{12}+(\mathfrak{t}+\frac{c_2}{2}){ϵ}^2{L}_{23}+\frac{\mathfrak{h}}{2}{ϵ}^3{L}_{13},\hfill \end{array}$$

(216)

with $V_{\mathrm{HE}}=e^{-A_u}{e}^{A_d}$. Since $A_u$ and $A_d$ are real antisymmetric, $V_{\mathrm{HE}}\in SO\left(3\right)$ and $J\left(V_{\mathrm{HE}}\right)=0$; this finite map determines the three mixing angles, while the CP phase is supplied by the separately marked norm-seven holonomy below. The leading 13 retention and vanishing connected order-four 23 correction give

$$\mathfrak{t}=c_2(\mathfrak{q}-3+{\mathfrak{q}}^{-1})+6c_3(1-{\mathfrak{q}}^{-1}),\mathfrak{h}=c_2(1-{\left(2\mathfrak{q}\right)}^{-1})-c_3(2-3{\mathfrak{q}}^{-1}).$$

(217)

Put $D=c_2-6c_3$ and $H_{\mathrm{jet}}=\mathfrak{h}-c_2+2c_3$. The common-jet branch obeys

$${\mathcal{R}}_H:=2H_{\mathrm{jet}}(\mathfrak{t}+3c_2-6c_3+2H_{\mathrm{jet}})+c_{2}D=0,{\mathfrak{q}}_H=-{\displaystyle \frac{D}{2H_{\mathrm{jet}}}}.$$

(218)

The second expression is used for $D{H}_{\mathrm{jet}}\ne 0$. On the exact primitive second-response branch of Corollary 11, $c_2=\sqrt{2/3}$ and $c_3=1/3$. On the typed source-Hodge family of Corollary 10, write $q:=q_{*}=\mathfrak{q}=4+{\kappa }_g^2\ge 4$; then $H_{\mathrm{jet}}=4H_{*}/q$ with $H_{*}=-D/8$. The exact finite product $V_{\mathrm{HE}}=e^{-A_u}{e}^{A_d}$ already fixes all higher Taylor coefficients, so they are not retained as independent fit or gate variables. On the strict $q=4$ branch the parameter-free leading limits are

$$\underset{\lambda \to 0}{\lim }{\displaystyle \frac{s_{23}}{{\lambda }^2}}=\sqrt{{\displaystyle \frac{2}{3}}},\underset{\lambda \to 0}{\lim }{\displaystyle \frac{s_{13}}{\lambda s_{23}}}={\displaystyle \frac{1}{\sqrt{6}}},\underset{\lambda \to 0}{\lim }{\displaystyle \frac{|J_q|}{{\lambda }^6}}={\displaystyle \frac{60\sqrt{2}}{343}}.$$

(219)

The last two limits in (219) are dependent readings of Proposition 41: the modulus $1/\sqrt{6}$ and the leading Jarlskog coefficient arise from the same complex quotient after the contraction and orientation locks have passed. Together with (220), $|J_q|/\left(s_{12}{s}_{23}{s}_{13}\right)\to 180\sqrt{3}/343=\left|\sin {\delta }_7\right|$. The map $ϵ\mapsto s_{12}\left(V_{\mathrm{HE}}\left(ϵ\right)\right)$ is locally invertible because $V_{\mathrm{HE}}^{\prime }\left(0\right)=L_{12}$; on the strict $q=4$ branch validated interval evaluation gives $d{s}_{12}/dϵ>0.3825545$ on $0\le ϵ\le 0.7743639666820934$, so the inverse is unique on the full normal-ordering interval used below. On the reduced saturation packet the common-cell chart supplies its argument without a flavor refit. A microscopic magnitude still requires the same normalized Parent baseline/numerator path. The real finite map has vanishing Jarlskog invariant, so its CP phase is supplied only by the separately marked norm-seven holonomy. Smith-type and BCH checks remain algebraic diagnostics rather than additional reconstruction conditions.

The primitive norm-seven phase is the oriented coefficient class already fixed in (53):

$$e^{i{\delta }_{\mathrm{CKM}}^{\min }}=e^{i{\delta }_7},\cos {\delta }_{\mathrm{CKM}}^{\min }={\displaystyle \frac{143}{343}},\sin {\delta }_{\mathrm{CKM}}^{\min }={\displaystyle \frac{180\sqrt{3}}{343}}.$$

(220)

With the three angles read from $V_{\mathrm{HE}}$, the reduced PDG-form complex completion is

$$V_{\mathrm{CKM}}^{\mathrm{PDG}}(u,\sigma )=R_{23}\left({\theta }_{23}\right)\left(\begin{array}{ccc}{c}_{13}& 0& s_{13}{e}^{-i\sigma {\delta }_7}\\ 0& 1& 0\\ -s_{13}{e}^{i\sigma {\delta }_7}& 0& c_{13}\end{array}\right)R_{12}\left({\theta }_{12}\right).$$

(221)

The matrix (221) supplies the complex completion of the HE angles, whereas row/column rephasing of $V_{\mathrm{HE}}$ leaves the Jarlskog invariant zero. The norm-seven spectrum (49) and (47) fix the two conjugate points in (220). If the physical determinant/primitive line and marked loop are typed to the same protected norm-seven cycle, the primitive winding is $n=\pm 1$ and Proposition 19 gives the no-backtracking/unique-crossing test. On this passed physical orientation lock, $H_2^3=e^{i{\sigma }_7^{\mathrm{phys}}{\delta }_7}$ makes the CKM phase the cubic reading of the same marked holonomy, with ${\sigma }_7^{\mathrm{phys}}={\sigma }_7^{\mathrm{ar}}$; before that incidence the conjugate sign remains the discrete branch datum ${\sigma }_7^{\mathrm{phys}}$. Conjugation reverses both the Wilson point and $J_{\mathrm{cen}}$, while $|J_{\mathrm{cen}}|=(9\sqrt{3}/2)\left|\sin {\delta }_{\mathrm{CKM}}^{\min }\right|$. Entanglement-based flavor selection gives an independent comparison [98]. For invertible neutral and direct susceptibilities ${\mathcal{S}}_N,{\mathcal{S}}_D$ and the marked triality permutation P, exact triality is a scalarity test of ${\mathcal{S}}_{N}P{\mathcal{S}}_D^{-1}$; its vanishing traceless part implies ${\mathcal{S}}_{N}P=\kappa {\mathcal{S}}_D$ and scales the full singular spectrum by $\left|\kappa \right|$, whereas determinant agreement alone is insufficient.

For the charged-lepton vertex polarization the sourced baseline satisfies $H_{\ell }={\mathbb{E}}_{\mathrm{vert}}\left(H_{\ell }\right)$, where ${\mathbb{E}}_{\mathrm{vert}}$ is the vertex conditional expectation, hence lies in the clock algebra after a torsor origin has been chosen. The positive amplitudes $x_{\ell ,r}=\sqrt{m_{\ell ,r}}$ have the real ${\mathbb{Z}}_3$ Fourier form $x_{\ell ,r}=C_{\ell }(a_{\ell }+\cos ({\varphi }_{\ell }+2\pi r/3))$. With ${\mathfrak{X}}_0={\sum }_r{x}_{\ell ,r}$ and ${\mathfrak{X}}_1={\sum }_r{\omega }^{-r}{x}_{\ell ,r}$, the single complex response ${\mathcal{Z}}_{\ell }=\log ({\mathfrak{X}}_0/\left(\sqrt{2}{\mathfrak{X}}_1\right))$ has real part equal to half the balance response and imaginary part $-{\varphi }_{\ell }$. Writing ${\Sigma }_{\ell }$ for the completed Schur correction to the invariant/complement norm balance, the finite Parseval relation is $2a_{\ell }^2=e^{{\Sigma }_{\ell }}$; the zero-correction branch therefore has $Q_{\ell }=2/3$. This is the Koide balance [99] in its geometric form [100]; the companion ${\mathbb{Z}}_3$ clock reference is represented by [101]. On the fixed $Q_{\ell }=2/3$ shell, the marked clock phase and normalized determinant are equivalent one-scalar spectral coordinates; ${\mathcal{Z}}_{\ell }$ retains the stronger parent-response reading. The charged clock reference $2/9$ is not identified with the neutral determinant coordinate $s_{\det }\left(0\right)=2/9$. Higher $V_3$ response enters only through the second-order Schur term of Proposition 33.

5.3. Neutral Schur Package and PMNS

For the complex-symmetric neutral block $K_N=\left(\begin{array}{cc}{A}_L& D_{\nu }^T\\ D_{\nu }& M_R{R}_N\end{array}\right)$, with $A_L^T=A_L$, $R_N^T=R_N$, $|\det R_N|=1$, the standard skew doubling gives

$$K_{\nu }^{\mathrm{eff}}=A_L-D_{\nu }^T{\left(M_R{R}_N\right)}^{-1}{D}_{\nu },\det K_{\nu }^{\mathrm{eff}}={\displaystyle \frac{\mathrm{Pf}\left({\mathbb{A}}_N\right)}{\mathrm{Pf}\left({\mathbb{A}}_H\right)}},m_1{m}_2{m}_3=\left|\det K_{\nu }^{\mathrm{eff}}\right|.$$

(222)

The seesaw interpretation is standard [102]. Since unrestricted $D_{\nu }$ makes the Schur map locally surjective onto complex symmetric matrices, PMNS predictivity requires an upstream restriction in the fixed charged frame. A finite list of Takagi jets likewise determines a finite endpoint only with an exact transfer or a certified remainder. Relative column phases remain required for $m_{\beta \beta }$.

On the normalized type-I mass-shape branch,

$${\mathcal{R}}_{\nu }\left(u\right)={\displaystyle \frac{{\left(\frac{3}{7}-\frac{3}{2}u\right)}^4-{\left(\frac{6}{49}\right)}^4}{{(1+16u)}^4-{\left(\frac{6}{49}\right)}^4}},$$

(223)

and elimination of u with $r=m_W^2/m_Z^2$ gives

$${\mathcal{R}}_{\nu }={\displaystyle \frac{6^4\left[r^4-{(1-r)}^4\right]}{{(273-337r)}^4-6^4{(1-r)}^4}}.$$

(224)

The oscillation comparison uses [103,104]. The K1-K3 terminal attachment is already supplied by Theorem 11 and Corollary 10.

The marked real-Fourier/TBM relay convention is

$$U_{\mathrm{TBM}}=\left(\begin{array}{ccc}\sqrt{2/3}& 1/\sqrt{3}& 0\\ -1/\sqrt{6}& 1/\sqrt{3}& 1/\sqrt{2}\\ -1/\sqrt{6}& 1/\sqrt{3}& -1/\sqrt{2}\end{array}\right),R_{ij}\left(\theta \right)=\exp \left(\theta L_{ij}\right).$$

(225)

For $H_0=H_S\left(0\right)$, the torsor directions obey

$$U_{\mathrm{TBM}}^{\mathsf{T}}G{U}_{\mathrm{TBM}}=-L_{13},U_{\mathrm{TBM}}^{\mathsf{T}}\left(-{\displaystyle \frac{\sqrt{2}}{3}}[K_Z,H_0]\right)U_{\mathrm{TBM}}=L_{23},U_{\mathrm{TBM}}^{\mathsf{T}}\left({\displaystyle \frac{\sqrt{2}}{3}}[[K_Z,H_0],G]\right)U_{\mathrm{TBM}}=L_{12},$$

(226)

and

$$[K_Z,H_0]={\displaystyle \frac{\sqrt{3}}{2}}\left(-\omega ZS+{\omega }^{2}Z{S}^2-{\omega }^2{Z}^{2}S+\omega Z^2{S}^2\right).$$

(227)

Thus the relay spans $\mathfrak{so}\left(3\right)$ and contains the marked Weil line of phase $4\pi /3$; a purely circulant correction would impose the excluded trimaximal solar/reactor relation.

The consecutive Eisenstein trace-norm ratios fix the first two rational relay coefficients, so

$$U_{\nu }=U_{\mathrm{TBM}}{R}_{12}\left({\beta }_{\nu }\right)R_{13}\left({\gamma }_{\nu }\right)R_{23}({\alpha }_{\nu },{\varphi }_{\nu }),{\alpha }_{\nu }={\displaystyle \frac{2}{\sqrt{3}}}{\mathsf{\Xi }}_{23}{\lambda }_{\mathrm{cen}},{\gamma }_{\nu }=-{\displaystyle \frac{13}{7}}{\lambda }_{\mathrm{cen}}^2,{\beta }_{\nu }=-{\displaystyle \frac{7}{5}}{\lambda }_{\mathrm{cen}}^3.$$

(228)

Exact PLSR attachment gives ${\mathsf{\Xi }}_{23}=\sqrt{3/8}$, while (227) supplies the coefficient-level phase origin. Off the exact connection locus write ${\varphi }_{\nu }=4\pi /3-{\alpha }_{\nu }-h_{\nu }$; Proposition 18 removes the continuous mismatch after the line bundle and curvature are matched. Identifying the quark and neutral orientation signs still requires the same physical marked cycle.

Writing $x={\sin }^2{\theta }_{13}$, $z={\sin }^2{\theta }_{12}$, $y={\sin }^2{\theta }_{23}$ and $V=U_{\mathrm{TBM}}{R}_{12}\left({\beta }_{\nu }\right)R_{13}\left({\gamma }_{\nu }\right)$, let $s=\sin {\alpha }_{\nu }$, $c=\cos {\alpha }_{\nu }$, $A_a=s^2{V}_{a2}^2+c^2{V}_{a3}^2$, $B_a=2sc{V}_{a2}{V}_{a3}$, $q_{\varphi }=B_{\mu }/B_e$, $p_{\varphi }=A_{\mu }-q_{\varphi }{A}_e$, and $J_{\mathrm{amp}}=V_{e1}{V}_{\mu 1}sc(V_{\mu 2}{V}_{e3}-V_{\mu 3}{V}_{e2})$. Eliminating the one phase gives

$$(1-z)(1-x)=V_{e1}^2,y+q_{\varphi }={\displaystyle \frac{p_{\varphi }+q_{\varphi }}{V_{e1}^2}}(1-z),{\left({\displaystyle \frac{x-A_e}{B_e}}\right)}^2+{\left({\displaystyle \frac{J_{\mathrm{PMNS}}}{J_{\mathrm{amp}}}}\right)}^2=1.$$

(229)

On the strict saturation branch $h_{\nu }=0$, all Dirac-PMNS observables are functions of the same common u; Majorana phases require the full complex endpoint.

The local relay jets provide an independent no-refit test. Their first three coefficients reconstruct the relay parameters, while the fourth-order closure is

$$q_{13}^{\left(3\right)}-{\displaystyle \frac{13}{14}}=-{\displaystyle \frac{147}{130}}\overline{\left(q_{12}^{\left(2\right)}+{\displaystyle \frac{21}{5}}\right)}.$$

(230)

A finite jet does not determine an unrestricted analytic endpoint. For microscopic and relay paths starting in the same frame, ${\mathfrak{R}}_{\mathrm{flow}}:={\int }_0^{{\lambda }_0}{\parallel \mathsf{\Omega }-\mathsf{\Gamma }\parallel }_{\mathrm{op}}d\lambda$ bounds the endpoint operator mismatch; each modulus changes by at most ${\mathfrak{R}}_{\mathrm{flow}}$, each squared modulus by at most $2{\mathfrak{R}}_{\mathrm{flow}}$, and a Jarlskog invariant by at most $4{\mathfrak{R}}_{\mathrm{flow}}$. This is the required precision certificate for a finite relay approximation.

5.4. Saturation Manifold, Prediction Protocol, and Completion Fibers

Proposition 42

(Reduced strict saturation packet). On the passed Parent image of Section 4, let ${\mathcal{L}}_{\mathrm{red}}$ contain the reduced PLSR conditions together with the active family gates (214), the same-object primitive contraction relay of Corollary 11, and the physical norm-seven orientation. The former scalar condition ${\rho }_{\mathrm{RP}}=1$ is then a dependent consequence of the contraction relay. The selected charge-fold point fixes $u_{*}$ through (167), with (165) retained as its same-response transport representation; the trace lift follows from $B_{\mathrm{fl}}\left(0\right)=B_0$ and (175) through (197), with (176) retained as a dependent checksum. Terminal/source-Hodge data are supplied by Theorem 11 and the same contraction provenance, while Proposition 18 supplies the neutral line attachment. With the two arithmetic norm-seven orientations retained as the discrete ${\sigma }_7^{\mathrm{ar}}$ fiber,

$${\mathcal{M}}_{\mathrm{sat}}^{\mathrm{red}}\simeq {\mathcal{M}}_{\mathrm{alg}}\times {\{+1,-1\}}_{{\sigma }_7^{\mathrm{ar}}},{\mathcal{M}}_{\mathrm{alg}}=[0,2/7).$$

(231)

The labeled normal-ordering neutrino formulas are restricted to ${\mathcal{M}}_{\mathrm{NO}}=[0,10/49)$. Each chart in (200) is globally injective on ${\mathcal{M}}_{\mathrm{alg}}$, so one predeclared global chart observable fixes u without sector-dependent retuning.

Equation (231) concerns the frozen reduced maps after the physical Parent certificate has passed; Proposition 12 supplies local class nonemptiness but does not replace the nonhomogeneous defect, constrained-Jacobi, terminal-Hodge, mixed-product, Callias/Dirac, or family tests. Physical identification of the arithmetic orientation requires the marked-cycle incidence ${\sigma }_7^{\mathrm{phys}}$. At regular points the twelve-coordinate scale-free packet formed by the common cell, neutral mass ratios, the three additional CKM coordinates with Cabibbo already contained in the common cell, and four Dirac-PMNS invariants has rank one and therefore eleven independent local no-refit relations after definitional identities are removed. Let G be a local complete relation map for this regular curve $M=G^{-1}\left(0\right)$. A completion field preserves M iff $DGV=0$, equivalently $V\left(Y\left(u\right)\right)=f\left(u\right)Y^{\prime }\left(u\right)$; its flow is then an exact reparametrization of u. A finite completion map C preserves the no-refit relations iff $G\left(C\right(Y\left(u\right)\left)\right)=0$.

External evaluation is separated from the analytic construction. One global chart observable may be predeclared as CAL while no certified Parent value is available; the remaining scale-free coordinates are HOLDOUTs and independently invertible observables give redundant reconstructions $u_i=F_i^{-1}\left(Y_i\right)$. An independently certified $u_{\mathrm{Parent}}$ from (167) replaces rather than supplements this CAL, so the calibration row is then promoted to HOLDOUT. The scorecard uses $r_{WZ}$ as the sole current calibration chart and collects numerical values, marginal pulls, redundant-u checks, and source versions in Appendix A. No external datum enters a structural pass/fail condition, and a calibrated coordinate is not counted again as a prediction.

Calibration nuisances are propagated only after the global chart has fixed u; local tests are counted as independent only in directions transverse to the calibration kernel. For the declared three-coordinate matching chart, let Q have orthonormal rows spanning the two-dimensional normal space of the affine prediction line and let microscopic controls have first response $J\theta$ with contravariant metric $C_{\mathrm{ctrl}}\succ 0$. Put $A_N=QJ$ and

$$S_N:=A_N{C}_{\mathrm{ctrl}}{A}_N^{†}.$$

(232)

For raw normal flow b, exact first-order closure is possible precisely when

$$(\mathbf{1}-S_N{S}_N^{+})b=0.$$

(233)

On this locus the minimum-$C_{\mathrm{ctrl}}^{-1}$-norm correction is ${\theta }_{*}=-C_{\mathrm{ctrl}}{A}_N^{†}{S}_N^{+}b$ with cost $b^{†}{S}_N^{+}b$; $S_N\succ 0$ is universal rank-two controllability, while a rank-deficient susceptibility may still close a particular reachable target. If the controls also move already fixed low/pole coordinates, the raw susceptibility is replaced by the Schur-reduced $A_{\mathrm{fb}}=A_N-B_x{J}_x^{-1}{E}_{\theta }$ and, away from a passed upstream point, b by $b-B_x{J}_x^{-1}{r}_x$. Thus the control quotient is chart-local and cannot be used as hidden retuning of the full observable packet. Across pole, family-tomography, and matching layers, closure loss of the corresponding gap or response rank is reported as loss of identifiability rather than absorbed into a fitted correction.

• Dimensionful scale and unresolved phases.

On the normal-ordered common-cell branch, the mass-shape ratios in (202) become absolute-mass targets after one atmospheric splitting $\Delta :=\Delta m_{31}^2$ is supplied. If $w_i={\left|{\left(U_{\mathrm{PMNS}}\right)}_{ei}\right|}^2$, then

$${\displaystyle \frac{{\sum }_i{m}_i}{\sqrt{\Delta }}}={\displaystyle \frac{1+x_{\nu }^2+y_{\nu }^2}{\sqrt{y_{\nu }^4-1}}},{\displaystyle \frac{m_{\beta }}{\sqrt{\Delta }}}=\sqrt{{\displaystyle \frac{w_1+w_2{x}_{\nu }^4+w_3{y}_{\nu }^4}{y_{\nu }^4-1}}}.$$

(234)

For $A_1=w_1/\sqrt{y_{\nu }^4-1}$, $A_2=w_2{x}_{\nu }^2/\sqrt{y_{\nu }^4-1}$, and $A_3=w_3{y}_{\nu }^2/\sqrt{y_{\nu }^4-1}$, the unresolved Majorana phases give the exact envelope

$${\displaystyle \frac{m_{\beta \beta }^{\max }}{\sqrt{\Delta }}}=A_1+A_2+A_3,{\displaystyle \frac{m_{\beta \beta }^{\min }}{\sqrt{\Delta }}}=\max \left\{2\underset{i}{\max }{A}_i-\sum _i{A}_i,0\right\}.$$

(235)

Current numerical scale-outs obtained from one atmospheric splitting, together with the corresponding Majorana envelope, are reported in Appendix A. Ratios among (234) and (235) remove that scale and define additional cross-observable FUTURE/HOLDOUT coordinates on the restricted normal-ordered branch. None fixes the absolute action normalization.

• Completion fibers.

Absolute action normalization remains independent of the scale-free manifold: $S_{\mathrm{prim}}=4{\pi }^2{\kappa }_{\mathsf{\Omega }}$ would give $u={\left(64{\pi }^2{\kappa }_{\mathsf{\Omega }}\right)}^{-1}$ only after an additional protected normalization. The quarter-action neutral completion is retained only as an ansatz, with $S_M=S_{\mathrm{prim}}/4$, $M_R=M_{\mathrm{Pl}}{e}^{-S_M}$, $R_N=\mathbf{1}$, $A_L=0$, ${\Sigma }_{\nu }=\mathrm{diag}(B_0,A,1+S_{\mathrm{prim}}^{-1})$, and $D_{\nu }=m_D{\Sigma }_{\nu }{V}^{†}$; the seesaw comparison follows [102]. Proton conversion and heavy matching remain completion problems represented by [105,106]; finite threshold separation is represented by [107], while standard Higgs decay formulae are used only for comparison [108]. Relative gauge normalization is structural, whereas an absolute common matching scale requires thresholds. On the declared family-central parity-paired adjoint completion, the conditional one-loop identities are ${\mu }_{23}/M_{\mathrm{adj},\mathrm{C}}={16}^2$, ${\mu }_{23}/M_{\mathrm{adj},\mathrm{W}}={16}^3$, and ${\Delta }_{12}^{\mathrm{th}}=48\log 2/\pi$. For normalized family amplitudes, only family-differential anomalous dimensions move the shape, as encoded by ${\dot{Q}}_{\mathrm{run}}={\sum }_i{r}_i(r_i-Q_{\mathrm{run}}){\gamma }_i$ and $d\log {\pi }_{\mathrm{run}}/dt={\displaystyle \frac{1}{2}}{\sum }_i(1-3r_i){\gamma }_i$.

The logical levels used below are therefore structural reconstruction, one Parent realization, active spectral attachment, the reduced one-u manifold, and completion. The first four carry the paper’s no-refit claims; absolute units, high-scale thresholds, continuum widths, and optional neutral completions remain downstream. The principal quantitative claim is the one-dimensional scale-free saturation manifold and its transverse no-retune relations, with the current experimental instantiation isolated in Appendix A.

6. Discussion and Conclusions

6.1. Results and Reconstruction Mechanism

PLSR fixes the least recoverable finite packet before microscopic realization. Primitive multiplication determines the $3+2$ carrier, rigid port, dark Veronese line, and positive-response geometry; determinant/exterior functoriality gives the gauge packet, while one CM triality organizes the norm-seven family frame. The Alena-Codazzi Parent realizes this structure on one nonhomogeneous defect. On regular multiplier patches its two-eigenvalue sector is locally conformal-product and one-sided affine-warp; source-Hodge, Callias/polar carrier, charge ensemble, family response, and transport remain typed microscopic gates. After common-root provenance, the all-order identities (178)-(179) and the coprime Bézout reconstruction (180) organize the dark/family weight-$2,3,5$ descendants without adding selectors.

6.2. Selection, Critical Export, and Spectral States

The master selector is the action-native discriminant (99). Theorem 7 transfers the retained reading to the full KKT problem, Theorem 8 certifies monotone crossing from directional Parent bounds, and Corollary 6 identifies the first protected wall on a certified chain from the physical stable anchor. The constrained Schur zero, VH condition (98), KKT pole (101), intrinsic curvature (114), and cubic Lyapunov-Schmidt jet (100) are dependent readings after their incidences pass.

The selected solution gives $u_{*}=\mathcal{U}\left({\mathsf{\Phi }}_{*}\right)$ through (167). The same fold also has the panel-covariant critical residue (116); Theorem 9 reconstructs it from two stable-side response boxes with a certified second-order remainder, and Corollary 8 converts probe-complete components into typed critical tangents. Rank-one, pair-consistency, projective-transport, and causal tests distinguish a common Parent germ from stronger directed mediation. Spectral energy remains a different coordinate: Proposition 28 gives at most one state per protected multiplicity-one gap, while minimal realization degree, one-pole arithmetic, Hankel reconstruction, symmetry-controlled resolvent bounds, and the dark-port/outgoing tests remain independent spectral checks.

6.3. Prediction Manifold and Completion

On the passed Parent image, Proposition 42 gives the one-dimensional manifold (231). The fixed-Q charged-family shapes are $A_2$ circles (204); their determinant, Cartan-angle, cubic, and conic descriptions are coordinate charts rather than independent constraints. The contraction-Wilson packet (215) packages the primitive gap and complex family quotient, so after the upstream lock $\gamma =1/3$, the hierarchy modulus, norm-seven phase, and leading Jarlskog coefficient are dependent readings. Physical pole closure is tested by (195)-(199). The same-arrow dark condition protects the structural line from the outgoing self-energy, while (149) and (134) control conditional widths and fold-dark exponents. Absolute scale, unresolved Majorana data, high-scale thresholds, and outgoing/domain completion remain downstream.

6.4. Relation to Geometric and Spectral Constructions

The geometric comparison class includes Rainich theory [109], Codazzi geometry [110], first-order/twistor constructions [111], and twistor methods [112]. The conditional monopole reading uses [80], with cyclic charge-three data represented by [85]. Almost-commutative and spectral-action comparisons are represented by [54,92]; dynamical principal bundles by [113]. The completion uses matrix Herglotz theory [76] and the moment problem [77]. Flavor comparisons include modular symmetry [114], generalized CP [115], stabilization near $\tau =\omega$ [116], and explicit quark models [117].

6.5. Further Directions

The immediate target is one validated nonhomogeneous Parent branch with a common gauge/graph-norm convention. The required quantitative data are the Parent/Feshbach tail gap, the directional seed bounds entering Theorem 8, the first two $\mathsf{\Omega }$-derivatives of the fixed centered observable, and the response-jet bounds entering (122). Certified boxes are then chained by Corollary 6; no wall location or fitted u enters the selector. Once the chain reaches the fold, (167) and (117) release respectively the global reduced coordinate and the critical response tensor. The contraction-Wilson and common-root packets are tested on the same solution, followed by an actual equivariant spectral channel and a same-arrow outgoing completion. If the constrained response has effective rank greater than one, the scalar susceptibility is replaced by the corresponding susceptibility matrix rather than by multiple scalar selectors. AIII heavy descent and outgoing/domain continuation remain the principal microscopic provenance problems. Boundary/open-index extensions are represented by [118,119]; optional Einstein-Weyl and minitwistor comparisons by [120,121]; tubular comparisons by [122,123].