Self-Reconstructing Codazzi Defects, CP¹ Quantization, and the Minimal Standard-Model Carrier

1. Introduction

The geometrization of interactions is commonly obtained by enlarging the geometric structure. Gauge variables are represented by higher-dimensional or bundle-metric data in Kaluza-Klein mechanisms [1]. Forced dynamics may be represented as geodesic motion on an extended space through the Eisenhart-Duval lift [2]. Randers and Finsler descriptions encode charged trajectories by an effective geometry [3], while finite-resolution curvature defects provide a related local comparison class [4]. A narrower inverse problem is considered here: which finite internal data are forced by a local defect when only its resolved boundary observables, their source grading, and their closure relations are retained? In each of the constructions above the connection belongs to the assumed geometric data. Here it does not. The retained datum is a local defect together with the grading of its boundary source; the carrier, its structure algebra, and the transport acting on it are selected by the same minimality, so that the connection appears as the projected transport of the isolated sector rather than as an added bundle. Enlargement of the ambient geometry is thereby replaced by closure of the retained boundary description.

The local datum is a codimension-three timelike defect in a four-dimensional Lorentzian branch. Its oriented rank-three normal bundle has a two-sphere link, read in the optical Codazzi setting as a projective spinor line ${\mathbb{CP}}_{\mathsf{\Gamma }}^1$. After the scalar trace has been separated, a normal source of transverse order at most two has two principal non-scalar components: an order-one current moment and an order-two trace-free Codazzi moment. The aim is to determine the finite carrier generated by these graded boundary data and to separate this structural result from its geometric realization and subsequent low-energy completion.

The organizing principle is self-reconstruction. A physical configuration determines an observable description, and the retained description is required to reconstruct the same physical configuration:

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

(1)

Here ${\mathcal{A}}_{\mathsf{\Phi }}$ is the generated observable algebra, $\mathfrak{S}\left({\mathcal{A}}_{\mathsf{\Phi }}\right)$ is its state space, ${\delta }_{\mathsf{\Phi }}$ is the retained dynamics, ${Sec}_{\mathrm{iso}}\left({\mathcal{A}}_{\mathsf{\Phi }}\right)$ denotes the isolated stable sectors, and ${\nabla }_{\mathsf{\Phi }}$ denotes their natural transport data. Equality of $\mathsf{\Phi }$ and $\widehat{\mathsf{\Phi }}$ is understood on the physical quotient, after gauge, diffeomorphism, and unitary equivalences have been removed.

Minimal reconstruction is characterized at each exact depth by a protected interface and the poset of admissible generated subextensions inside a fixed ambient envelope. Closure under intersections gives a unique least admissible object. The ambient slots and canonical operations are fixed by depth, while enlargement of downstream protected data enlarges the least object monotonically. Finite-core multiplicity reduction, Borel-Weil witness descent, finite-shadow orbit closure, and protected Schur descent are consequences of this property. Auxiliary normal or high-sector coordinates may subsequently be eliminated by stationary or Schur reduction while their induced effective and determinant data are exported before descent. Physical inputs fixed independently at an earlier closure depth label the reconstruction class.

Known physical equations retain their usual variational, geometric, constraint, or spectral origin and enter the reconstruction as corresponding closure classes. A first-variation residual gives the Euler-Lagrange equation. The Einstein residual is accompanied by its Bianchi compatibility and covariant source conservation. Gauss closure fixes the boundary superselection data, while Fredholm and determinant closures control the isolated spectral sector and its global consistency. At the state level, positivity gives the Robertson-Schrödinger inequality. The trace-preserving expectation onto the reconstructible algebra generates a completely positive semigroup for positive reconstruction time, with monotone relative entropy and decay of information outside the reconstructible subalgebra. These properties place the finite selection mechanism within the usual uncertainty and second-law constraints.

The decisive structural datum is the normal order carried by the two source types. Their $SU\left(2\right)$ representation content is standard [5], while faithful Gauss reconstruction retains the two types in orthogonal central sectors. On a finite Borel-Weil support, a marked sector carrying $V_{\ell }$ has rank at least $\ell +1$. Minimality therefore places the two sectors on $E_2$ and $E_3$, excludes the one-block visibility alias, and gives the graded Hermitian space $E_3\oplus E_2\simeq {\mathbb{C}}^3\oplus {\mathbb{C}}^2$. The normal $SU\left(2\right)$ action and the section multiplication complete their role in selecting this support. They are then reduced as selection witnesses, while the split Hermitian grading and determinant line remain protected. Their automorphism group is $S\left(U\right(3)\times U(2\left)\right)$. Its centered integral grading gives the hypercharge direction, and its exterior second quantization gives the even one-generation Fock package. Twistor-space Standard-Model constructions provide a representation-theoretic comparison [6]; the carrier group and module used here are selected by the reconstructed degree filtration.

The same integral grading has a finite cocharacter shadow. Its intersection with the derived block group is ${\mathbb{Z}}_6$, whose parity-selected affine quotient carries a three-dimensional induced module. Finite permutation and Clifford family constructions provide comparison models [7]. In the present construction the three-dimensional module is first obtained as a structural finite-shadow object. Its interpretation as a physical family factor additionally requires exact covariance with the primitive low sector, parity closure, and an open Callias-Schur gap. The associated projective-color torsor carries a finite differential, Wilson defect, Laplacian, non-circulant response, and subalgebra entropy. These are different readings of one finite edge complex.

Under the realization hypotheses, the Alena-Codazzi collar supplies the geometric inputs: a compact-leaf source, a primitive thin-core worldline, a moment-resolved normal grading, two Gauss-local charges, and their calibrated boundary transport. The moment map preserves the source grading, and minimal marked Borel-Weil reconstruction identifies it with the carrier level. The microscopic normal family is then formed from the retained current-Codazzi realization residuals. Its Gram-Riesz splitting separates stationary obstruction from motion of the isolated Riesz bundle. The faithful blockwise finite-core image commutes with principal Schur reduction, so the multiplier pattern has support $C|W_{\sigma }|C$ without an independent finite Hessian-factorization condition; the normalized trace then fixes the corresponding neutral weights.

The argument is arranged in four layers: filtered minimal reconstruction, structural reconstruction from the primitive graded defect, Alena-Codazzi realization with microscopic normal descent, and completed Schur analysis with quantitative tests. The first two determine the generated finite content. The third supplies the source model, the isolated low bundle, and the finite principal support on the realized interface. The fourth is ordered from chiral and determinant data to spectral response, relative family motion, neutral completion, and scale matching. In the direct branch the second angular jet separates the generated square from the source-marked primitive response. Its norm is independent of Picard polarization. The one-line and balanced Picard subbranches belong to the relative Hermitian packet used for mixing. The direct Hodge ladder supplies an integral incidence packet compatible with the least simple Eisenstein torsor branch. A completed Hodge-Eisenstein jet bridge identifies its surviving-to-lost support ratio with the common-to-relative first family jet; the subsequent matrix corrections are then fixed by unitary completion rather than by identifying Rees path norms directly with exact CKM angles. The Weyl cubic algebra separates its symmetric and antisymmetric cubic combinations, while the strong determinant requires separate pre-Hermitian lifts in the up and down sectors because $argdet\left(Y_u{Y}_d\right)$ is sectorwise additive. The unit Rees-Picard normalization, the Hodge-to-jet bridge, and microscopic preservation of the commensurate arithmetic class retain their stated completed statuses. Neutral Lie directions are fixed algebraically. The completed high-block branch retains the full parity pair before low compression, fixes its adjoint gap by a Clifford-Picard functional of the protected level, and uses determinant centering of the projected weak-leg connection for the neutral phase attachment. The remaining neutral reduced coefficients are retained at completed depth.

2. Filtered Minimal Reconstruction

The self-reconstruction loop of (1) is used in a filtered form. The filtration separates geometric, algebraic, determinant, finite, spectral, and completed closures according to the data required to define them. Standard physical equations enter only through their corresponding retained residual classes; finite minimality acts on their represented data and does not replace their usual physical origin. The same framework retains the positivity of states, the standard uncertainty relation, and a completely positive reduction onto reconstructible observables. These state-level properties are applied after a finite reconstruction class has been fixed.

The principle is local on the physical quotient. A slice, Hilbert chart, or Kuranishi chart is fixed whenever derivatives on the quotient by gauge, diffeomorphism, or unitary equivalence are used. Spectral projections are understood in the perturbative sense of [8]. The open-space Fredholm comparison used later is supplied by the Callias mechanism [9] and its geometric form [10].

2.1. Observation, Reconstruction, and Physical Fixed Points

At reconstruction depth i, the retained description associated with $\mathsf{\Phi }$ is

$$X_{\le i}\left(\mathsf{\Phi }\right)=\left({\mathcal{A}}_{\mathsf{\Phi }};{\mathfrak{A}}_{\le i}^{\mathrm{rec}},{\pi }_i,{\mathcal{H}}_i,{\mathcal{M}}_{\le i}\right),$$

(2)

where ${\mathfrak{A}}_{\le i}^{\mathrm{rec}}$ is the finite algebra generated at that depth, ${\pi }_i$ is its faithful representation, and ${\mathcal{M}}_{\le i}$ is the marking already defined by the preceding reconstruction entries. The marking contains sector projections, grading and level operators, determinant tensors, and covariant transports. Selection witnesses, local frames, and auxiliary multipliers are not included unless a downstream dependence is retained through a protected invariant. The retained coordinates are exactly the entries displayed in (2); $\mathsf{\Phi }$ labels the source point of the observation map.

Let $\mathfrak{D}$ be a local space of admissible full retained descriptions of the form (2), containing the observed descriptions $X\left(\mathsf{\Phi }\right):=X_{\le N}\left(\mathsf{\Phi }\right)$. Observation and reconstruction are written as

$$\mathcal{O}:\mathfrak{M}⟶\mathfrak{D},\mathsf{\Phi }⟼X\left(\mathsf{\Phi }\right),\mathcal{R}:\mathfrak{D}⟶\mathfrak{M}.$$

(3)

Exact local reconstruction means $\mathcal{R}\circ \mathcal{O}={id}_{\mathfrak{M}}$. Hence $\mathcal{O}$ is injective, and $\mathcal{O}\circ \mathcal{R}=id$ on $im\mathcal{O}$. With $\mathcal{P}=\mathcal{O}\circ \mathcal{R}$, one has ${\mathcal{P}}^2=\mathcal{P}$ and $im\mathcal{P}=im\mathcal{O}$. Reconstructible descriptions are therefore the fixed points of $\mathcal{P}$, with their finite entries represented in the reconstruction core defined below.

The physical content of the principle lies in the choice of observable generators, canonical actions, and closure relations. Once these data are fixed, exact reconstruction removes descriptions which cannot be recovered from their retained observables. Variational, geometric, Gauss, spectral, determinant, and state-level laws enter as distinct types of closure.

2.2. Filtered Residuals and Lexicographic Closure

A reconstructed configuration determines a residual section over $\mathfrak{C}/\mathcal{G}$. At finite depth it is written as

$$\mathcal{R}\left(\mathsf{\Phi }\right)=\left(R_0\left(\mathsf{\Phi }\right),R_1\left(\mathsf{\Phi }\right),\dots ,R_N\left(\mathsf{\Phi }\right)\right),$$

(4)

where $R_i\left(\mathsf{\Phi }\right)$ takes values in a Hilbert obstruction bundle ${\mathcal{E}}_i\to \mathfrak{C}/\mathcal{G}$. The index records definability depth. The residuals are grouped into structural, realization, and completed classes,

$$\mathcal{R}=\left({\mathcal{R}}_{\mathrm{str}};{\mathcal{R}}_{\mathrm{real}};{\mathcal{R}}_{\mathrm{comp}}\right).$$

(5)

The structural entries fix the finite reconstructed object. The realization entries attach it to geometric and analytic data. The completed entries act on the already selected low representative.

For the primitive branch used below, the active ordered residual is

$${\mathcal{R}}_{\mathrm{prim}}=\left(R_{\mathrm{geom}},R_{\mathrm{link}},R_{\mathrm{Gauss}},R_{\mathrm{vis}},R_{\mathrm{ext}},R_{\det },R_{B-L},R_{\mathrm{fin}},R_{\mathrm{par}},R_{\mathrm{tor}},R_{\mathrm{gap}},R_{\mathrm{cov}},R_{\mathrm{Schur}},R_{\mathrm{RG}}\right).$$

(6)

The geometric and link entries fix the resolved defect. The Gauss, visibility, minimal finite, exterior, determinant, and finite-shadow entries fix the structural carrier and its finite shadow. Degree fidelity is derived on the primitive minimal support. The gap and covariance entries attach the shadow to an isolated low bundle. The Schur and scale entries belong to the completed branch.

For each component of (4), introduce an auxiliary multiplier ${\mathsf{\Lambda }}_i\in {\mathcal{E}}_i$ and put

$${\mathfrak{L}}_i(\mathsf{\Phi },{\mathsf{\Lambda }}_i)={\displaystyle \frac{1}{2}}{\parallel R_i\left(\mathsf{\Phi }\right)\parallel }_{{\mathcal{E}}_i}^2+Re{\langle {\mathsf{\Lambda }}_i,R_i\left(\mathsf{\Phi }\right)\rangle }_{{\mathcal{E}}_i}.$$

(7)

On geometric obstruction bundles the Hilbert structure attached to the corresponding analytic problem is retained. On finite matrix obstruction modules represented in a multiplicity-free reconstruction core, an independent finite kinetic form is retained only when it is supplied as protected physical data. In its absence, the minimal generated finite metric is the normalized represented trace, ${\langle X,Y\rangle }_{\mathrm{fin}}={\tau }_{\mathrm{rep}}\left(X^{†}Y\right)$. This convention is applied after finite-core descent. At a closed residual, the quadratic part of $J_i={\displaystyle \frac{1}{2}}{\parallel R_i\parallel }^2$ has Hessian

$$H_i^{\mathrm{Gram}}=D{R}_i^{†}{G}_{i}D{R}_i,$$

(8)

where $G_i$ is the retained Hilbert metric on ${\mathcal{E}}_i$. Stationary elimination is applied to this quadratic form after the protected quotient has been taken.

Variation in ${\mathsf{\Lambda }}_i$ imposes $R_i=0$. The exact strata are defined recursively by

$$Z_{-1}:=\mathfrak{C}/\mathcal{G},Z_i=\left\{\left[\mathsf{\Phi }\right]\in Z_{i-1}:R_i\left(\mathsf{\Phi }\right)=0\right\},0\le i\le N.$$

(9)

A configuration is self-reconstructing to depth N when $\left[\mathsf{\Phi }\right]\in Z_N$.

Assume that $Z_{i-1}$ is represented near $\mathsf{\Phi }$ by a clean submanifold or a Kuranishi chart, and let ${\mathsf{\Pi }}_i$ be the orthogonal projection onto $T_{\mathsf{\Phi }}{Z}_{i-1}$ in the chosen slice. The Euler-reconstruction generator at depth i is

$${\mathfrak{E}}_i(\mathsf{\Phi },{\mathsf{\Lambda }}_i)=\left(R_i\left(\mathsf{\Phi }\right),{\mathsf{\Pi }}_{i}D{R}_i{\left(\mathsf{\Phi }\right)}^{*}\left(R_i\left(\mathsf{\Phi }\right)+{\mathsf{\Lambda }}_i\right)\right).$$

(10)

The ordered Euler-reconstruction equations are

$${\mathfrak{E}}_i(\mathsf{\Phi },{\mathsf{\Lambda }}_i)=0,0\le i\le N.$$

(11)

Their projection to the configuration variable is $Z_N$. Every point of $Z_i$ solves the i-th pair with ${\mathsf{\Lambda }}_i=0$, while multiplier variation imposes the residual relation itself.

For a local regular estimate, put $J_i={\displaystyle \frac{1}{2}}{\parallel R_i\parallel }^2$. If $A_i=D\left(R_i{|}_{Z_{i-1}}\right)$ and $A_i{A}_i^{*}\ge {\kappa }_i\mathbf{1}$ on the relevant obstruction space, then

$${∥A_i^{*}{R}_i∥}^2\ge {\kappa }_i{∥R_i∥}^2.$$

(12)

Thus no nonzero stationary residual occurs in this regular neighborhood. Weighted penalty functionals may be used after the exact order (9) has been fixed. The order prevents a later completion from changing an earlier structural selection.

Table 1. Reconstruction depths and their logical status in the primitive branch.

Depth	Principal entries	Reconstructed output	Status
geometric	$R_{\mathrm{geom}},R_{\mathrm{link}}$	resolved defect, projective link, and positive class	structural
source	$R_{\mathrm{Gauss}}$ and fidelity	central labels and covariant boundary charges	structural
graded finite	$R_{\mathrm{vis}}$ and finite descent	minimal marked support, derived level projections, and generated core	structural
Clifford-determinant	$R_{\mathrm{ext}},R_{\det },R_{B-L}$	exterior package, determinant group, and Abelian grading plane	structural
finite shadow	$R_{\mathrm{fin}},R_{\mathrm{par}},R_{\mathrm{tor}}$	full coefficient shadow, parity sector, and torsor cycle	structural and conditional
analytic	$R_{\mathrm{gap}},R_{\mathrm{cov}}$	isolated covariant low bundle	conditional
completed	$R_{\mathrm{Schur}}$ with spectral and relative-basis subdepths, and $R_{\mathrm{RG}}$	effective matrix elements, reduced basis response, and scale comparison	completed

Table 1. Reconstruction depths and their logical status in the primitive branch.

Depth	Principal entries	Reconstructed output	Status
geometric	$R_{\mathrm{geom}},R_{\mathrm{link}}$	resolved defect, projective link, and positive class	structural
source	$R_{\mathrm{Gauss}}$ and fidelity	central labels and covariant boundary charges	structural
graded finite	$R_{\mathrm{vis}}$ and finite descent	minimal marked support, derived level projections, and generated core	structural
Clifford-determinant	$R_{\mathrm{ext}},R_{\det },R_{B-L}$	exterior package, determinant group, and Abelian grading plane	structural
finite shadow	$R_{\mathrm{fin}},R_{\mathrm{par}},R_{\mathrm{tor}}$	full coefficient shadow, parity sector, and torsor cycle	structural and conditional
analytic	$R_{\mathrm{gap}},R_{\mathrm{cov}}$	isolated covariant low bundle	conditional
completed	$R_{\mathrm{Schur}}$ with spectral and relative-basis subdepths, and $R_{\mathrm{RG}}$	effective matrix elements, reduced basis response, and scale comparison	completed

For the primitive branch, the structural and completed dependencies are represented by

$$\begin{array}{c}{\mathfrak{E}}_{\mathrm{link}}⟶{\mathfrak{E}}_{\mathrm{Gauss}}⟶{\mathfrak{E}}_{\mathrm{vis}}⟶{\mathfrak{E}}_{\mathrm{red}}⟶{\mathfrak{E}}_{\mathrm{ext}}⟶{\mathfrak{E}}_{\det }⟶{\mathfrak{E}}_{B-L}⟶{\mathfrak{E}}_{\mathrm{fin}}⟶{\mathfrak{E}}_{\mathrm{par}}⟶{\mathfrak{E}}_{\mathrm{tor}},\\ {\mathfrak{E}}_{\mathrm{gap}}⟶{\mathfrak{E}}_{\mathrm{cov}}⟶{\mathfrak{E}}_{\mathrm{Schur}}⟶{\mathfrak{E}}_{\mathrm{RG}}.\end{array}$$

(13)

The first line fixes the finite boundary cycle. The second acts after this cycle has been selected. Here ${\mathfrak{E}}_{\mathrm{red}}$ denotes finite descent of the marked visible support. On the primitive minimal core, degree fidelity follows from Theorem 1. Within ${\mathfrak{E}}_{\mathrm{Schur}}$, sectoral spectral data are closed before relative-basis data; ${\mathfrak{E}}_{\mathrm{mix}}$ and ${\mathfrak{E}}_{\mathrm{bal}}^{\ell }$ denote channel subentries at those completed subdepths.

2.3. Standard Physical Laws as Reconstruction Closures

Standard physical equations are admitted as closure classes when their defining data are retained by the observation map. Their physical content is inherited from the corresponding variational, geometric, constraint, or spectral law; the reconstruction acts on the retained representation and its closure. A variational closure is obtained by taking a residual to be the first variation of an action on the chosen slice. An Einstein closure is obtained from the geometric residual $G_{\mu \nu }-\kappa T_{\mu \nu }$, while a Gauss closure is obtained from a moment-map or charge constraint. These examples may be written schematically as

$$R_{\mathrm{var}}={\displaystyle \frac{\delta S}{\delta \mathsf{\Phi }}},R_{\mathrm{Ein},\mu \nu }=G_{\mu \nu }-\kappa T_{\mu \nu },R_{\mathrm{Gauss}}=\mu -{\mu }_0.$$

(14)

The equations $R_{\mathrm{var}}=0$, $R_{\mathrm{Ein}}=0$, and $R_{\mathrm{Gauss}}=0$ are then entries of (11). The choice of action, source tensor, and constraint surface remains part of the reconstructed physical description.

For a Levi-Civita geometric entry, the differential Bianchi identities are relations of the generated curvature algebra. With the contracted identity ${\nabla }^{\mu }{G}_{\mu \nu }=0$, the Einstein residual satisfies

$${\nabla }^{\mu }{R}_{\mathrm{Ein},\mu \nu }=-\kappa {\nabla }^{\mu }{T}_{\mu \nu }.$$

(15)

Hence covariant source conservation is the compatibility condition for exact Einstein closure. The curvature conventions are those of [11]. The Bianchi identities remain geometric identities, while the field equation and source conservation occupy the dynamical and compatibility entries of the reconstruction.

Gauss constraints are treated on the reduced charge surface. In a local observable representation, the equivalent boundary-charge form is $[A,Q]=0$ for every retained local observable A and boundary charge Q. The corresponding sector projections are then central in the relative observable algebra, in the standard superselection reading of [12]. This form is used for the primitive defect below.

Fredholm closure fixes a stable index class and an isolated low sector. Determinant closure fixes the vanishing of the relevant local curvature obstruction and, when imposed globally, the corresponding holonomy obstruction. The determinant-family comparison is supplied by [13], while the differential-K refinement is represented by [14]. Odd finite insertions in the completed operator admit the superconnection organization of [15]. These constructions are retained as residual classes rather than as additional independent principles.

Table 2. Standard physical laws represented as reconstruction closures.

Physical structure	Reconstructed datum	Closure or compatibility	Status
least action	action and first variation	$\delta S/\delta \mathsf{\Phi }=0$	variational
Einstein equation	metric and source tensor	$G_{\mu \nu }-\kappa T_{\mu \nu }=0$	geometric and variational
Bianchi compatibility	Levi-Civita curvature algebra	${\nabla }^{\mu }{G}_{\mu \nu }=0$	geometric identity
source conservation	covariant source data	${\nabla }^{\mu }{T}_{\mu \nu }=0$	compatibility
Gauss law	moment map or boundary charge	constraint or central sector closure	constraint
Fredholm stability	operator family and low window	index and gap closure	analytic
determinant consistency	determinant line and connection	curvature and holonomy closure	determinant

Table 2. Standard physical laws represented as reconstruction closures.

Physical structure	Reconstructed datum	Closure or compatibility	Status
least action	action and first variation	$\delta S/\delta \mathsf{\Phi }=0$	variational
Einstein equation	metric and source tensor	$G_{\mu \nu }-\kappa T_{\mu \nu }=0$	geometric and variational
Bianchi compatibility	Levi-Civita curvature algebra	${\nabla }^{\mu }{G}_{\mu \nu }=0$	geometric identity
source conservation	covariant source data	${\nabla }^{\mu }{T}_{\mu \nu }=0$	compatibility
Gauss law	moment map or boundary charge	constraint or central sector closure	constraint
Fredholm stability	operator family and low window	index and gap closure	analytic
determinant consistency	determinant line and connection	curvature and holonomy closure	determinant

The common role of these entries is exact recoverability from the retained physical data. Their specific form is inherited from the corresponding variational, geometric, constraint, or spectral structure. The reconstruction framework supplies their order, compatibility, and minimal reconstructed representation.

2.4. Faithful Reconstruction and Protected Descent

Let ${\mathsf{\Lambda }}_{\mathrm{pr}}$ be the finite set of active principal charge types. Their label algebra and covariant coefficient module are

$${\mathcal{L}}_{\partial }^{\mathrm{pr}}=\underset{a\in {\mathsf{\Lambda }}_{\mathrm{pr}}}{⨁}\mathbb{C}{e}_a,{\mathcal{Q}}_{\partial ,\mathrm{cov}}^{\mathrm{pr}}=\underset{a\in {\mathsf{\Lambda }}_{\mathrm{pr}}}{⨁}{V}_a.$$

(16)

Let ${\mathfrak{A}}_F$ be the finite support algebra at the Gauss depth and let $Z_F=Z\left({\mathfrak{A}}_F\right)$. The marked sector map is a possibly nonunital *-homomorphism

$${\zeta }_F:{\mathcal{L}}_{\partial }^{\mathrm{pr}}⟶Z_F,{\zeta }_F\left(e_a\right)=p_a.$$

(17)

The images are orthogonal central projections. For each active type, let ${\iota }_{F,a}:V_a\to p_a{\mathfrak{A}}_F{p}_a$ be the equivariant coefficient encoding and let ${\rho }_{F,a}:p_a{\mathfrak{A}}_F{p}_a\to V_a$ be its equivariant reconstruction map. With a marked label reconstruction ${\lambda }_F:Z_F\to {\mathcal{L}}_{\partial }^{\mathrm{pr}}$, the fidelity defect is

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

(18)

Thus ${\delta }_{\mathrm{fid}}\left(F\right)=0$ gives faithful sector labeling and faithful reconstruction of every retained coefficient module. No preferred support block follows from fidelity alone.

2.4.1. Admissible Generated Extensions and Protected Descent

At structural depth i, let ${\mathcal{G}}_{\le i}$ be the retained observable coordinates and canonical finite actions. The reconstructed algebra is

$${\mathfrak{A}}_{\le i}^{\mathrm{rec}}=C^{*}\left({\mathcal{G}}_{\le i}|R_0=\dots =R_i=0\right).$$

(19)

No additional finite generator is inserted. A faithful representation ${\pi }_i:{\mathfrak{A}}_{\le i}^{\mathrm{rec}}\to End\left({\mathcal{H}}_i\right)$ is reconstructively complete when

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

(20)

A presentation before finite descent is written as ${\tilde{X}}_i=(X_i,w_i)$, where $X_i$ denotes the coordinates proposed for retention and $w_i$ collects selection frames, coefficient embeddings, intertwiners, and other auxiliary presentation data.

Definition 1

(Protected reconstruction interface). On the exact stratum at depth i, the protected interface is

$${\mathcal{P}}_i\left({\tilde{X}}_i\right)=\left({\mathfrak{A}}_{\le i}^{\mathrm{rec}},{\mathcal{M}}_{\le i},{\mathcal{I}}_i^{\mathrm{down}}\right),$$

(21)

where ${\mathcal{I}}_i^{\mathrm{down}}$ is the specified canonical data exported to the later entries of the closure cascade. This list is fixed before descent. Two presentations are protected-equivalent when their interfaces are related by a marking-preserving *-isomorphism intertwining the exported data. A later residual or canonical operator is basic when it factors through this equivalence, up to the induced unitary covariance in a finite-core representation.

Definition 2

(Admissible generated extensions). Fix an exact stratum at depth i and the ambient envelope ${\mathcal{U}}_{i+1}$ specified for the next reconstruction depth in Table 3. Open analytic conditions, including spectral isolation and non-degeneracy, are imposed in the choice of stratum or spectral corner before the finite descent. Canonical graph or image data required for later basicness are included in ${\mathcal{I}}_i^{\mathrm{down}}$ before the admissible class is formed.

Let ${\mathbf{Adm}}_{i+1}\left({\mathcal{P}}_i\right)$ be the poset of marking-preserving protected subextensions E with ${\mathcal{P}}_i\subseteq E\subseteq {\mathcal{U}}_{i+1}$, regarded as a thin category. An object is admissible when it contains the required source and observable slots, is closed under the canonical operations available at depth $i+1$, satisfies the closed residual relations on the chosen stratum, and preserves the external physical data defining the reconstruction class. A morphism $E\to E^{\prime }$ is present precisely when $E\subseteq E^{\prime }$.

Proposition 1

(Initial minimal reconstruction). On a fixed exact stratum, every nonempty family in ${\mathbf{Adm}}_{i+1}\left({\mathcal{P}}_i\right)$ has meet given by componentwise intersection. Hence every nonempty admissible poset has a unique least element,

$${\mathcal{P}}_{i+1}^{min}=Init{\mathbf{Adm}}_{i+1}\left({\mathcal{P}}_i\right)=\bigcap _{E\in Ob{\mathbf{Adm}}_{i+1}\left({\mathcal{P}}_i\right)}E.$$

(22)

An empty admissible poset records incompatibility of the closure data on the chosen stratum.

Proof. 

The protected interface, the required new data, and the fixed external data belong to every admissible object. Canonical operations preserve intersections, the closed residual relations restrict to protected subextensions, and the required downstream graph or image data have already been included in the protected interface. The intersection is therefore admissible and is contained in every admissible object. □

For fixed ${\mathcal{U}}_{i+1}$, minimal reconstruction is the associated closure operation: if D is a protected datum, ${Cl}_{i+1}\left(D\right)$ is the intersection of the admissible objects containing D. It is extensive, monotone, and idempotent. In particular, enlarging ${\mathcal{I}}_i^{\mathrm{down}}$ can only enlarge the retained object. If a later map is not basic, its required graph or image datum is added to ${\mathcal{I}}_i^{\mathrm{down}}$ and the least admissible extension is recomputed. The choice of action, source tensor, constraint surface, or an explicitly retained external coupling fixes the reconstruction class and hence the admissible poset.

Definition 3

(Finite reconstruction core). The marked generated algebra $\left({\mathfrak{A}}_{\le i}^{\mathrm{rec}},{\mathcal{M}}_{\le i}\right)$ together with a faithful representation satisfying (20) and representing the initial object (22) is called the finite reconstruction core at depth i. Equivalence is marking-preserving unitary equivalence inducing the same protected interface.

Proposition 2

(Finite-core uniqueness). Every finite-dimensional $C^{*}$-algebra has, up to marking-preserving unitary equivalence, a unique faithful representation satisfying (20).

Proof. 

For $\mathfrak{A}\simeq {⨁}_a{M}_{n_a}\left(\mathbb{C}\right)$ and $\mathcal{H}\simeq {⨁}_a{\mathbb{C}}^{n_a}\otimes {\mathbb{C}}^{m_a}$, faithfulness gives $m_a\ge 1$. The commutant on the a-th block is ${\mathbf{1}}_{n_a}\otimes M_{m_a}\left(\mathbb{C}\right)$, while the represented center is scalar on the multiplicity factor. Equation (20) is therefore equivalent to $m_a=1$ for every labeled simple block. □

Proposition 3

(No unsourced multiplicity on a fixed stratum). Let $E^{min}$ be the least admissible extension on a fixed exact or isolated spectral stratum. An additional copy of a represented irreducible channel which is not generated by the retained source and canonical operations, is not required by ${\mathcal{I}}^{\mathrm{down}}$, and may be deleted without leaving the fixed stratum is absent from $E^{min}$.

Proof. 

Deleting such a copy gives an admissible protected subextension. A least admissible object cannot contain a proper removable extension of this type. □

Proposition 4

(Protected presentation descent). Assume that the entries preceding depth i are closed. An auxiliary presentation datum whose required invariants are already contained in (21) and whose deletion leaves the initial object (22) unchanged is absent from the finite reconstruction core.

Proof. 

Such a datum occurs only in admissible presentations strictly larger than the initial object. The finite reconstruction core represents the initial object, so the auxiliary datum is absent after protected equivalence has been taken. □

Remark 1

(Integral reconstruction refinement). At a depth where the protected interface carries a canonical integral or cyclotomic form ${\mathsf{\Lambda }}_i$, the admissible category may be formed before complexification. Exact preservation of ${\mathsf{\Lambda }}_i$ is used when compatible with the closed residuals. If a fixed downstream spectral corner requires a commensurate enlargement, the finite-index enlargement of least admissible index is retained. This refinement is invoked only when the integral form itself is protected.

Lemma 1

(Penalty reduction after protected descent). Let x be retained coordinates and h auxiliary coordinates eliminated by stationary reduction on a local slice after the protected quotient has been taken. For a twice differentiable penalty $J_i(x,h)$, define

$$J_i^{\mathrm{red}}\left(x\right)=\underset{h}{inf}{J}_i(x,h).$$

(23)

Write the Hessian along the minimizing branch in the $(x,h)$ split as $H=\left(H_{\alpha \beta }\right)$. Assume that $h_{*}\left(x\right)$ is regular and that $H_{hh}$ is positive definite. At a critical point of $J_i^{\mathrm{red}}$,

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

(24)

Proof. 

The stationary equation in the eliminated direction gives $\delta h=-H_{hh}^{-1}{H}_{hx}\delta x$ to first order. Substitution into the quadratic part of $J_i$ gives (24). □

Minimal reconstruction is fixed before stationary elimination. An auxiliary normal or high-sector coordinate may then be eliminated while every induced datum required downstream is exported to the protected interface. In particular, the reduced Hessian (24), a Schur-compressed operator, and any required determinant, Pfaffian, threshold, or holonomy datum generated by the eliminated block are retained.

The completed direct entry is filtered before Hermitian spectral reduction. In the branch used below the internal order is

$${\mathfrak{E}}_{\mathrm{dir},\mathrm{ch}}\prec {\mathfrak{E}}_{\mathrm{dir},\det }\prec {\mathfrak{E}}_{\mathrm{dir},\mathrm{spec}}\prec {\mathfrak{E}}_{\mathrm{dir},\mathrm{tr}}\prec {\mathfrak{E}}_{\mathrm{neu}}\prec {\mathfrak{E}}_{\mathrm{mix}}.$$

(25)

The chiral direct maps are retained before their determinant data are exported at ${\mathfrak{E}}_{\mathrm{dir},\det }$. Hermitian spectral data are then reduced before an independent family orientation is retained. Relative-basis data are added only at the later transport or mixing depth. This refinement does not alter the structural carrier.

A transition to depth $i+1$ uses the least admissible extension on the exact stratum, or a spectral corner selected by the next closed residual before the finite descent. Sector projections remain central at the depth at which they label superselection sectors. After covariant transports have been adjoined, they remain marked projections in the corresponding covariant algebra.

The filtration fixes the order in which residual classes, ambient slots, and canonical operations become available. On each preceding exact stratum, the least admissible extension is formed before the next residual is evaluated. Degree fidelity is imposed before finite descent when independent; in the primitive two-channel Borel-Weil class it is the derived closure of Theorem 1.

On an open set $\mathcal{U}\subset \mathfrak{M}$ on which the finite marked support type is constant, put ${\mathfrak{D}}_{\mathcal{U}}:={\mathcal{R}}^{-1}\left(\mathcal{U}\right)$. Let $q_a:\mathcal{U}\to V_a$ be the coefficient coordinate of the active principal type a, and let ${pr}_{F,a}:{\mathfrak{D}}_{\mathcal{U}}\to p_a{\mathfrak{A}}_F{p}_a$ be its observed finite coordinate. A marked Gauss reconstruction chart is one in which the principal label algebra is retained as a coordinate summand, with observation and reconstruction restrictions ${\zeta }_F$ and ${\lambda }_F$, and in which

$${\left.{pr}_{F,a}\circ \mathcal{O}\right|}_{\mathcal{U}}={\iota }_{F,a}\circ q_a,{\left.q_a\circ \mathcal{R}\right|}_{{\mathfrak{D}}_{\mathcal{U}}}={\rho }_{F,a}\circ {pr}_{F,a},a\in {\mathsf{\Lambda }}_{\mathrm{pr}}.$$

(26)

Proposition 5

(Gauss fidelity from marked local reconstruction). Assume that the self-reconstruction loop is a local identity on a marked Gauss reconstruction chart and that every active coefficient map $q_a$ is nonzero at some point of the chart. Then ${\lambda }_F\circ {\zeta }_F={id}_{{\mathcal{L}}_{\partial }^{\mathrm{pr}}}$ and

$${\rho }_{F,a}\circ {\iota }_{F,a}={id}_{V_a},a\in {\mathsf{\Lambda }}_{\mathrm{pr}},$$

(27)

and consequently

$$ker{\zeta }_F=0,ker{\iota }_{F,a}=0,{\delta }_{\mathrm{fid}}\left(F\right)=0.$$

(28)

Proof. 

Restriction of $\mathcal{R}\circ \mathcal{O}=id$ to the marked label summand gives ${\lambda }_F\circ {\zeta }_F=id$. Put $T_a={\rho }_{F,a}\circ {\iota }_{F,a}$. Equation (26) and exact reconstruction give $T_a{q}_a=q_a$. Equivariance and irreducibility give $T_a=c_a{id}_{V_a}$. A nonzero value of $q_a$ gives $c_a=1$, and the remaining statements follow from (18). □

Proposition 5 supplies faithful reconstruction of the active principal coefficient modules on a marked Gauss chart. At later finite depths, initiality gives ${Vis}_{\mathrm{pr}}\left(F_i\right)\subseteq {Src}_{\le i}$ for independently marked visibility channels not generated by the retained operations. Representation types generated by retained products or commutators remain in the generated algebra and do not define additional principal channels. The completed specialization is applied after the low window and its grading data have been fixed.

2.5. Filtration-Preserving Reconstruction

Let $Q={⨁}_{\ell \in \mathsf{\Lambda }}{Q}_{\ell }$ be a finite graded source module with spectral projections ${\mathsf{\Pi }}_{\ell }$ and degree operator $N_Q={\sum }_{\ell \in \mathsf{\Lambda }}\ell {\mathsf{\Pi }}_{\ell }$. Let F be a finite reconstructed support with a self-adjoint level operator $N_F$, and let $\zeta$ represent the source labels in the center of the finite support algebra. Exact filtered reconstruction may be imposed by

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

(29)

The condition $R_{\mathrm{fil}}=0$ identifies the represented source-grade projections with the spectral projections of the reconstructed level. Functional calculus then preserves every function of the retained degree operator.

Where independent, this closure is imposed before finite descent. Visibility determines where a source type can act, while degree fidelity determines which spectral block carries its retained order. Selection witnesses may then be quotiented after the degree-faithful support has been fixed. In the primitive two-channel Borel-Weil class, central sector separation and minimality instead imply (29); the resulting spectral projections and their functional calculus belong to the protected interface.

The same ordering applies to later filtered completions. Once the carrier grading, determinant degree, parity sector, and low spectral projection have been fixed, Schur terms may deform the effective low operator while preserving these marked data. Every later map must factor through the earlier protected quotient. A coordinate removed at an earlier depth cannot be reintroduced as an independent later channel without enlarging the earlier interface and defining a different reconstruction class.

2.6. State-Level Reconstruction, Uncertainty, and Thermodynamic Direction

Let $\mathfrak{S}\left(\mathcal{A}\right)$ be the state space of a unital observable algebra. For $\omega \in \mathfrak{S}\left(\mathcal{A}\right)$ and $A=A^{†}$, put ${Var}_{\omega }\left(A\right)=\omega \left(A^2\right)-\omega {\left(A\right)}^2$ and $A^{\circ }=A-\omega \left(A\right)\mathbf{1}$. Positivity and the Cauchy-Schwarz inequality give

$${Var}_{\omega }\left(A\right){Var}_{\omega }\left(B\right)\ge {\displaystyle \frac{1}{4}}{\left|\omega \left(\right[A,B\left]\right)\right|}^2+{\displaystyle \frac{1}{4}}{\left|\omega \left(\{A^{\circ },B^{\circ }\}\right)\right|}^2.$$

(30)

This is the Robertson uncertainty relation [16] with the covariance refinement of [17]. Every represented finite reconstruction core is therefore subject to the standard state-space uncertainty bound.

Let ${\mathcal{A}}_X\subseteq End\left({\mathcal{H}}_X\right)$ be the represented algebra of a finite reconstructed description, and let ${\mathcal{N}}_X\subseteq {\mathcal{A}}_X$ be the unital *-subalgebra generated by the coordinates and sector labels retained by the already closed residuals. The unique expectation preserving the represented matrix trace is denoted by ${\mathbb{E}}_X:{\mathcal{A}}_X\to {\mathcal{N}}_X$. It is completely positive in the standard conditional-expectation sense of [18] and satisfies

$${\mathbb{E}}_X^2={\mathbb{E}}_X,{\mathbb{E}}_X\left(\mathbf{1}\right)=\mathbf{1},{\mathbb{E}}_X\left(B_{1}A{B}_2\right)=B_1{\mathbb{E}}_X\left(A\right)B_2$$

(31)

for $A\in {\mathcal{A}}_X$ and $B_1,B_2\in {\mathcal{N}}_X$. Its dual on density matrices is denoted by ${\mathbb{E}}_{X,*}$.

For $\gamma >0$, the expectation determines

$${\mathcal{T}}_t^X=exp\left(\gamma t({\mathbb{E}}_X-id)\right)={\mathbb{E}}_X+e^{-\gamma t}\left(id-{\mathbb{E}}_X\right),t\ge 0.$$

(32)

This is a norm-continuous quantum dynamical semigroup in the sense of [19]. Its dual is trace preserving.

Proposition 6

(State-level reconstruction semigroup). The maps ${\mathcal{T}}_t^X$, $t\ge 0$, form a unital completely positive semigroup. If ${\mathbb{E}}_X\ne id$, no extension of (32) to $t<0$ is positive. If a density matrix σ is invariant under ${\mathcal{T}}_{t,*}^X$, then

$$D\left({\mathcal{T}}_{t,*}^X\left(\rho \right)\parallel \sigma \right)\le D\left({\mathcal{T}}_{s,*}^X\left(\rho \right)\parallel \sigma \right),0\le s\le t.$$

(33)

If ${\mathcal{A}}_X=End\left({\mathcal{H}}_X\right)$, then

$$S\left({\mathcal{T}}_{t,*}^X\left(\rho \right)\right)\ge S\left({\mathcal{T}}_{s,*}^X\left(\rho \right)\right),0\le s\le t.$$

(34)

Proof. 

Idempotence of ${\mathbb{E}}_X$ gives (32). For $t\ge 0$, the map is a convex combination of id and ${\mathbb{E}}_X$, hence it is unital and completely positive.

Assume ${\mathbb{E}}_X\ne id$. Choose a nonzero self-adjoint $K\in ker{\mathbb{E}}_X$. Trace preservation gives $TrK=0$, so K has positive and negative spectrum. After normalization, $A=\mathbf{1}+K$ may be chosen positive with a zero eigenvalue. For $t<0$, the coefficient $e^{-\gamma t}$ is greater than one, and ${\mathcal{T}}_t^X\left(A\right)=\mathbf{1}+e^{-\gamma t}K$ has a negative eigenvalue. Positivity therefore fails.

Equation (33) is the data-processing inequality of [20]. The normalized trace state is invariant in the full matrix case, and (34) follows from its relative-entropy representation. □

For density matrices, put $D(\rho \parallel \sigma )=Tr\left(\rho \right(log\rho -log\sigma \left)\right)$ and $S\left(\rho \right)=-Tr(\rho log\rho )$. The entropy defect from the reconstructible algebra is

$${\mathcal{C}}_X\left(\rho \right)=D\left(\rho \parallel {\mathbb{E}}_{X,*}\rho \right).$$

(35)

This is the finite-dimensional subalgebra-relative entropy used in [21].

Proposition 7

(Entropic reconstruction defect). For every density matrix ρ,

$${\mathcal{C}}_X\left(\rho \right)=S\left({\mathbb{E}}_{X,*}\rho \right)-S\left(\rho \right)\ge 0.$$

(36)

If σ is faithful and ${\mathbb{E}}_{X,*}\sigma =\sigma$, then

$$D(\rho \parallel \sigma )={\mathcal{C}}_X\left(\rho \right)+D({\mathbb{E}}_{X,*}\rho \parallel \sigma ).$$

(37)

Along (32),

$${\mathcal{C}}_X\left({\mathcal{T}}_{t,*}^X\rho \right)\le e^{-\gamma t}{\mathcal{C}}_X\left(\rho \right),t\ge 0.$$

(38)

Proof. 

The trace-preserving expectation is self-adjoint for the represented matrix trace. The bimodule property gives $Tr(\rho log\left({\mathbb{E}}_{X,*}\rho \right))=Tr(\left({\mathbb{E}}_{X,*}\rho \right)log\left({\mathbb{E}}_{X,*}\rho \right))$, which proves (36). The same identity with $log\sigma \in {\mathcal{N}}_X$ gives (37). Finally, (32) writes the evolved state as a convex combination of $\rho$ and ${\mathbb{E}}_{X,*}\rho$, with the same expectation image. Convexity of relative entropy gives (38). States of non-full support follow by regularization with the normalized trace state. □

The inclusion ${\iota }_X:{\mathcal{N}}_X\hookrightarrow {\mathcal{A}}_X$ and the expectation form the exact retraction

$${\mathcal{N}}_X\stackrel{{\iota }_X}{\to }{\mathcal{A}}_X\stackrel{{\mathbb{E}}_X}{\to }{\mathcal{N}}_X,{\mathbb{E}}_X\circ {\iota }_X={id}_{{\mathcal{N}}_X}.$$

(39)

The algebra ${\mathcal{N}}_X$ is the least unital *-subalgebra of ${\mathcal{A}}_X$ containing the retained observable coordinates and sector labels, hence the operator-level initial admissible subsystem. The map in (39) is its canonical trace-preserving completely positive retraction. The entropy defect vanishes precisely on the reconstructible range. Approximate-recovery estimates are supplied by the relative-entropy recovery framework of [22]. Entropy decay for quantum Markov semigroups with nontrivial fixed algebras is developed in [23], while the general von Neumann algebra setting is treated in [24].

Equations (33)-(38) give the thermodynamic direction of the reconstruction semigroup. In the full matrix case the entropy is nondecreasing, while information outside the reconstructible algebra decays. This is the second-law-compatible state reduction associated with ${\mathbb{E}}_X$. The parameter t is operational reconstruction time. Its scale $\gamma$ and any identification with Lorentzian proper time belong to completed dynamical data. The reconstruction depth in (4) remains a definability order. Only the induced conditional expectation and its entropy/Dirichlet form are used in the finite-sector diagnostics below; the semigroup rate and reconstruction-time evolution do not enter the numerical completed branch.

2.7. Reconstruction Interface for the Primitive Branch

The preceding framework is applied below in three stages. The first stage reconstructs the primitive graded finite object. The second attaches it to an Alena-Codazzi collar and to a boundary-admissible low operator. The third completes the isolated low sector by graded Schur data. The interface is summarized in Table 3.

The visibility entry used in the next section is the ${\mathbb{CP}}^1$ Berezin-Toeplitz cutoff of [25], with the review convention of [26]. The exterior and determinant package is compared with the standard finite organization of [27]. These comparison structures are applied after the corresponding generators have been selected by the filtered reconstruction.

The state-level extension is independent of the carrier selection. It applies to every represented finite core and later specializes to the projective-color family algebra. The next section reconstructs the finite Standard-Model structure from the primitive graded defect datum.

3. Standard-Model Structure from a Primitive Graded Defect

The structural reconstruction is applied to a codimension-three timelike defect whose resolved link carries a primitive positive line and two nonzero principal normal grades. The source grading is first reconstructed on a finite Borel-Weil support. Its least marked section-space levels supply the carrier, while the determinant-preserving stabilizer of the resulting graded Hermitian space supplies the compact carrier group. Clifford completion is applied to the same integral grading; the protected Picard translation then generates the finite cocharacter orbit as the initial invariant extension, which is subsequently reduced by the parity lock. The analytic attachment to an isolated low sector is stated separately at the end of the section.

The construction is organized by the structural entries listed in Table 4. Conditions on an Alena-Codazzi collar which realize the geometric and source entries are given in Section 4. Masses, mixing matrices, Pfaffian data, running, and contact coefficients belong to the completed Schur problem.

3.1. Primitive Graded Defect Datum and Reconstruction Assumptions

Let $\mathsf{\Gamma }$ be a smooth embedded future-directed timelike worldline in an oriented and time-oriented four-dimensional Lorentzian branch. Its orthogonal normal bundle $N\mathsf{\Gamma }={\left(T\mathsf{\Gamma }\right)}^{\perp }$ carries the induced orientation and Euclidean metric and has rank three. The real blow-up along $\mathsf{\Gamma }$ has boundary fiber

$$\tilde{X}=[M;\mathsf{\Gamma }],\partial \tilde{X}=S\left(N\mathsf{\Gamma }\right),S{\left(N\mathsf{\Gamma }\right)}_t\simeq S_{\mathsf{\Gamma }}^2.$$

(40)

In an optical Codazzi branch, the two-eigenvalue splitting gives the corresponding pair of shear-free geodesic null directions in the Lorentzian principal plane. The Riemannian two-eigenvalue mechanism is represented by [28], while Lorentzian Codazzi structures and the related $2+2$ optical geometry are represented by [29] and [30]. The projective-spinor reading identifies the link with

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

(41)

The spinor conventions are those of [31] and [32]. The twistor comparison is supplied by [33].

The local complement has

$${\mathbb{R}}^4\setminus \mathbb{R}\simeq \mathbb{R}\times ({\mathbb{R}}^3\setminus \left\{0\right\}),H^2({\mathbb{R}}^4\setminus \mathbb{R};\mathbb{Z})\simeq \mathbb{Z}.$$

(42)

A positive resolved transverse class therefore determines

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

(43)

Positive classes form a monoid under tensor product. Atomicity means that the class is not a sum of two nonzero positive classes. Hence the atomic positive representative is

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

(44)

Higher positive degrees define composite filtered link data. The regularity and removable-singularity background is represented by [34]; the local elliptic convention is the one of [35].

Let $N_{\mathsf{\Gamma }}$ be the oriented rank-three normal fiber. A natural scalar-sector source of transverse order at most two has associated graded in ${Sym}^0\left(N_{\mathsf{\Gamma }}^{*}\right)\oplus {Sym}^1\left(N_{\mathsf{\Gamma }}^{*}\right)\oplus {Sym}^2\left(N_{\mathsf{\Gamma }}^{*}\right)$. After the scalar traces have been separated, its non-scalar part is

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

(45)

The brackets record normal order. For a particular source J, put

$$I\left(J\right)=\left\{\ell \in \{1,2\}:Q_{\ell }\ne 0\right\}.$$

(46)

The spherical tensor convention is that of [36]; the multipole terminology follows [37].

A principal boundary charge is read by a surface pairing on a linking sphere and is independent of the linking radius in a source-free annulus. In a curved collar the pairing is transported by the frozen collar connection and the corresponding adjoint link mode.

Definition 4

(Relative Gauss-local observable algebra). Let ${\mathcal{Q}}_{\partial }$ be the finite boundary-charge algebra generated by the retained principal charges and their quadratic Casimir ${\mathcal{C}}_{\partial }$. A relative Gauss-local observable algebra ${\mathcal{A}}_{\mathrm{loc}}$ is generated by annulus-supported observables A satisfying $[A,Q]=0$ for every $Q\in {\mathcal{Q}}_{\partial }$ and by the marked spectral projections of ${\mathcal{C}}_{\partial }$. The boundary data are fixed and no singular charge density is present in the annulus interior.

Definition 5

(Primitive graded defect datum). A possibly degenerate primitive graded defect datum consists of the link (41), a positive line as in (43), the graded source (45), charges $Q_1\in V_1$ and $Q_2\in V_2$ which may vanish, and a relative Gauss-local algebra in the sense of Definition 4. It is two-channel when $I\left(J\right)=\{1,2\}$. It is primitive after finite reconstruction when the positive class is atomic and its finite structural representation is the core of Definition 3.

The superselection interpretation is the boundary-charge form of the local sector reconstruction represented by [38]. In the realization of Section 4, $Q_1$ is supplied by the first normal current moment and $Q_2$ by the trace-free second Codazzi-Gauss moment.

Lemma 2

(Second-jet reduction). For a natural scalar-sector source of transverse order at most two, removal of the scalar traces gives (45).

Proof. 

The associated graded is contained in the first three symmetric powers of the normal cotangent fiber. Their $Spin\left(3\right)\simeq SU\left(2\right)$ types are $V_0$, $V_1$, and $V_0\oplus V_2$, respectively. Removing the scalar summands gives the stated module. □

Corollary 1

(Second-order branch exhaustion). The possible active sets are

$$I\left(J\right)\in \left\{⌀,\left\{1\right\},\left\{2\right\},\{1,2\}\right\}.$$

(47)

Lemma 3

(Relative Gauss-local boundary centrality). In every finite sector representation of a source-free relative Gauss-local annulus, the marked Casimir projections corresponding to the retained principal charges lie in the center of the represented local algebra.

Proof. 

The Stokes or Green identity identifies the boundary-charge algebra on all linking spheres in the annulus. Exact Gauss closure gives $[A,Q]=0$ for every represented local observable and every $Q\in {\mathcal{Q}}_{\partial }$. Functional calculus gives commutation with the spectral projections of ${\mathcal{C}}_{\partial }$, which are elements of the represented algebra by Definition 4. □

On $V_1\oplus V_2$, the Casimir eigenvalues are 2 and 6. The type projections are

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

(48)

The source Euler operator is

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

(49)

Under marked Gauss reconstruction, ${\mathsf{\Pi }}_a$ is represented by $p_a={\zeta }_F\left(e_a\right)$ in (17).

Lemma 4

(Reconstructive charge-sector separation). Assume that both principal charges are retained and that the fidelity defect (18) vanishes. Then $p_1$ and $p_2$ are nonzero orthogonal central projections and

$${\iota }_{F,a}\left(V_a\right)\subset p_a{\mathfrak{A}}_F{p}_a,a=1,2.$$

(50)

Proof. 

Centrality follows from Lemma 3. Orthogonality follows because ${\zeta }_F$ is a *-homomorphism, while $ker{\zeta }_F=0$ follows from vanishing fidelity. The coefficient support is part of the marked encoding used in (18). □

Toeplitz visibility below is understood equivariantly on the marked finite support. Since $p_a$ reconstructs the Casimir projection ${\mathsf{\Pi }}_a$, each marked sector projection is fixed by the Borel-Weil $SU\left(2\right)$ action.

The structural data used below are collected in Table 4. Minimal reconstruction is fixed by Proposition 1 and is not an additional structural datum. Degree fidelity is derived on the resulting carrier. The full ${\mu }_6$ shadow is the initial invariant extension generated by the protected primitive Picard translation, while parity selection is imposed by the parity-lock residual.

3.2. Rees Grading and the Borel-Weil Section-Space Carrier

For the primitive line, let

$${\mathcal{R}}_{\mathsf{\Gamma }}=\underset{k\ge 0}{⨁}{R}_{\mathsf{\Gamma },k},R_{\mathsf{\Gamma },k}=H^0({\mathbb{CP}}_{\mathsf{\Gamma }}^1,L_{\mathsf{\Gamma }}^k),{\mathsf{E}}_{\mathsf{\Gamma }}{|}_{R_{\mathsf{\Gamma },k}}=k\mathbf{1}.$$

(51)

The source filtration may equivalently be represented by the Rees terms $V_{1}t\oplus V_2{t}^2$, for which $t{\partial }_t$ is represented by (49). The source and section Euler gradings are initially retained separately. Their agreement on the primitive minimal support is established by Theorem 1. The $SU\left(2\right)$ action, chosen section frames, coefficient embeddings, and multiplication of ${\mathcal{R}}_{\mathsf{\Gamma }}$ form the witnessed Borel-Weil presentation used for visibility and support selection. After the least marked support has been fixed, the initial protected extension retains the positive line class, the level projections, the split support, and the Picard translation induced by tensoring with $L_{\mathsf{\Gamma }}$. A chosen section basis or multiplication tensor is a witness unless a downstream canonical operator depends on one of its invariants.

With the Borel-Weil indexing used throughout the paper,

$$E_q=H^0({\mathbb{CP}}_{\mathsf{\Gamma }}^1,L_{\mathsf{\Gamma }}^{q-1})\simeq H^0({\mathbb{CP}}^1,\mathcal{O}(q-1)),q\ge 1.$$

(52)

Thus $E_q\simeq {Sym}^{q-1}{\mathbb{C}}^2$ and $dim{E}_q=q$. The homogeneous-bundle statement is the Borel-Weil result represented by [39]. The same count is obtained from monopole harmonics as in [40]; the Taub-NUT Dirac comparison is supplied by [41].

For the Hermitian endomorphisms of $E_q$,

$${End}_0\left(E_q\right)={\displaystyle \underset{\ell =1}{\overset{q-1}{⨁}}}{V}_{\ell }.$$

(53)

Hence $V_{\ell }$ first occurs on $E_{\ell +1}$. In quantization language this is the ${\mathbb{CP}}^1$ Berezin-Toeplitz cutoff of [25]. The same finite-mode content appears in the fuzzy-sphere model of [42] and in the finite-matrix brane comparison of [43].

Definition 6

(Jet-Borel-Weil degree reconstruction). Let $F={⨁}_q{m}_q{E}_q$ be a finite positive support and let $P_q^F$ be its isotypic projections. Its level operator is

$${\mathsf{N}}_F=\sum _{q\ge 1}(q-1)P_q^F.$$

(54)

For the principal labels in (17), put

$$R_{\mathrm{deg}}(F,{\zeta }_F)={\left({\zeta }_F\left(e_{\ell }\right)-{\mathbf{1}}_{\{\ell \}}\left({\mathsf{N}}_F\right)\right)}_{\ell \in {\mathsf{\Lambda }}_{\mathrm{pr}}}.$$

(55)

The support is degree-faithful when ${\zeta }_F$ is injective, (55) vanishes, the coefficient support satisfies (50), and every active type is Toeplitz visible.

The residual in (55) is the primitive-link instance of (29). It is retained as a diagnostic of the marked level projections.

Lemma 5

(Marked Borel-Weil rank bound). Let $F={⨁}_q{m}_q{E}_q$ and let p be an $SU\left(2\right)$-invariant projection on F. If an injective equivariant map $V_{\ell }\to pEnd\left(F\right)p$ exists, then $rankp\ge \ell +1$. Equality implies $pF\simeq E_{\ell +1}$.

Proof. 

Let $E_{q_{max}}$ be a highest-spin summand of $pF$. The Clebsch-Gordan decomposition bounds the degree of every type in $End\left(pF\right)$ by $q_{max}-1$. The occurrence of $V_{\ell }$ gives $q_{max}\ge \ell +1$, and hence $rankp\ge \ell +1$. Equality leaves the single summand $E_{\ell +1}$. □

Theorem 1

(Initial marked Borel-Weil carrier). Let Γ satisfy entries A and B of Table 4. Atomicity fixes the positive line (44). The marked Borel-Weil rank bound fixes the least admissible sector levels, and the least faithful marked extension has support

$$V_{\mathsf{\Gamma }}=C_{\mathsf{\Gamma }}\oplus W_{\mathsf{\Gamma }},C_{\mathsf{\Gamma }}=E_3=R_{\mathsf{\Gamma },2},W_{\mathsf{\Gamma }}=E_2=R_{\mathsf{\Gamma },1}.$$

(56)

On this support, (55) vanishes and $rank{V}_{\mathsf{\Gamma }}=5$.

Proof. 

The two active source types are retained in nonzero orthogonal central sectors. Lemma 5 gives ranks at least 2 and 3, so the least marked levels are $E_2$ and $E_3$. Their nonzero $V_1$ and $V_2$ images generate $M_2\left(\mathbb{C}\right)$ and $M_3\left(\mathbb{C}\right)$, respectively. The equality case of the rank bound identifies the two marked summands, while initiality excludes an unmarked remainder and Proposition 2 removes representation multiplicity. The level projections are therefore $P_2^F$ and $P_3^F$, which gives the degree closure. □

Corollary 2

(Borel-Weil witness descent). On the minimal marked stratum of Theorem 1, Borel-Weil presentations which induce the same marked split, level operator, positive line class, Picard translation, and exported coefficient invariants are protected-equivalent. Chosen section frames, coefficient embeddings, and multiplication intertwiners are absent from the retained carrier marking. The $SU\left(2\right)$ representation class and any isotypic projection required downstream are retained only through the equivariant data in Definition 1.

Proof. 

The split support and level projections are fixed by Theorem 1. Presentation data absent from the least protected object are removed by Proposition 4; the line class and its Picard translation are retained downstream. □

Proposition 8

(Integer-source block parity and low-block rigidity). The mixed blocks between $E_2$ and $E_3$ carry only half-integer $SU\left(2\right)$ types. Hence the retained integer principal channels act diagonally on (56). No integer $V_{\ell }$ with $\ell \ge 3$ occurs in ${End}_0(E_2\oplus E_3)$, so every such principal moment has zero direct compression to the selected support.

Proof. 

One has ${End}_0\left(E_2\right)=V_1$, ${End}_0\left(E_3\right)=V_1\oplus V_2$, and $Hom(E_2,E_3)\simeq V_{1/2}\oplus V_{3/2}$. The result follows by equivariance. □

3.3. The Determinant-Preserving Carrier Group

The Borel-Weil $SU\left(2\right)$ action and the multiplication of (51) select the support at the witnessed level. By Corollary 2, a chosen Borel-Weil frame or multiplication intertwiner is not retained pointwise. The determinant package is defined on the resulting split Hermitian carrier by its level projections and split top exterior line. The group considered below is the automorphism group of the retained carrier component of this interface; the exported equivariant data are transported rather than pointwise stabilized. Its attachment to an isolated low bundle is governed by the covariance and gap conditions of SubSection 3.7.

Put

$$C:=C_{\mathsf{\Gamma }},W:=W_{\mathsf{\Gamma }},V:=C\oplus W.$$

(57)

The retained level operator is

$$D_{\mathsf{\Gamma }}:={\mathsf{N}}_F{|}_V=2P_C+P_W.$$

(58)

Thus $dimC=3$, $dimW=2$, and $Tr{D}_{\mathsf{\Gamma }}=8$. The primitive trace-free integral degree is

$$H_{\mathsf{\Gamma }}^{\mathrm{deg}}=5D_{\mathsf{\Gamma }}-8\mathbf{1}=2P_C-3P_W.$$

(59)

For $t>0$, writing the carrier Rees dilation as $R_{\mathsf{\Gamma }}\left(t\right)=t^{D_{\mathsf{\Gamma }}}$, its determinant-normalized representative is

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

(60)

Thus the centered integral degree is the infinitesimal generator of the unimodular Rees flow on the selected carrier. It defines the cocharacter ${\lambda }_{\mathsf{\Gamma }}\left(z\right)=z^2{P}_C+z^{-3}{P}_W$ and spans the trace-free central line in $u\left(C\right)\oplus u\left(W\right)$. Stabilization of the split top exterior line imposes

$$det\left(g_C\right)det\left(g_W\right)=1.$$

(61)

The determinant-preserving carrier automorphisms are

$${Aut}_{\mathrm{u},\mathrm{gr},\det }(V,D_{\mathsf{\Gamma }})=\left\{g\in U\left(V\right):[g,D_{\mathsf{\Gamma }}]=0,detg=1\right\}=S(U\left(C\right)\times U\left(W\right)).$$

(62)

The compact carrier group is therefore

$$G_{\mathsf{\Gamma }}=S(U\left(C\right)\times U\left(W\right))\simeq S(U\left(3\right)\times U\left(2\right)).$$

(63)

Corollary 3

(Faithful global carrier form). The effective global form of (63) is

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

(64)

Proof. 

The covering action $(A,B,z)\mapsto (z^{2}A,z^{-3}B)$ has kernel ${\mu }_6$. The finite carrier representation is faithful after quotienting this ineffective kernel, which gives (64). □

The sensitivity of line operators to the global form is discussed in [44].

Remark 2

(Tannakian/Rees reading). The carrier $V_{\mathsf{\Gamma }}$ is a tensor generator of the rigid tensor subcategory generated by $V_{\mathsf{\Gamma }}$, $V_{\mathsf{\Gamma }}^{*}$, the retained grading and determinant tensors, and their subquotients. Its fiber functor to finite-dimensional complex vector spaces carries the Rees grading induced by (59); graded and filtered fiber functors of this type are treated in [45]. The tensor automorphisms preserving the retained grading and determinant marking have Hermitian-preserving subgroup (63). The distinguished Rees cocharacter therefore gives a Mumford-Tate-type reading of the structural reconstruction without introducing another closure datum.

Let $t\in {\mathbb{Z}}_3$ be color triality and $p\in {\mathbb{Z}}_2$ the parity of the weak representation. With the normalization fixed below, descent through (64) is equivalent to

$$Y\equiv {\displaystyle \frac{p}{2}}-{\displaystyle \frac{t}{3}}(mod\mathbb{Z}).$$

(65)

The congruence is the trivial-action condition for the generator of the covering kernel.

3.4. Fock Completion and Canonical Abelian Gradings

Let $\mathcal{F}$ be a finite ${\mathbb{Z}}_2$-graded module and let c be an odd action of $V\oplus V^{*}$. The exterior residual is

$$R_{\mathrm{ext}}\left(c\right)={\left(\left\{c\right(v),c(w\left)\right\},\left\{c\right(\alpha ),c(\beta \left)\right\},\left\{c\right(v),c(\alpha \left)\right\}-\alpha \left(v\right)\mathbf{1}\right)}_{v,w\in V,\alpha ,\beta \in V^{*}}.$$

(66)

Its closure gives a graded representation of $Cl(V\oplus V^{*})$.

Proposition 9

(Commutant-closed Clifford completion). Every finite graded solution of (66) has the form

$$\mathcal{F}\simeq {\mathsf{\Lambda }}^{•}V\otimes {\mathbb{C}}^m$$

(67)

for some $m\ge 1$. Finite-core closure gives $m=1$.

Proof. 

The standard identification $Cl(V\oplus V^{*})\simeq End\left({\mathsf{\Lambda }}^{•}V\right)$ gives the displayed form. Its multiplicity commutant is $M_m\left(\mathbb{C}\right)$, so (20) gives $m=1$. □

Remark 3

(Super-Tannakian Clifford reading). After Clifford completion the reconstructed tensor category is naturally ${\mathbb{Z}}_2$-graded and its fiber functor takes values in finite-dimensional complex supervector spaces. The decomposition into ${\mathsf{\Lambda }}^{\mathrm{even}}V$ and ${\mathsf{\Lambda }}^{\mathrm{odd}}V$ is intrinsic to this grading. The physical chiral half is fixed separately by the chirality orientation used in (68).

The Clifford-module convention is the one of [46]. The two chiral halves are ${\mathsf{\Lambda }}^{\mathrm{even}}V$ and ${\mathsf{\Lambda }}^{\mathrm{odd}}V$. Under the split top-form trivialization they are dual. With the even chirality/exterior-parity orientation, the finite chiral module is

$$F_{\mathsf{\Gamma }}^{\mathrm{even}}={\mathsf{\Lambda }}^{\mathrm{even}}V.$$

(68)

Its dimension is 16. This is the standard exterior package in the $SU\left(5\right)$ and $Spin\left(10\right)$ comparison represented by [27]. Clifford-ideal comparisons are supplied by [47] and [48].

For later finite-Hessian readings, the multiplicity-free Clifford core carries the normalized represented trace. Let $\epsilon \left(e_i\right)$ and $\iota \left(e_i^{*}\right)$ denote creation and contraction in an orthonormal one-particle basis of V. The CAR relations on (68) give

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

(69)

with vanishing mixed trace. Thus all one-particle directions have the same tracial weight. On the finite Clifford-odd obstruction spaces this trace realizes the finite metric specified after (7) when no independent kinetic form is retained. No additional directional weights are introduced at this stage.

Let $N_C$ and $N_W$ be the exterior number operators associated with C and W. The second quantization of $-H_{\mathsf{\Gamma }}^{\mathrm{deg}}/6$ is

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

(70)

The electric charge is $Q=T_3+Y$. The weak-hypercharge superselection comparison is represented by [49].

The exterior completion also has the total number operator $N_{\mathrm{F}}=N_C+N_W$. The $B-L$ degree is

$$B-L=1-{\displaystyle \frac{2}{3}}{N}_C.$$

(71)

The centered Fock number is

$$X_{\mathrm{F}}:=5\mathbf{1}-2N_{\mathrm{F}}=5(B-L)-4Y.$$

(72)

Thus $X_{\mathrm{F}}$ is the $Spin\left(10\right)$-type degree denoted by $X_{\mathrm{Spin}}$, and

$$B-L={\displaystyle \frac{1}{5}}\left(X_{\mathrm{F}}+4Y\right).$$

(73)

The canonical Abelian plane is therefore generated by the second-quantized Borel-Weil degree and the centered Fock number. The basis $(Y,B-L)$ is adapted to the low Schur channels. Related ladder and internal-space comparisons are represented by [50]; algebraic unification variants are represented by [51,52].

The separate baryon and lepton numbers on the finite package are polynomial functions of the color number:

$$B={\displaystyle \frac{1}{6}}{N}_C(3-N_C)(3-2N_C),L=B-(B-L).$$

(74)

On the ordered summands of Table 5, these operators give the standard chiral values for $(N^c,u^c,Q,e^c,d^c,L)$.

The corresponding local readings are the neutral singlet, up-type conjugate, quark doublet, charged singlet, down-type conjugate, and lepton doublet. Their hypercharge-square trace is

$${Tr}_{{\mathsf{\Lambda }}^{\mathrm{even}}V}\left(Y^2\right)={\displaystyle \frac{10}{3}}.$$

(75)

With the fundamental Dynkin index normalized by $T\left(\mathbf{fund}\right)=1/2$, the nonabelian traces are

$$I_3=2,I_2=2,$$

(76)

and hence

$$k_Y={\displaystyle \frac{{Tr}_{{\mathsf{\Lambda }}^{\mathrm{even}}V}\left(Y^2\right)}{I_2}}={\displaystyle \frac{5}{3}}.$$

(77)

Proposition 10

(Finite-trace orthogonal Abelian channel). The centered Fock number is finite-trace orthogonal to hypercharge:

$$\begin{array}{c}{Tr}_{{\mathsf{\Lambda }}^{\mathrm{even}}V}\left(Y(B-L)\right)={\displaystyle \frac{8}{3}},{Tr}_{{\mathsf{\Lambda }}^{\mathrm{even}}V}\left(Y{X}_{\mathrm{F}}\right)=0,\\ {Tr}_{{\mathsf{\Lambda }}^{\mathrm{even}}V}\left(X_{\mathrm{F}}\right)=0,{Tr}_{{\mathsf{\Lambda }}^{\mathrm{even}}V}\left(X_{\mathrm{F}}^2\right)=80.\end{array}$$

(78)

For a split one-particle degree,

$$Y_{a,b}=a{N}_C+b{N}_W.$$

(79)

The infinitesimal top-form condition is

$$3a+2b=0.$$

(80)

The local determinant-obstruction factors on (68) are

$$\begin{array}{cc}\mathrm{obstruction}& \mathrm{degree}\mathrm{factor}\\ SU{\left(3\right)}^{2}U\left(1\right)& 3a+2b\\ SU{\left(2\right)}^{2}U\left(1\right)& 3a+2b\\ {\mathrm{grav}}^{2}U\left(1\right)& 8(3a+2b)\\ U{\left(1\right)}^3& 4(3a+2b)(9a^2+6ab+5b^2)\end{array}$$

(81)

The last quadratic factor is positive definite.

Proposition 11

(Centered-degree determinant kernel). The common local determinant kernel is

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

(82)

Its central direction is generated by (59), and its maximal connected subgroup is (63).

Proof. 

The nonabelian factors vanish on the completed package. On the center, the common local kernel of (81) is (80). The primitive integral solution is (59). This is also the infinitesimal stabilizer of the split top exterior line, and exponentiation gives (63). □

Proposition 12

(Determinant trivialization). Assume that the determinant datum contains a gauge-invariant trivialization of the determinant line of the chiral Clifford completion, compatible with its determinant connection. Then the local vertical determinant obstruction and the determinant holonomy along gauge loops vanish.

Proof. 

The local obstruction is the vertical curvature of the determinant connection and the global obstruction is its gauge-loop holonomy. Both vanish under the stated trivialization. The determinant-family reading is the one of [13]. □

The global $SU\left(2\right)$ parity obstruction is controlled separately by the mod-two index. On the even chiral module (68), it vanishes because the weak-doublet multiplicity is $3+1=4$, as displayed in Table 6.

Proposition 13

(Anomaly-compatible second-order branch check). Among the four branches in (47), with atomic positive class, faithful minimal Borel-Weil reconstruction, commutant-closed exterior completion, and the even chirality/exterior-parity orientation, the two-channel branch is the only branch which contains both nontrivial rank-three and rank-two non-Abelian factors and satisfies the anomaly and $SU\left(2\right)$ parity checks displayed in Table 6. Its line, carrier, group, and chiral module are given by (44), (56), (63), and (68).

Proof. 

For $I\left(J\right)=⌀$, no non-scalar generated support remains. For $I\left(J\right)=\left\{1\right\}$, the carrier is $E_2$ and no rank-three non-Abelian factor is present. For $I\left(J\right)=\left\{2\right\}$, the carrier is $E_3$ and the retained even chiral half has a nonzero cubic $SU{\left(3\right)}^3$ coefficient. The two-channel branch is supplied by Theorem 1; its anomaly coefficients and weak parity are those of Table 6. Finite-core uniqueness excludes an additional source-invisible multiplicity. □

The local characteristic-class convention is represented by [53]. Secondary flat and torsion information belongs to the global determinant problem represented by [54]. The determinant kernel, top-form stabilizer, and graded determinant automorphism group therefore give the same local carrier, while residual flat holonomy remains completed data. The axial phase of the completed quark determinant first enters at completed Schur depth and remains outside this structural determinant closure.

3.5. The Finite Cocharacter Shadow and the Family Module

The cocharacter defined by (59) has finite intersection

$${\mu }_6:={\lambda }_{\mathsf{\Gamma }}\left(U\left(1\right)\right)\cap \left(SU\left(C\right)\times SU\left(W\right)\right)\simeq {\mathbb{Z}}_6.$$

(83)

Indeed, the conditions ${\left(z^2\right)}^3=1$ and ${\left(z^{-3}\right)}^2=1$ are equivalent to $z^6=1$. The primitive generator of $Pic\left({\mathbb{CP}}^1\right)$ is fixed by (44), and its finite coefficient reduction is

$$c_1\left(L_{\mathsf{\Gamma }}\right)⟼{\left[c_1\left(L_{\mathsf{\Gamma }}\right)\right]}_6\in H^2({\mathbb{CP}}^1,{\mathbb{Z}}_6)\simeq {\mathbb{Z}}_6.$$

(84)

Its projective-color projection is

$${\left[c_1\left(L_{\mathsf{\Gamma }}\right)\right]}_6⟼{\left[c_1\left(L_{\mathsf{\Gamma }}\right)\right]}_3\in H^2({\mathbb{CP}}^1,{\mathbb{Z}}_3)\simeq {\mathbb{Z}}_3.$$

(85)

The complementary projection is the coefficient class in $H^2({\mathbb{CP}}^1,{\mathbb{Z}}_2)$.

Let ${\mathcal{T}}_{\mathsf{\Gamma }}^{\left(6\right)}$ be the affine congruence space underlying $Pic\left({\mathbb{CP}}^1\right)/6Pic\left({\mathbb{CP}}^1\right)$. Its vertex algebra is ${\mathcal{B}}_6=C\left({\mathcal{T}}_{\mathsf{\Gamma }}^{\left(6\right)}\right)$. Tensoring by $L_{\mathsf{\Gamma }}$ gives the primitive translation ${\tau }_6$, and tensoring by $L_{\mathsf{\Gamma }}^{*}$ gives its inverse. This tensor translation is a protected monoidal output of the line class; it is independent of the section frame and multiplication intertwiner removed in Corollary 2.

Proposition 14

(Generated finite-shadow orbit). Let $p_x$, $x\in {\mathcal{T}}_{\mathsf{\Gamma }}^{\left(6\right)}$, be the minimal projections of ${\mathcal{B}}_6$ and let $S_6$ implement the protected primitive translation, $S_6{p}_x{S}_6^{*}=p_{{\tau }_6\left(x\right)}$ and $S_6^6=\mathbf{1}$. The least invariant extension of one nonzero finite-shadow sector is the complete six-element orbit. Its covariant algebra is $M_6\left(\mathbb{C}\right)$, and its finite reconstruction core is the regular module

$${\mathcal{H}}_{\mathsf{\Gamma }}^{\left(6\right)}={\ell }^2\left({\mathcal{T}}_{\mathsf{\Gamma }}^{\left(6\right)}\right)\simeq \mathbb{C}\left[{\mathbb{Z}}_6\right].$$

(86)

Proof. 

The primitive translation acts transitively on the six vertices, so every invariant hull containing one nonzero vertex contains the complete orbit. Cyclic covariance makes the vertex ranks equal. The crossed product $C\left({\mathcal{T}}_{\mathsf{\Gamma }}^{\left(6\right)}\right)⋊{\mathbb{Z}}_6$ is $M_6\left(\mathbb{C}\right)$, and finite-core closure removes its representation multiplicity. □

With $S_3=S_6^2$ and $S_2=S_6^3$, the coprime decomposition is

$${\mathcal{H}}_{\mathsf{\Gamma }}^{\left(6\right)}\simeq \mathbb{C}\left[{\mathbb{Z}}_3\right]\otimes \mathbb{C}\left[{\mathbb{Z}}_2\right].$$

(87)

Let $q_j$ denote coefficient reduction to the affine ${\mathbb{Z}}_j$ factor.

Proposition 15

(Fixed-parity torsor fiber). For each $p\in {\mathcal{T}}_{\mathsf{\Gamma }}^{\left(2\right)}$, the fiber

$${\mathcal{T}}_{\mathsf{\Gamma },p}^{\left(6\right)}:=q_2^{-1}\left(p\right)$$

(88)

is an affine torsor for $2{\mathbb{Z}}_6\simeq {\mathbb{Z}}_3$. The restriction

$${\left.q_3\right|}_{{\mathcal{T}}_{\mathsf{\Gamma },p}^{\left(6\right)}}:{\mathcal{T}}_{\mathsf{\Gamma },p}^{\left(6\right)}⟶{\mathcal{T}}_{\mathsf{\Gamma }}$$

(89)

is an equivariant affine bijection.

Proof. 

The kernel of $q_2:{\mathbb{Z}}_6\to {\mathbb{Z}}_2$ is $2{\mathbb{Z}}_6$. It acts freely and transitively on every fiber. The coprime map $(q_3,q_2)$ is an affine bijection, which gives (89). □

Let $\epsilon \in \{1,-1\}$ be a character of the subgroup ${\mu }_2\subset {\mu }_6$. The associated induced module is

$${\mathcal{H}}_{\mathrm{fam}}^{\left(\epsilon \right)}:={Ind}_{{\mu }_2}^{{\mu }_6}\epsilon \simeq {\ell }^2({\mu }_6/{\mu }_2;\epsilon ),dim{\mathcal{H}}_{\mathrm{fam}}^{\left(\epsilon \right)}=3.$$

(90)

The order-three translation generated by $S_6^2$ acts on this module in its regular representation. The construction is intrinsic to the affine quotient and does not require a chosen origin.

Proposition 16

(Parity-character corner). Let ${\mathcal{A}}_6=C\left({\mathcal{T}}_{\mathsf{\Gamma }}^{\left(6\right)}\right)⋊{\mathbb{Z}}_6$ and $P_{\epsilon }={\displaystyle \frac{1}{2}}(\mathbf{1}+\epsilon S_2)$. Then

$$P_{\epsilon }{\mathcal{A}}_6{P}_{\epsilon }\simeq M_3\left(\mathbb{C}\right),P_{\epsilon }{\mathcal{H}}_{\mathsf{\Gamma }}^{\left(6\right)}\simeq {\mathcal{H}}_{\mathrm{fam}}^{\left(\epsilon \right)}.$$

(91)

The compressed vertex algebra is $C\left({\mathcal{T}}_{\mathsf{\Gamma }}\right)$ and the compressed order-three translation is regular.

Proof. 

Under (87), $S_2$ acts on the regular two-dimensional factor. Its character projections have rank one. Compression of $M_3\left(\mathbb{C}\right)\otimes M_2\left(\mathbb{C}\right)$ therefore gives $M_3\left(\mathbb{C}\right)$ and the induced three-dimensional module. □

The fixed-parity fiber, the corner (91), and the induced representation (90) are equivalent realizations of the same projective-color module. Its interpretation as a physical family factor additionally requires the low-sector covariance and gap conditions stated below.

The sign $\epsilon$ is fixed by the parity-lock datum. Once the parity residual (151) is imposed, the corresponding nonzero corner is retained and the complementary regular parity mode is absent from the protected corner.

The parity-character corner is fixed before the connection phase is chosen and does not determine ${\theta }_2$. If phase lifts are retained, the factor phases obey

$${\theta }_3\equiv 2{\theta }_6,{\theta }_2\equiv 3{\theta }_6,3{\theta }_3-2{\theta }_2\equiv 0(mod2\pi ).$$

(92)

Their coherent holonomies satisfy

$${\mathcal{W}}_{\mathrm{ph}}^{\left(6\right)}:=e^{-6i{\theta }_6}=e^{-3i{\theta }_3}=e^{-2i{\theta }_2}.$$

(93)

Corollary 4

(Coherent factor holonomy). When the projective-color coefficient transport is cycle-trivial, its scalar Wilson holonomy is the common value in (93).

Proof. 

The relations in (92) give the equalities in (93). Cycle-trivial coefficient transport leaves this scalar phase as the Wilson holonomy. □

The parity factor has differential $d_2=S_2-e^{i{\theta }_2}\mathbf{1}$ and Laplacian

$${\mathsf{\Delta }}_2=d_2^{†}{d}_2=2\mathbf{1}-2cos\left({\theta }_2\right)S_2,\sigma \left({\nu }_2{\mathsf{\Delta }}_2\right)=\left\{2{\nu }_2(1-cos{\theta }_2),2{\nu }_2(1+cos{\theta }_2)\right\}.$$

(94)

Lemma 6

(Character-adapted weak-parity factor). Let $\epsilon \in \{1,-1\}$ be the retained parity character. For the character-adapted flat phase $e^{i{\theta }_2}=\epsilon$, the retained character has eigenvalue 0 and the complementary character has eigenvalue $4{\nu }_2$.

Proof. 

On the s-character of the regular shift, $s\in \{1,-1\}$, the eigenvalue in (94) is $2{\nu }_2(1-scos{\theta }_2)$. For $e^{i{\theta }_2}=\epsilon$, one has $cos{\theta }_2=\epsilon$. Substitution of $s=\epsilon$ and $s=-\epsilon$ gives the stated values. □

3.6. Torsor Hodge-Spectral Module and Information Geometry

The projective-color torsor is the image of the full affine shadow under $q_3$,

$${\mathcal{T}}_{\mathsf{\Gamma }}=q_3\left({\mathcal{T}}_{\mathsf{\Gamma }}^{\left(6\right)}\right),{\tau }_{\mathsf{\Gamma }}\left(a\right)=a+2{\left[c_1\left(L_{\mathsf{\Gamma }}\right)\right]}_3.$$

(95)

The factor translation is free and transitive. After an auxiliary origin has been chosen, it is represented by $a\mapsto a+1$ on ${\mathbb{Z}}_3$. Let ${\mathcal{B}}_{\mathrm{cen}}=C\left({\mathcal{T}}_{\mathsf{\Gamma }}\right)$ and let $e_a$ be its minimal projections.

Proposition 17

(Finite-core torsor representation). Let a faithful representation of ${\mathcal{B}}_{\mathrm{cen}}$ be equipped with a unitary U implementing ${\tau }_{\mathsf{\Gamma }}$. Its dimension is $3m$ for some $m\ge 1$. If the covariant algebra generated by ${\mathcal{B}}_{\mathrm{cen}}$ and U is represented in the finite reconstruction core, then $m=1$.

Proof. 

The three vertex projections have a common positive rank m by cyclic covariance. The crossed product $C\left({\mathcal{T}}_{\mathsf{\Gamma }}\right)⋊{\mathbb{Z}}_3$ is $M_3\left(\mathbb{C}\right)$. Its multiplicity commutant is removed by (20). □

The resulting central response carrier is

$${\mathcal{H}}_{\mathrm{cen}}={\ell }^2\left({\mathcal{T}}_{\mathsf{\Gamma }}\right)\simeq \mathbb{C}\left[{\mathbb{Z}}_3\right]\simeq {\mathcal{H}}_{\mathrm{fam}}^{\left(\epsilon \right)}.$$

(96)

Here central means that this factor commutes with the retained charge and exterior algebra before the completed family blocks are inserted.

Let ${\mathcal{E}}_{\mathsf{\Gamma }}$ be a finite Hermitian coefficient system over ${\mathcal{T}}_{\mathsf{\Gamma }}$. Its edge resolution is

$$0⟶C^0({\mathcal{T}}_{\mathsf{\Gamma }},{\mathcal{E}}_{\mathsf{\Gamma }})\stackrel{d_{\mathsf{\Gamma }}}{\to }{C}^1({\mathcal{T}}_{\mathsf{\Gamma }},{\mathcal{E}}_{\mathsf{\Gamma }})⟶0.$$

(97)

With the orientation induced by ${\tau }_{\mathsf{\Gamma }}$,

$${\left(d_{\mathsf{\Gamma }}\xi \right)}_{a+1}=U_a{\xi }_a-e^{i{\theta }_{\mathsf{\Gamma }}}{\xi }_{a+1},a\in {\mathbb{Z}}_3.$$

(98)

The projective-color Hodge-spectral module is

$${\mathfrak{H}}_{\mathsf{\Gamma }}^{\mathrm{tor}}=\left(C^{•}({\mathcal{T}}_{\mathsf{\Gamma }},{\mathcal{E}}_{\mathsf{\Gamma }}),d_{\mathsf{\Gamma }},D_{\mathsf{\Gamma }}^{\mathrm{tor}},{\mathsf{\Delta }}_{\mathsf{\Gamma }},{\mathcal{A}}_{\mathrm{fam}}\right),{\mathcal{A}}_{\mathrm{fam}}:=End\left({\mathcal{H}}_{\mathrm{cen}}\right),$$

(99)

where

$$D_{\mathsf{\Gamma }}^{\mathrm{tor}}=\left(\begin{array}{cc}0& d_{\mathsf{\Gamma }}^{†}\\ d_{\mathsf{\Gamma }}& 0\end{array}\right),{\mathsf{\Delta }}_{\mathsf{\Gamma }}={\left(D_{\mathsf{\Gamma }}^{\mathrm{tor}}\right)}^2.$$

(100)

The graph-Hodge comparison is represented by [55]; unitary vector-bundle Laplacians are represented by [56]. Finite spectral-module comparisons are supplied by [57].

Fix a reference vertex and transport the represented boundary-charge algebra to its coefficient fiber. The finite Wilson defect is

$${\mathcal{W}}_{\mathsf{\Gamma }}=e^{-3i{\theta }_{\mathsf{\Gamma }}}{U}_2{U}_1{U}_0.$$

(101)

Let ${\mathfrak{A}}_{\mathsf{\Gamma }}^{\mathrm{tr}}$ be the finite $C^{*}$-algebra generated on this fiber by the transported boundary-charge algebra and ${\mathcal{W}}_{\mathsf{\Gamma }}$. A different reference vertex gives a unitarily conjugate algebra. Let ${\mathbb{E}}_{Z,\mathsf{\Gamma }}:{\mathfrak{A}}_{\mathsf{\Gamma }}^{\mathrm{tr}}\to Z\left({\mathfrak{A}}_{\mathsf{\Gamma }}^{\mathrm{tr}}\right)$ be the conditional expectation preserving the represented matrix trace. The structural torsor residual is

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

(102)

Its closure retains the central Wilson holonomy. The flat subbranch is defined by the additional residual

$$R_{\mathrm{flat}}^{\mathsf{\Gamma }}={\mathcal{W}}_{\mathsf{\Gamma }}-\mathbf{1}.$$

(103)

When (102) vanishes, ${\mathcal{W}}_{\mathsf{\Gamma }}$ lies in $Z\left({\mathfrak{A}}_{\mathsf{\Gamma }}^{\mathrm{tr}}\right)$ and acts by a scalar unitary on each simple summand. The scalar spectral formulas below are evaluated on one such summand. In a cycle-trivial coefficient frame, $U_a=\mathbf{1}$ and

$$d_{\theta }=S-e^{i{\theta }_{\mathsf{\Gamma }}}\mathbf{1}.$$

(104)

The scalar vertex Laplacian is

$${\mathsf{\Delta }}_{\theta }=d_{\theta }^{†}{d}_{\theta }=2\mathbf{1}-e^{-i{\theta }_{\mathsf{\Gamma }}}S-e^{i{\theta }_{\mathsf{\Gamma }}}{S}^{†}.$$

(105)

Let $P_{\mathbf{1}}=(\mathbf{1}+S+S^2)/3$ and $P_{\mathrm{T}}=\mathbf{1}-P_{\mathbf{1}}$. These are the orthogonal projections onto the invariant line and the real torsor complement, and they commute with (105).

Lemma 7

(Torsor Hodge quotient). Let x be a real vector on ${\mathcal{T}}_{\mathsf{\Gamma }}$ with $P_{\mathbf{1}}x\ne 0$ and $P_{\mathrm{T}}x\ne 0$, and put ${\mathsf{\Sigma }}_{\mathrm{H}}\left(x\right):=log(\parallel P_{\mathbf{1}}{x\parallel }^2/\parallel P_{\mathrm{T}}x{\parallel }^2)$. Then

$${\mathcal{Q}}_{\mathrm{H}}\left(x\right):={\displaystyle \frac{{\parallel x\parallel }^2}{{\left|{\sum }_{a\in {\mathbb{Z}}_3}{x}_a\right|}^2}}={\displaystyle \frac{1+e^{-{\mathsf{\Sigma }}_{\mathrm{H}}\left(x\right)}}{3}}.$$

(106)

Proof. 

One has $\parallel P_{\mathbf{1}}{x\parallel }^2={\displaystyle \frac{1}{3}}{\left|{\sum }_a{x}_a\right|}^2$ and ${\parallel x\parallel }^2=\parallel P_{\mathbf{1}}{x\parallel }^2+{\parallel P_{\mathrm{T}}x\parallel }^2$. Substitution gives (106). □

The quadratic boundary form is

$$S_{\partial ,\mathrm{cen}}\left(\xi \right)={\nu }_{\mathsf{\Gamma }}\sum _{a\in {\mathbb{Z}}_3}{\left|{\xi }_a-e^{i{\theta }_{\mathsf{\Gamma }}}{\xi }_{a+1}\right|}^2,$$

(107)

and the eigenvalues of ${\nu }_{\mathsf{\Gamma }}{\mathsf{\Delta }}_{\theta }$ are

$${\lambda }_k=2{\nu }_{\mathsf{\Gamma }}\left(1-cos\left({\displaystyle \frac{2\pi k}{3}}-{\theta }_{\mathsf{\Gamma }}\right)\right),k=0,1,2.$$

(108)

Proposition 18

(Wilson-Laplacian determinant). For the scalar complex,

$$det{\mathsf{\Delta }}_{\theta }={\left|1-{\mathcal{W}}_{\mathsf{\Gamma }}\right|}^2.$$

(109)

For a rank-m coefficient system,

$$det\left(d_{\mathsf{\Gamma }}^{†}{d}_{\mathsf{\Gamma }}\right)={\left|det(\mathbf{1}-{\mathcal{W}}_{\mathsf{\Gamma }})\right|}^2.$$

(110)

Proof. 

Vertex gauge transformations place the full transport on one cyclic edge. The resulting block determinant is $det(\mathbf{1}-{\mathcal{W}}_{\mathsf{\Gamma }})$ up to a unit-modulus factor. □

Corollary 5

(Torsor cohomology). The cohomology dimensions are

$$dim{H}^0({\mathcal{T}}_{\mathsf{\Gamma }},{\mathcal{E}}_{\mathsf{\Gamma }})=dim{H}^1({\mathcal{T}}_{\mathsf{\Gamma }},{\mathcal{E}}_{\mathsf{\Gamma }})=dimker(\mathbf{1}-{\mathcal{W}}_{\mathsf{\Gamma }}).$$

(111)

Proof. 

A closed section is determined by one vertex value fixed by ${\mathcal{W}}_{\mathsf{\Gamma }}$. This gives $H^0$. Since $C^0$ and $C^1$ have the same finite dimension, the Euler characteristic gives the same dimension for $H^1$. □

Proposition 19

(Scalar torsor spectral invariants). For the unscaled scalar spectrum ${\mu }_k={\lambda }_k/{\nu }_{\mathsf{\Gamma }}$,

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

(112)

Hence

$${\chi }_{{\mathsf{\Delta }}_{\theta }}\left(\mu \right)={\mu }^3-6{\mu }^2+9\mu -{\left|1-{\mathcal{W}}_{\mathsf{\Gamma }}\right|}^2.$$

(113)

Proof. 

The first two identities follow from the cubic-root sums in (108); the product is (109). These elementary symmetric functions give the characteristic polynomial. □

Proposition 20

(Wilson spectral discriminant). With $w_{\mathsf{\Gamma }}={|1-{\mathcal{W}}_{\mathsf{\Gamma }}|}^2$,

$${Disc}_{\mu }{\chi }_{{\mathsf{\Delta }}_{\theta }}=27w_{\mathsf{\Gamma }}(4-w_{\mathsf{\Gamma }}).$$

(114)

For scalar unitary holonomy, the spectrum is simple precisely for ${\mathcal{W}}_{\mathsf{\Gamma }}\ne \pm 1$.

Proof. 

With $\mu =x+2$, the polynomial in (113) becomes $x^3-3x+2-w_{\mathsf{\Gamma }}$. Its discriminant is $27w_{\mathsf{\Gamma }}(4-w_{\mathsf{\Gamma }})$. □

An additional finite arithmetic subbranch is obtained by requiring the singular values of $d_{\theta }$ to be commensurate. After removal of their common scale, this means

$${\mu }_1={\displaystyle \frac{p^2}{n}},{\mu }_2={\displaystyle \frac{q^2}{n}},{\mu }_3={\displaystyle \frac{r^2}{n}},0<p\le q\le r,gcd(p,q,r)=1,$$

(115)

with $p,q,r,n\in \mathbb{N}$. This arithmetic closure is imposed after reconstruction of the torsor complex and is not used in the carrier selection.

Proposition 21

(Primitive commensurate torsor spectrum). Under (115), the first two spectral invariants in (112) imply

$$r=p+q,n={\displaystyle \frac{p^2+pq+q^2}{3}}.$$

(116)

Let $\omega =e^{2\pi i/3}$ and ${\mathsf{\Lambda }}_E=\mathbb{Z}\left[\omega \right]$. Every primitive solution determines

$$\xi ={\displaystyle \frac{p-q\omega }{1-\omega }}\in {\mathsf{\Lambda }}_E,N_E\left(\xi \right)=\xi \overline{\xi }=n,$$

(117)

and, after a cyclic labeling of the three characters and a choice of orientation,

$$e^{i{\theta }_{\mathsf{\Gamma }}}=-{\omega }^2{\displaystyle \frac{\xi }{\overline{\xi }}},{\left|1-e^{i{\theta }_{\mathsf{\Gamma }}}\right|}^2={\displaystyle \frac{p^2}{n}},{\left|\omega -e^{i{\theta }_{\mathsf{\Gamma }}}\right|}^2={\displaystyle \frac{q^2}{n}},{\left|{\omega }^2-e^{i{\theta }_{\mathsf{\Gamma }}}\right|}^2={\displaystyle \frac{r^2}{n}}.$$

(118)

Hence the commensurate holonomies lie in the unit-circle commensurator of the Eisenstein lattice,

$${Comm}_{U\left(1\right)}\left({\mathsf{\Lambda }}_E\right)=U\left(1\right)\cap \mathbb{Q}\left(\omega \right).$$

(119)

The least-denominator primitive solution is the nonsimple ${\mathcal{W}}_{\mathsf{\Gamma }}=-1$ endpoint. Among simple primitive spectra, the least denominator is $n=7$, and

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

(120)

For this spectrum $w_{\mathsf{\Gamma }}=400/343$.

Proof. 

Substitution in the first two identities of (112) gives $p^2+q^2+r^2=6n$ and $p^2{q}^2+p^2{r}^2+q^2{r}^2=9n^2$. Eliminating n gives $(p+q+r)(-p+q+r)(p-q+r)(p+q-r)=0$, the vanishing Heron product for a degenerate triangle with side lengths $p,q,r$. The ordering in (115) gives $r=p+q$, and substitution gives (116). Integrality of n then gives $p\equiv q(mod3)$, so (117) belongs to $\mathbb{Z}\left[\omega \right]$; direct evaluation of the Eisenstein norm gives $N_E\left(\xi \right)=n$. Equation (118) follows by substitution and reproduces the three scalar Laplacian eigenvalues. A unitary scalar maps ${\mathsf{\Lambda }}_E$ to a commensurate lattice precisely when it belongs to the $\mathbb{Q}$-span $\mathbb{Q}\left(\omega \right)$, which gives (119). The least-denominator primitive solution is $(p,q,r;n)=(1,1,2;1)$. For a simple spectrum one has $p<q$. If $n<7$, then $q\le 3$; the primitive pairs $(1,2)$, $(1,3)$, and $(2,3)$ give numerators 7, 13, and 19 in (116), none divisible by 3. The next pair $(1,4)$ gives $(p,q,r;n)=(1,4,5;7)$ and (120). □

The endpoint spectra are $(0,3,3)$ at ${\mathcal{W}}_{\mathsf{\Gamma }}=1$ and $(1,1,4)$ at ${\mathcal{W}}_{\mathsf{\Gamma }}=-1$. Their spectral-entropy distinction is retained in the following form. For $H_{\mathsf{\Gamma }}={\nu }_{\mathsf{\Gamma }}{\mathsf{\Delta }}_{\theta }$ and ${\rho }_{\beta ,\mathsf{\Gamma }}=e^{-\beta H_{\mathsf{\Gamma }}}/Tr\left(e^{-\beta H_{\mathsf{\Gamma }}}\right)$,

$$S_{\beta ,\mathsf{\Gamma }}=log3-{\left(\beta {\nu }_{\mathsf{\Gamma }}\right)}^2+{\displaystyle \frac{{\left(\beta {\nu }_{\mathsf{\Gamma }}\right)}^3}{3}}(w_{\mathsf{\Gamma }}-2)+O\left({\left(\beta {\nu }_{\mathsf{\Gamma }}\right)}^4\right).$$

(121)

Corollary 6

(Wilson spectral entropy). The entropy depends on the scalar holonomy through $w_{\mathsf{\Gamma }}$. As $\beta \to \infty$, it tends to 0 at ${\mathcal{W}}_{\mathsf{\Gamma }}=1$ and to $log2$ at ${\mathcal{W}}_{\mathsf{\Gamma }}=-1$.

Proof. 

The uniform spectral mean, variance, and third centered moment are 2, 2, and $w_{\mathsf{\Gamma }}-2$. Expansion of $S_{\beta ,\mathsf{\Gamma }}$ gives (121); the low-temperature limits follow from the endpoint spectra. □

The relation of Wilson observables to entropic order parameters is represented by [58].

With $\omega$ as in Proposition 21, let $Z{e}_a={\omega }^a{e}_a$. The Weyl operators satisfy $ZS=\omega SZ$. Every $T\in {\mathcal{A}}_{\mathrm{fam}}$ has the expansion

$$T=\sum _{a,b\in {\mathbb{Z}}_3}{t}_{ab}{Z}^a{S}^b,t_{ab}={\displaystyle \frac{1}{3}}{Tr}_{\mathrm{cen}}\left(S^{-b}{Z}^{-a}T\right).$$

(122)

Its non-circulant part is

$${\mathsf{\Pi }}_{\mathrm{noncirc}}\left(T\right)=\sum _{\begin{array}{c}a=1,2\\ b=0,1,2\end{array}}{t}_{ab}{Z}^a{S}^b.$$

(123)

For $T_{ab}=Z^a{S}^b$,

$$[d_{\theta },T_{ab}]=({\omega }^{-a}-1)Z^a{S}^{b+1}.$$

(124)

Consequently,

$${Tr}_{\mathrm{cen}}\left({[d_{\theta },T]}^{†}[d_{\theta },T]\right)=3{Tr}_{\mathrm{cen}}\left({\mathsf{\Pi }}_{\mathrm{noncirc}}{\left(T\right)}^{†}{\mathsf{\Pi }}_{\mathrm{noncirc}}\left(T\right)\right).$$

(125)

Proposition 22

(Torsor spectral seminorm). The quadratic form

$${\mathcal{L}}_{\mathsf{\Gamma }}{\left(T\right)}^2:={Tr}_{\mathrm{cen}}\left({[d_{\theta },T]}^{†}[d_{\theta },T]\right)$$

(126)

is a spectral seminorm on ${\mathcal{A}}_{\mathrm{fam}}$. Its kernel is the circulant algebra $\mathbb{C}\left[S\right]$.

Proof. 

Equation (125) identifies the seminorm with the Hilbert-Schmidt norm of the non-circulant projection. Its kernel is therefore $\mathbb{C}\left[S\right]$. □

The Hermitian family responses used below are represented by

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

(127)

The Weyl relation gives

$$\begin{array}{cc}\hfill [K_Z,H_S\left(\vartheta \right)]={\displaystyle \frac{\sqrt{3}}{2}}(& -\omega e^{-i\vartheta }ZS+{\omega }^2{e}^{i\vartheta }Z{S}^2\hfill \\ \hfill & -{\omega }^2{e}^{-i\vartheta }{Z}^{2}S+\omega e^{i\vartheta }{Z}^2{S}^2).\hfill \end{array}$$

(128)

Thus the commutator is supported in ${\mathcal{A}}_1\oplus {\mathcal{A}}_2$; in particular, the oriented $Z{S}^2$ coefficient has phase ${\omega }^2{e}^{i\vartheta }$. The clock-sine spectrum is $\{0,\pm \sqrt{3}/2\}$. The generator G is real skew, has spectrum $\{0,\pm i\}$, and fixes the invariant line of Lemma 7.

The same Weyl pair is subject to (30) in every state of ${\mathcal{A}}_{\mathrm{fam}}$. The uncertainty bound follows from state positivity, while (126) tests the non-circulant component of an operator.

The relative transport of the two source channels has a canonical low-symbol decomposition after an order-three normal-frame calibration has been fixed.

Proposition 23

($V_1/V_2$ low-symbol decomposition). In standard weight bases,

$$Hom(V_1,V_2)={\mathcal{H}}_0\oplus {\mathcal{H}}_1\oplus {\mathcal{H}}_2,dim{\mathcal{H}}_r=5,$$

(129)

where ${\mathcal{H}}_r$ consists of transitions with weight difference r modulo three. Moreover,

$$Hom(V_1,V_2)\simeq V_1\oplus V_2\oplus V_3.$$

(130)

Let ${\sigma }_{\mathrm{low}}$ be the orthogonal $SU\left(2\right)$-equivariant projection $Hom(V_1,V_2)\to V_1\oplus V_2\simeq {End}_0\left(E_3\right)$. After an order-three normal-frame calibration has been fixed, let $j_{\mathsf{\Gamma }}:E_3\to {\mathcal{H}}_{\mathrm{cen}}$ be a unitary intertwiner between the restricted order-three action on $E_3$ and the regular translation on ${\mathcal{H}}_{\mathrm{cen}}$. Then

$${\kappa }_{\mathsf{\Gamma }}={Ad}_{j_{\mathsf{\Gamma }}}\circ {\sigma }_{\mathrm{low}},ker{\kappa }_{\mathsf{\Gamma }}=V_3,$$

(131)

and its graded image satisfies

$${\kappa }_{\mathsf{\Gamma }}\left({\mathcal{H}}_0\right)={\mathcal{A}}_0\cap {End}_0\left({\mathcal{H}}_{\mathrm{cen}}\right),{\kappa }_{\mathsf{\Gamma }}\left({\mathcal{H}}_r\right)={\mathcal{A}}_r,r=1,2,$$

(132)

where ${\mathcal{A}}_r=\{T:ST{S}^{-1}={\omega }^{-r}T\}$.

Proof. 

The weight count gives five transitions in each residue class. The Clebsch-Gordan decomposition gives the orthogonal projection ${\sigma }_{\mathrm{low}}$ and its kernel $V_3$. Restriction of $E_3$ to the order-three subgroup contains each character once, so the calibrated intertwiner $j_{\mathsf{\Gamma }}$ exists and is unique up to diagonal character phases. Equivariance with respect to the order-three subgroup gives the stated graded image. □

Lemma 8

(Picard translation and low-symbol degree). For $T\in {\mathcal{A}}_r$ and $b\in {\mathbb{Z}}_3$, one has $S^{b}T,T{S}^b\in {\mathcal{A}}_r$. Hence the regular Picard translation has degree zero with respect to (132). If $X\in {\mathcal{A}}_1$ is invertible, then

$$XS{X}^{-1}{S}^{-1}=\omega \mathbf{1}.$$

(133)

Proof. 

The first statement follows directly from the defining covariance of ${\mathcal{A}}_r$ and $S^3=\mathbf{1}$. For $X\in {\mathcal{A}}_1$, (132) gives $SX{S}^{-1}={\omega }^{-1}X$, which is equivalent to (133). □

A first-order relative transport contributes to the mixed torsor curvature precisely when its low symbol has degree one or two. The $V_3$ component enters only through higher Schur terms. The normalized trace of the scalar torsor Laplacian is

$${tr}_{\mathrm{cen}}\left({\mathsf{\Delta }}_{\theta }\right)=2,$$

(134)

and hence

$${tr}_{\mathrm{cen}}\left({\mathsf{\Delta }}_{\theta }\right){Tr}_{{\mathsf{\Lambda }}^{\mathrm{even}}V}\left(Y^2\right)={\displaystyle \frac{20}{3}}.$$

(135)

The first quantity is flux independent, while the determinant in (109) records the Wilson obstruction.

The same Weyl pair carries the dual vertex algebra ${\mathcal{A}}_{\mathrm{vert}}=\mathbb{C}\left[Z\right]$ and transport algebra ${\mathcal{A}}_{\mathrm{tr}}=\mathbb{C}\left[S\right]$. The trace-preserving vertex expectation is

$${\mathbb{E}}_{\mathrm{vert}}\left(T\right)=\sum _{a\in {\mathbb{Z}}_3}{e}_{a}T{e}_a={\displaystyle \frac{1}{3}}{\displaystyle \sum _{r=0}^2}{Z}^{r}T{Z}^{-r}.$$

(136)

For the Weyl expansion (122), direct evaluation gives

$${Tr}_{\mathrm{cen}}\left({[Z,T]}^{†}[Z,T]\right)=3{Tr}_{\mathrm{cen}}\left({(T-{\mathbb{E}}_{\mathrm{vert}}T)}^{†}(T-{\mathbb{E}}_{\mathrm{vert}}T)\right).$$

(137)

Hence the vertex seminorm has kernel $\mathbb{C}\left[Z\right]$, while (126) has kernel $\mathbb{C}\left[S\right]$. Their traceless parts are Hilbert-Schmidt orthogonal and $\mathbb{C}\left[Z\right]\cap \mathbb{C}\left[S\right]=\mathbb{C}\mathbf{1}$.

The circulant algebra is the range of the trace-preserving expectation

$${\mathbb{E}}_{\mathrm{circ}}\left(T\right)={\displaystyle \frac{1}{3}}{\displaystyle \sum _{r=0}^2}{S}^{r}T{S}^{-r}.$$

(138)

The corresponding subalgebra entropy is

$${\mathcal{C}}_{\mathsf{\Gamma }}\left(\rho \right)=D\left(\rho \parallel {\mathbb{E}}_{\mathrm{circ}}\rho \right).$$

(139)

The finite-dimensional subalgebra framework is represented by [21].

Proposition 24

(Entropic torsor curvature). For every density matrix ρ,

$${\mathcal{C}}_{\mathsf{\Gamma }}\left(\rho \right)=S\left({\mathbb{E}}_{\mathrm{circ}}\rho \right)-S\left(\rho \right),0\le {\mathcal{C}}_{\mathsf{\Gamma }}\left(\rho \right)\le log3.$$

(140)

If $X=X^{†}$, $TrX=0$, and ${\rho }_{\epsilon }=\mathbf{1}/3+\epsilon X$ is positive, then

$${\mathcal{C}}_{\mathsf{\Gamma }}\left({\rho }_{\epsilon }\right)={\displaystyle \frac{{\epsilon }^2}{2}}{\mathcal{L}}_{\mathsf{\Gamma }}{\left(X\right)}^2+O\left({\epsilon }^3\right).$$

(141)

For ${\mathcal{G}}_{\mathsf{\Gamma }}=\gamma ({\mathbb{E}}_{\mathrm{circ}}-id)$,

$$-{Tr}_{\mathrm{cen}}\left(T^{†}{\mathcal{G}}_{\mathsf{\Gamma }}\left(T\right)\right)={\displaystyle \frac{\gamma }{3}}{\mathcal{L}}_{\mathsf{\Gamma }}{\left(T\right)}^2.$$

(142)

Proof. 

The expectation (138) is the trace-orthogonal projection onto $\mathbb{C}\left[S\right]$. Equation (36) gives (140). The second-order expansion at $\mathbf{1}/3$, together with (125), gives (141); the same orthogonal decomposition gives (142). □

The relative-entropy asymmetry comparison is supplied by [59]. Let $|k\rangle$ be the Fourier basis of S and let $V_{\mathsf{\Gamma }}|k\rangle =|k\rangle \otimes |k\rangle$.

Corollary 7

(Entanglement lift). For every density matrix ρ on ${\mathcal{H}}_{\mathrm{cen}}$,

$$E_R\left(V_{\mathsf{\Gamma }}\rho V_{\mathsf{\Gamma }}^{†}\right)={\mathcal{C}}_{\mathsf{\Gamma }}\left(\rho \right),$$

(143)

where $E_R$ is the relative entropy of entanglement. Its maximal value is $log3$.

Proof. 

The lifted state is maximally correlated. The coherence-entanglement identity is represented by [60]; the resource-theoretic convention is reviewed in [61]. □

On the full shadow, define

$${\mathbb{E}}_j^{\left(6\right)}\left(T\right)={\displaystyle \frac{1}{j}}{\displaystyle \sum _{r=0}^{j-1}}{S}_j^{r}T{S}_j^{-r},j=2,3,{\mathbb{E}}_6^{\left(6\right)}={\mathbb{E}}_2^{\left(6\right)}{\mathbb{E}}_3^{\left(6\right)}.$$

(144)

Proposition 25

(Finite-shadow information split). For every density matrix ρ on $End\left({\mathcal{H}}_{\mathsf{\Gamma }}^{\left(6\right)}\right)$,

$$\begin{array}{cc}\hfill D(\rho \parallel {\mathbb{E}}_6^{\left(6\right)}\rho )=& D(\rho \parallel {\mathbb{E}}_2^{\left(6\right)}\rho )\hfill \\ \hfill & +D({\mathbb{E}}_2^{\left(6\right)}\rho \parallel {\mathbb{E}}_6^{\left(6\right)}\rho ),\hfill \end{array}$$

(145)

where the two terms are bounded by $log2$ and $log3$. On one parity character,

$$D(\rho \parallel {\mathbb{E}}_6^{\left(6\right)}\rho )=D(\rho \parallel {\mathbb{E}}_3^{\left(6\right)}\rho )\le log3.$$

(146)

Proof. 

The commuting expectations in (144) have nested ranges. Equation (37) gives (145). Averaging over j unitaries increases entropy by at most $logj$, and the parity character is fixed by ${\mathbb{E}}_2^{\left(6\right)}$. □

The full information capacity is therefore split according to the coprime factors of the same finite cocharacter shadow. Parity closure removes the active $log2$ contribution and leaves the projective-color capacity.

The normalized trace state on the even exterior module has the parity decomposition

$${\rho }_{\mathrm{ext}}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{P_C^{+}}{4}}\otimes {\displaystyle \frac{P_W^{+}}{2}}+{\displaystyle \frac{P_C^{-}}{4}}\otimes {\displaystyle \frac{P_W^{-}}{2}}\right).$$

(147)

Its color-weak mutual information is $log2$, and the state is separable. Coherent entanglement belongs to completed data.

Table 7. Holonomy-controlled spectral regimes of the projective-color torsor Laplacian.

Holonomy	Scalar spectrum of ${\mathit{\Delta }}_{\mathit{\theta }}$	Ground-state multiplicity
${\mathcal{W}}_{\mathsf{\Gamma }}=1$	$(0,3,3)$	1
${\mathcal{W}}_{\mathsf{\Gamma }}\ne \pm 1$	simple	1
${\mathcal{W}}_{\mathsf{\Gamma }}=-1$	$(1,1,4)$	2

Table 7. Holonomy-controlled spectral regimes of the projective-color torsor Laplacian.

Holonomy	Scalar spectrum of ${\mathit{\Delta }}_{\mathit{\theta }}$	Ground-state multiplicity
${\mathcal{W}}_{\mathsf{\Gamma }}=1$	$(0,3,3)$	1
${\mathcal{W}}_{\mathsf{\Gamma }}\ne \pm 1$	simple	1
${\mathcal{W}}_{\mathsf{\Gamma }}=-1$	$(1,1,4)$	2

3.7. Conditional Low-Sector Attachment and Standard-Model Channel Structure

The finite objects constructed above precede the choice of a low spectral representative. Their attachment to a completed operator requires a boundary-admissible family, an isolated primitive window, exact finite-shadow covariance, and parity closure. The projected connection on an isolated Borel-Weil block is

$${\nabla }_{B,q}=P_{q}d{P}_q.$$

(148)

This is the Berry-Wilczek-Zee connection represented by [62] and [63]. Stability of the Riesz projection is understood in the sense of [8]. The open normal operator is compared with the Callias framework of [9] and its geometric form in [10]. Local source-to-carrier intertwiners and Borel-Weil frames are treated as sections of the admissible reconstruction-frame torsor. The microscopic Jacobian is required to obey the unitary covariance (196) under marking-preserving frame changes. When the selected window is isolated, Theorem 4 gives the corresponding covariance of the normal family, Riesz projection, projected connection, and reduced normal form. Their rank, gap, index, and determinant holonomy therefore descend to the protected quotient.

Let ${\gamma }_{\mathrm{br}}^5$ be branch chirality and let ${\mathsf{\Gamma }}_V={(-1)}^{N_C+N_W}$ be exterior parity. The locked involution is

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

(149)

Here ${\mathsf{\Gamma }}_V$ is defined on the full Clifford module before the locked chiral projection; $F_{\mathsf{\Gamma }}^{\mathrm{even}}$ is the selected internal chiral half. With the corresponding locked projections $P_{\pm }$, a completed low operator has the finite odd form

$$Q_F=P_{+}\left(D_E+c(\mathsf{\Phi }+{\mathsf{\Phi }}^{†})+Q_{\mathrm{hol}}+Q_C\right)P_{-}.$$

(150)

Here $D_E$ is the projected even-package Dirac part, $c(\mathsf{\Phi }+{\mathsf{\Phi }}^{†})$ is the exterior-odd bridge, and $Q_{\mathrm{hol}}$ and $Q_C$ are completed transport and Codazzi-Schur terms. Odd insertions admit the superconnection organization of [15]. Defect-localized and domain-wall zero-mode comparisons are supplied by [64] and [65].

Let ${\mathcal{H}}_{\mathrm{lock}}^{\epsilon }$ be the retained $\epsilon$-eigenspace of ${\mathsf{\Gamma }}_{\mathrm{lock}}$. The diagonal parity residual is

$$R_{\mathrm{par}}={\left.\left(S_2\otimes {\mathsf{\Gamma }}_{\mathrm{lock}}-\mathbf{1}\right)\right|}_{{\mathcal{H}}_{\mathrm{par}}\otimes {\mathcal{H}}_{\mathrm{lock}}^{\epsilon }}.$$

(151)

Its closure selects the corresponding one-dimensional character of the regular parity factor and excludes the complementary character. In the character-adapted flat reference, the two characters are spectrally separated by Lemma 6.

Proposition 26

(Parity-paired odd high lift). Assume that the completed high lift is formed on the full finite shadow before parity compression, is odd with respect to ${\mathsf{\Gamma }}_{\mathrm{lock}}$, and preserves the diagonal parity condition in (151). Put ${\mathfrak{a}}_W={End}_0\left(W\right)$ and ${\mathfrak{a}}_C={End}_0\left(C\right)$. On each ${\mathbb{Z}}_3$ fiber of (87), the two matched parity/chirality lines are exchanged by the odd high operator. The adjoint high carrier therefore has the factorization

$${\mathcal{H}}_{H,\mathrm{adj}}\simeq {\mathcal{H}}_{\mathrm{cen}}\otimes {\mathbb{C}}_{\mathrm{par}-\mathrm{lock}}^2\otimes \left({\mathfrak{a}}_W\oplus {\mathfrak{a}}_C\right),$$

(152)

where ${\mathbb{C}}_{\mathrm{par}-\mathrm{lock}}^2$ is the sum of the two matched lines. Hence the full shadow supplies three Dirac-form pairs in each adjoint channel.

Proof. 

Oddness reverses the ${\mathsf{\Gamma }}_{\mathrm{lock}}$ eigenvalue. Preservation of $S_2{\mathsf{\Gamma }}_{\mathrm{lock}}$ then reverses the $S_2$ character simultaneously. The two regular parity characters are therefore paired off-diagonally at fixed ${\mathbb{Z}}_3$ vertex. Equation (87) gives three such vertices. The central $U\left(1\right)$ acts trivially on ${\mathfrak{a}}_W$ and ${\mathfrak{a}}_C$, so both coefficient spaces have zero hypercharge. □

The normal quadratic form used below is first reduced along presentation fibers as in Lemma 1. For a low-high splitting $\mathcal{H}={\mathcal{H}}_L\oplus {\mathcal{H}}_H$, with $K_{HH}$ invertible, the remaining effective operator is

$$K_{\mathrm{eff}}=K_{LL}-K_{LH}{K}_{HH}^{-1}{K}_{HL}+R_{\ge 3}.$$

(153)

Charge-compatible completions preserve the central projections of Lemma 3. Higher integer moments have zero direct compression by Proposition 8 and first enter through the Schur term.

Proposition 27

(Schur suppression of higher integer moments). Let $K_0=K_0^{†}$ and let P be an isolated Riesz projection of $K_0$, with $Q=\mathbf{1}-P$. Assume that $K_{HH}^{\left(0\right)}:=Q{K}_{0}Q$ is invertible and put $g_{\mathrm{CSch}}={\parallel {\left(K_{HH}^{\left(0\right)}\right)}^{-1}\parallel }^{-1}$. Let $T_{\ell }=T_{\ell }^{†}$ be a charge-compatible principal moment of integer type $V_{\ell }$, $\ell \ge 3$, whose direct compression vanishes as in Proposition 8. If $\parallel T_{\ell }\parallel <g_{\mathrm{CSch}}/2$, then for $K\left(\epsilon \right)=K_0+\epsilon T_{\ell }$ and $\left|\epsilon \right|\le 1$,

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

(154)

Proof. 

Since P is a spectral projection of $K_0$, one has $[K_0,P]=0$, so the unperturbed low-high blocks vanish. The high block of $K\left(\epsilon \right)$ is $K_{HH}^{\left(0\right)}+\epsilon Q{T}_{\ell }Q$. Its inverse is expanded by the Neumann series under the stated norm bound. Substitution in (153), together with $P{T}_{\ell }P=0$, gives (154). □

Let $P_{\mathrm{prim}}$ be the isolated primitive locked projection before the finite-shadow action is attached. Write $Ran{P}_{\mathrm{prim}}\simeq {\mathcal{H}}_{\mathrm{ch}}\otimes M_{\mathrm{prim}}$, where ${\mathcal{H}}_{\mathrm{ch}}$ carries the generated charge-exterior core.

Definition 7

(Finite-core primitive normal representative). A primitive locked normal representative is finite-core when its represented charge-exterior algebra satisfies (20) on ${\mathcal{H}}_{\mathrm{ch}}\otimes M_{\mathrm{prim}}$.

Finite-core uniqueness gives

$$dim{M}_{\mathrm{prim}}=1.$$

(155)

Let $K_{\mathrm{pre}}$ be the corresponding pre-torsor representative on the protected quotient. Exact finite-shadow covariance is imposed by

$$R_{\mathrm{cov}}={\left([p_x,B],[S_6,B],[p_x,K_{\mathrm{pre}}],[S_6,K_{\mathrm{pre}}]\right)}_{\begin{array}{c}x\in {\mathcal{T}}_{\mathsf{\Gamma }}^{\left(6\right)}\\ B\in {\mathfrak{A}}_{\mathrm{ch}}\end{array}}.$$

(156)

This residual is evaluated after the primitive window has been isolated.

Proposition 28

(Locked central low-sector factorization). Assume exact monoidal closure, exact finite-shadow covariance, the parity closure (151), and a finite-core primitive projection separated by a gap ${\mathsf{\Delta }}_{\mathrm{CSch}}$. Then the generated unperturbed low algebra and module are

$${\mathfrak{A}}_{\mathrm{low}}^{\left(0\right)}={\mathfrak{A}}_{\mathrm{ch}}\otimes P_{\epsilon }{\mathcal{A}}_6{P}_{\epsilon },Ran{P}_{\mathrm{low}}^{\left(0\right)}\simeq {\mathcal{H}}_{\mathrm{ch}}\otimes {\mathcal{H}}_{\mathrm{cen}}.$$

(157)

If a charge-compatible central Schur perturbation satisfies $\parallel R_{\mathrm{cen}}\parallel <{\mathsf{\Delta }}_{\mathrm{CSch}}/2$, then

$$rank{P}_{\mathrm{low}}=rank{P}_{\mathrm{prim}}dim{\mathcal{H}}_{\mathrm{cen}}.$$

(158)

Proof. 

Proposition 14 supplies ${\mathcal{A}}_6$. Parity closure reduces it to the corner (91). Exact covariance makes this corner commute with the pre-torsor charge-exterior core. The product window follows from finite-core uniqueness, and the perturbation bound preserves its Riesz projection. □

Corollary 8

(Weak-parity closure). A nonzero parity sector satisfying (151) is one-dimensional. Retaining both regular parity modes violates the residual.

Proof. 

On ${\mathcal{H}}_{\mathrm{lock}}^{\epsilon }$, the residual reduces to $(\epsilon S_2-\mathbf{1}){|}_{{\mathcal{H}}_{\mathrm{par}}}=0$. Each eigenspace of the regular two-dimensional shift is one-dimensional. □

Corollary 9

(Three-dimensional family factor). Under Proposition 28, the completed low rank is $3rank{P}_{\mathrm{prim}}$. In each fixed simple primitive charge-exterior channel,

$$rank{P}_{\mathrm{low},\mathrm{ch}}=3.$$

(159)

Proof. 

Equation (158) and $dim{\mathcal{H}}_{\mathrm{cen}}=3$ give the rank formula. A simple primitive channel has rank one before the torsor factor is attached. □

The statement concerns the rank and spectral transport class of the low bundle. A tensor-sum form of the completed operator is not required. The determinant-line reading of the moving low family is the one of [13].

Corollary 10

(No additional light family on the isolated branch). Assume the fixed Callias-Schur isolated stratum of Proposition 28 and an invertible heavy neutral denominator in (290). The rank-three active low factor is preserved by Schur elimination. A fourth light family factor requires a new protected source, a gap crossing, or another downstream datum.

Proof. 

The rank statement is (159). On the fixed isolated stratum the heavy neutral elimination does not add a light factor, and an unsourced additional copy is excluded by Proposition 3. □

The linear Clifford-odd tangents are the one-particle insertions C, $C^{*}$, W, and $W^{*}$. Their canonical degrees determine the structural low channels.

Using (74), the four-fermion components of the determinant class $2(3,2)$ split into $B,L$-preserving contractions and baryon-violating contractions. The latter are represented by the $QQQL$ and $u^c{u}^c{d}^c{e}^c$ types and satisfy

$$\mathsf{\Delta }B=\mathsf{\Delta }L=\pm 1.$$

(160)

Thus a $\mathsf{\Delta }B=2$ transition is outside this dimension-six top-form class.

The unique color-singlet, $B-L$-preserving linear odd tangent is $W\oplus W^{*}$.

Corollary 11

(No second unsourced weak doublet). On the fixed linear weak-bridge reconstruction stratum, the retained direct channel contains one fundamental weak-doublet class. A second independent fundamental weak-doublet channel requires an additional protected source or downstream datum.

Proof. 

The retained color-singlet, $B-L$-preserving odd channel is $W\oplus W^{*}$. An additional independent copy is excluded by Proposition 3 unless it is generated or required downstream. □

The distinction between Dirac, Majorana, and determinant-contact classes is the usual one in exterior unification language represented by [27]; related Clifford-ideal channels are represented by [47].

At this stage the carrier, carrier-group representation, rank of the parity-selected finite-shadow module, and channel grading are fixed. Its attachment as a physical rank-three family factor remains conditional on the covariance and gap hypotheses stated below. The matrix elements of (150), the singular values of the direct blocks, the neutral Schur denominator, and the contact coefficients remain completed data. Their quantitative analysis is carried out in Section 5.

3.8. Primitive Graded Standard-Model Reconstruction Theorem

Theorem 2

(Primitive graded Standard-Model reconstruction). Let Γ satisfy entries A and B of Table 4. Then the minimal finite carrier is (56), its degree fidelity is automatic, and the Borel-Weil presentation is reduced as in Corollary 2. Under entry C, its determinant-preserving carrier group is (63), its fixed chiral finite module is (68), and the canonical Abelian gradings are (70) and (72). The global form is (64). The protected primitive Picard translation generates the finite shadow (86) as its initial invariant extension. Under entry D, its parity-selected module is (90). If the analytic hypotheses of Proposition 28 also hold, this module is attached as the rank-three factor (159) in each simple primitive low channel. The direct, Majorana, and contact channels are those of Table 8.

Proof. 

The carrier statement is Theorem 1. The carrier group follows from (62), and the Clifford and determinant statements follow from Proposition 9 and Proposition 11. Proposition 14 gives the generated finite-shadow orbit, while Proposition 16 gives the parity corner. The low attachment follows from Proposition 28 and Corollary 9. □

Table 9. Status of the outputs of the primitive graded reconstruction.

Output	Status	Required entries
primitive line, rank-five carrier, and degree fidelity	derived	A and B
graded determinant group and faithful global form	derived	C
even one-generation Fock package	derived	C and fixed chirality orientation
hypercharge and canonical Abelian plane	derived	centered carrier degree and Fock number
local anomaly and parity compatibility	derived check	determinant-compatible even package
full ${\mu }_6$ finite shadow	minimal-reconstruction consequence	protected primitive Picard translation
three-dimensional parity-selected module	derived algebraically	generated finite shadow and D
physical family attachment	conditional	finite-shadow covariance and Callias-Schur gap
weak-bridge, Majorana, and contact classes	structural channel decomposition	locked exterior and $B-L$ filtration
masses, mixing, running, and contact coefficients	completed	Schur and scale data

Table 9. Status of the outputs of the primitive graded reconstruction.

Output	Status	Required entries
primitive line, rank-five carrier, and degree fidelity	derived	A and B
graded determinant group and faithful global form	derived	C
even one-generation Fock package	derived	C and fixed chirality orientation
hypercharge and canonical Abelian plane	derived	centered carrier degree and Fock number
local anomaly and parity compatibility	derived check	determinant-compatible even package
full ${\mu }_6$ finite shadow	minimal-reconstruction consequence	protected primitive Picard translation
three-dimensional parity-selected module	derived algebraically	generated finite shadow and D
physical family attachment	conditional	finite-shadow covariance and Callias-Schur gap
weak-bridge, Majorana, and contact classes	structural channel decomposition	locked exterior and $B-L$ filtration
masses, mixing, running, and contact coefficients	completed	Schur and scale data

The Alena-Codazzi realization in the next section supplies the geometric and source data in entry A and the relative Gauss-local part of entry B, together with a conditional mechanism for the transported boundary data. The preserved moment grading is identified with the Borel-Weil level by the minimal marked reconstruction of Theorem 1. The completed spectral and quantitative problem is treated after the realization layer has been fixed.

4. Alena-Codazzi Realization of the Reconstruction Data

The structural theorem of Section 3 starts from a primitive projective link, a degree-resolved two-channel normal source, relative Gauss-local charges, and admissible boundary transport. The present section gives an explicit compact-leaf source model and formulates a geometric realization criterion for these inputs. Within this criterion, a current-residual Alena-Codazzi collar supplies the compact-leaf source, the thin-core worldline component, the source Euler grading, and the two principal boundary charges. Torsor-admissible transport supplies the finite edge data required by the projective-color complex.

When the realization criterion is satisfied, the resulting collar fixes the source side of the reconstruction. The source Euler grading is preserved by the moment map and agrees with the Borel-Weil level on the minimal marked support by Theorem 1. The determinant, Clifford, and finite-shadow constructions are then those of Section 3. The completed matrix elements are treated in Section 5.

4.1. Current-Codazzi Collar and Multiplier Closure

The residual part of the Alena-type current branch is written as

$${\mathcal{L}}_{\mathrm{cr}}=\phi p_{\mathsf{\Lambda }},\phi =1-{\zeta }^2-{\mu }_{\zeta }{R}_{\omega }.$$

(161)

Here $\zeta$ is the translational-current amplitude, ${\mu }_{\zeta }={\mu }_{\zeta }\left({\rho }_{\zeta }\right)$ is positive, and $R_{\omega }$ is the normalized vorticity response. The branch-stress interpretation is supplied by the Alena Tensor identification [66]. The current and vorticity slots are those of [67]. The continuum, variational, and branch-potential inputs are represented by [68,69], and [70]. Only the current, vorticity, stress, and Codazzi coefficients are used in the realization below.

The translational current is

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

(162)

The frozen scalar block is assumed non-degenerate:

$$V_{\zeta }^{\prime \prime }\left({\rho }_{\zeta }\right)+R_{\omega }{\mu }_{\zeta }^{\prime \prime }\left({\rho }_{\zeta }\right)>0.$$

(163)

This condition gives local persistence of the scalar collar inside the selected current branch.

The product-rule Hilbert response of (161) is decomposed as

$$T_{\mu \nu }^{\mathrm{cr}}={\mathsf{\Xi }}_{\mu \nu }+\phi Y_{\mu \nu }.$$

(164)

The first term records the response of the residual scalar density. The second carries the branch-response coefficient associated with $p_{\mathsf{\Lambda }}$. The punctured collar is split-conserved when

$${\nabla }_{\mu }^{\left(k\right)}{\mathsf{\Xi }}^{\mu \nu }=0,{\nabla }_{\mu }^{\left(k\right)}\left(\phi Y^{\mu \nu }\right)=0.$$

(165)

The split is used before the normal two-jet is extracted.

Let $\tau =k^{\mu \nu }{Y}_{\mu \nu }$ and put

$$B_{\mu \nu }=Y_{\mu \nu }-{\displaystyle \frac{1}{3}}\tau k_{\mu \nu }.$$

(166)

The indices in this subsection range over the four-dimensional collar. The tensor B is the trace-adjusted branch coefficient used in the multiplier equation. Its trace-free principal normal projection supplies the order-two source coefficient below. A nonzero scalar $\phi$ is a Codazzi multiplier when

$$A_{\mu \nu }=\phi B_{\mu \nu }$$

(167)

satisfies the trace-adjusted Codazzi equation on the source-free collar. Define

$$C_{\alpha \mu \nu }^B={\nabla }_{\alpha }^{\left(k\right)}{B}_{\mu \nu }-{\nabla }_{\mu }^{\left(k\right)}{B}_{\alpha \nu },\theta =dlog\left|\phi \right|.$$

(168)

The multiplier equation is

$$C_{\alpha \mu \nu }^B+{\theta }_{\alpha }{B}_{\mu \nu }-{\theta }_{\mu }{B}_{\alpha \nu }=0.$$

(169)

On a connected open set where $B_{\mu \nu }$ is invertible, let ${\beta }^{\mu \nu }$ denote its inverse. Contraction of (169) determines

$${\theta }_{\alpha }^B=-{\displaystyle \frac{1}{3}}{\beta }^{\mu \nu }{C}_{\alpha \mu \nu }^B.$$

(170)

The coefficient $1/3$ in (170) is ${(dim\mathsf{\Omega }-1)}^{-1}$ for the four-dimensional collar.

Proposition 29

(Current-Codazzi closure). Let Ω be a connected source-free collar set on which $B_{\mu \nu }$ is invertible. A nonzero scalar Codazzi multiplier exists if and only if (169) holds with $\theta ={\theta }^B$ and ${\theta }^B$ is exact on Ω. If ${\theta }^B=d{f}_B$, then

$$\phi =C_{\phi }{e}^{f_B},{\zeta }^2=1-{\mu }_{\zeta }{R}_{\omega }-C_{\phi }{e}^{f_B},$$

(171)

where $C_{\phi }\ne 0$. The remaining current compatibility is

$$\begin{array}{cc}\hfill U\left({\mu }_{\zeta }{R}_{\omega }\right)+C_{\phi }{e}^{f_B}{\theta }^B\left(U\right)=& \left(1-{\mu }_{\zeta }{R}_{\omega }-C_{\phi }{e}^{f_B}\right)\hfill \\ \hfill & \times \left({\nabla }_{\mu }^{\left(k\right)}{U}^{\mu }+U(log{p}_{\mathsf{\Lambda }})\right).\hfill \end{array}$$

(172)

Proof. 

Substitution of (167) into the Codazzi equation gives (169). Its contraction with ${\beta }^{\mu \nu }$ gives (170). Exactness gives $\phi =C_{\phi }{e}^{f_B}$. Equation (171) follows from (161), and current conservation gives (172). □

Lemma 9

(Multiplier rescaling covariance). Write the left-hand side of (169) as $R_B(B,\theta )$. For every smooth scalar f,

$$R_B(e^{f}B,\theta -df)=e^f{R}_B(B,\theta ).$$

(173)

Consequently, on the multiplier zero locus the infinitesimal direction $(\delta B,\delta \theta )=(fB,-df)$ is a presentation redundancy of the multiplier equation.

Proof. 

The product rule gives $C^{e^{f}B}=e^f(C^B+df\wedge B)$. Substitution in (169) cancels the two $df\wedge B$ terms and gives (173). □

Proposition 30

(Principal multiplier Schur type pattern). Let $B_0\in {Sym}_0^2\left(N_{\mathsf{\Gamma }}\right)$ be a non-degenerate frozen principal normal coefficient and let $0\ne \xi \in N_{\mathsf{\Gamma }}^{*}$. On $b\in {Sym}_0^2\left(N_{\mathsf{\Gamma }}\right)$ and $\eta \in N_{\mathsf{\Gamma }}^{*}$, define the principal maps

$${\left({\mathcal{A}}_{\xi }b\right)}_{ijk}={\xi }_i{b}_{jk}-{\xi }_j{b}_{ik},{\left({\mathcal{M}}_{B_0}\eta \right)}_{ijk}={\eta }_i{\left(B_0\right)}_{jk}-{\eta }_j{\left(B_0\right)}_{ik}.$$

(174)

Both maps are injective and

$$Im{\mathcal{A}}_{\xi }\cap Im{\mathcal{M}}_{B_0}=span\left\{{\mathcal{A}}_{\xi }{B}_0\right\}=span\left\{{\mathcal{M}}_{B_0}\xi \right\}.$$

(175)

For any positive residual metric, let ${\mathsf{\Pi }}_{B_0}^{\perp }$ denote the orthogonal projection onto the complement of $Im{\mathcal{M}}_{B_0}$. Then

$$rank\left({\mathsf{\Pi }}_{B_0}^{\perp }{\mathcal{A}}_{\xi }\right)=4,ker\left({\mathsf{\Pi }}_{B_0}^{\perp }{\mathcal{A}}_{\xi }\right)=span\left\{B_0\right\}.$$

(176)

Thus the stationary principal normal Schur block has the type pattern $V_2\left|V_1\right|V_2$: a $V_2$ source is compensated through the $V_1$ multiplier channel and returns as a reduced quadratic form on $V_2$. Its one-dimensional removed direction is the infinitesimal rescaling of Lemma 9.

Proof. 

If ${\mathcal{A}}_{\xi }b=0$, each one-form $b(·,v)$ is proportional to $\xi$. Symmetry gives $b=c\xi \otimes \xi$, and trace-freeness gives $c=0$. Hence ${\mathcal{A}}_{\xi }$ is injective. If ${\mathcal{M}}_{B_0}\eta =0$, contraction with $B_0^{-1}$ in the three-dimensional normal fiber gives $2\eta =0$, so ${\mathcal{M}}_{B_0}$ is injective.

Suppose ${\mathcal{A}}_{\xi }b={\mathcal{M}}_{B_0}\eta$ and put $C=b{B}_0^{-1}$. Using the oriented cross-product identification on $N_{\mathsf{\Gamma }}$, this equation is ${\left[\xi \right]}_{\times }C={\left[\eta \right]}_{\times }$. Choose an orthonormal frame with $\xi =\left|\xi \right|e_1$. Antisymmetry of ${\left[\xi \right]}_{\times }C$ then gives $C_{21}=C_{31}=C_{23}=C_{32}=0$ and $C_{22}=C_{33}$. Hence $C=a\mathbf{1}+\xi \otimes v$ for a covector v. Symmetry of $b=C{B}_0$ requires the rank-one term $\xi \otimes \left(v{B}_0\right)$ to be symmetric, so $v{B}_0=c\xi$. Since both b and $B_0$ are trace-free, $trb={c\left|\xi \right|}^2=0$, and therefore $c=0$. Invertibility of $B_0$ gives $v=0$, so $b=a{B}_0$ and $\eta =a\xi$. This proves (175). The rank and kernel statements in (176) follow because $rank{\mathcal{A}}_{\xi }=5$ and the intersection in (175) is one-dimensional. □

The multiplier criterion is local on the connected non-degenerate collar. Global existence additionally requires the vanishing of the periods of ${\theta }^B$. The realization theorem below uses exact multiplier closure on the selected punctured collar.

4.2. Compact-Leaf Source and Conditional Thin-Core Extraction

An explicit compact-leaf source model is supplied by a warped collar. Let ${\mathsf{\Sigma }}_g$ be a compact hyperbolic surface of genus $g\ge 2$, with metric $\gamma$. On a frozen collar interval, let $s=tanh\chi \in (0,1)$ be the leaf anisotropy and let $a\left(s\right)>0$ be the warp factor. The two-eigenvalue Codazzi first integrals contain a constant $F_0$. The warped-product curvature convention is the one of [11].

The normalized vorticity closure has the leaf equation

$${\mathsf{\Delta }}_{\gamma }\alpha ={\sigma }_{\omega }{\displaystyle \frac{2a{\left(s\right)}^2{D}_o^2}{c^{2}s}},{\sigma }_{\omega }=\pm 1,$$

(177)

where $D_o\ne 0$ is the frozen vorticity-flux scale. The right-hand side has nonzero mean. A one-core regularization is obtained by choosing a non-negative mollifier ${\rho }_{\epsilon }$ with unit integral and setting

$${\mathsf{\Delta }}_{\gamma }{\alpha }_{\epsilon }={\sigma }_{\omega }{\displaystyle \frac{2a{\left(s\right)}^2{D}_o^2}{c^{2}s}}+q_{\epsilon }\left(s\right){\rho }_{\epsilon },$$

(178)

where

$$q_{\epsilon }\left(s\right)=-{\sigma }_{\omega }{\displaystyle \frac{2a{\left(s\right)}^2{D}_o^2}{c^{2}s}}{Area}_{\gamma }\left({\mathsf{\Sigma }}_g\right).$$

(179)

The right-hand side of (178) has zero integral. The resulting compact Poisson problem is therefore solvable, uniquely after the additive constant has been fixed, by the standard elliptic result represented by [35]. If ${\rho }_{\epsilon }\rightharpoonup {\delta }_p$, the limit has one distributional leaf source at p. Transport of p along a timelike integral curve gives the local core.

Source-free will mean singular-source-free. The smooth zero-mode-compensating term in (178) is retained as part of the frozen leaf geometry. The principal boundary data are read after subtraction of a smooth local particular solution.

The same frozen sector has Codazzi gap

$${\mathsf{\Delta }}_C=-{\displaystyle \frac{2F_0}{3-s}}.$$

(180)

For $F_0\ne 0$ and $0<s<1$, the collar lies in the non-degenerate two-eigenvalue sector used by the projective-link construction.

Proposition 31

(Resolved compact-leaf source collar). Let ${\mathsf{\Sigma }}_g$ be a compact hyperbolic leaf with $g\ge 2$. For every frozen range with $0<s<1$, $D_o\ne 0$, and $F_0\ne 0$, equation (178) has a unique smooth solution with zero mean. As ${\rho }_{\epsilon }\rightharpoonup {\delta }_p$, its source converges to a one-core distributional closure. Away from the core, after subtraction of the smooth background, the collar is singular-source-free and has the nonzero gap (180).

Proof. 

The zero-mean condition follows from (179). Compact elliptic solvability gives the smooth solution. Weak convergence of the mollifier gives the distributional limit, and the gap statement is immediate from (180). □

Smooth frozen-background terms do not alter the finite carrier when the singular associated-graded charges, their source grading, and the nonzero gap are unchanged. The support selection depends on the primitive link and the degree-resolved non-scalar source.

The compact-leaf model is compatible with the current-residual scalar under the following local condition.

Proposition 32

(Compatibility with the current-residual form). Let Ω be a connected non-degenerate punctured collar on which $B_{\mu \nu }$ is invertible. Assume that a smooth nonzero scalar ${\varphi }_C$ makes ${\varphi }_C{B}_{\mu \nu }$ trace-adjusted Codazzi. Let $p_{\mathsf{\Lambda }}>0$ and let U be smooth. Suppose that positive initial data may be chosen for

$$U\left(y\right)+y\left({\nabla }_{\mu }^{\left(k\right)}{U}^{\mu }+U(log{p}_{\mathsf{\Lambda }})\right)=0$$

(181)

so that its solution satisfies $1-{\varphi }_C-y={\mu }_{\zeta }{R}_{\omega }$ with the required sign. After restriction to a smaller flow collar if required, the calibrated tensor has the representation

$${\varphi }_C=1-{\zeta }^2-{\mu }_{\zeta }{R}_{\omega },{\zeta }^2>0,{\nabla }_{\mu }^{\left(k\right)}\left(p_{\mathsf{\Lambda }}{\zeta }^2{U}^{\mu }\right)=0.$$

(182)

Moreover, ${\varphi }_C=C_{\varphi }{e}^{f_B}$ and ${\theta }^B=d{f}_B$.

Proof. 

Proposition 29 gives

$${\theta }^B=dlog\left|{\varphi }_C\right|.$$

(183)

Thus ${\varphi }_C=C_{\varphi }{e}^{f_B}$. Positive initial data for (181) remain positive along the U-flow. With $y={\zeta }^2$, define

$${\mu }_{\zeta }{R}_{\omega }=1-C_{\varphi }{e}^{f_B}-y.$$

(184)

Equations (181) and (184) give (182) and (172). □

Under the regularity hypothesis stated below, the thin-core limit identifies the primitive worldline component. The compactness input is supplied by integral-current compactness [71], in the geometric-measure form represented by [72]. Vortex concentration gives the comparison with [73]; the Jacobian-current formulation is represented by [74]. A smooth auxiliary Riemannian metric h is fixed on the collar. All mass bounds and local flat compactness statements below are taken with respect to h, while timelikeness and orthogonal normal bundles are defined by the Lorentzian metric.

Theorem 3

(Conditional thin-core extraction). Let $\left\{{\mathcal{C}}_{\epsilon }\right\}$ be regularized Alena-Codazzi collars and let $T_{\epsilon }$ be the integral 1-current carried by their regularized cores. Assume:

(i)

the local masses of $T_{\epsilon }$ and $\partial T_{\epsilon }$ are uniformly bounded;

(ii)

$\partial T_{\epsilon }=0$ on the singular-source-free part of the collar;

(iii)

the integral linking charge converges to a primitive degree-one charge;

(iv)

the trace-adjusted Codazzi residual converges to zero in $H_{\mathrm{loc}}^{-1}$ away from every subsequential current limit;

(v)

the absolute frozen Codazzi gap is uniformly bounded below on compact subsets disjoint from the cores;

(vi)

the component selected by the primitive linking charge is a regular timelike multiplicity-one component of the limiting current.

After passage to a subsequence, $T_{\epsilon }$ converges locally to an integral 1-current T without interior boundary on the source-free collar. The selected regular component is a timelike worldline Γ. Its oriented normal bundle gives (41), and the limiting charge gives (44). The trace-adjusted Codazzi equation holds distributionally away from $sptT$.

Proof. 

Integral-current compactness gives a locally convergent subsequence. Boundary continuity gives the absence of interior boundary. The residual convergence gives the distributional Codazzi equation away from the limit. The selected regular multiplicity-one component is represented by a timelike worldline. The limiting primitive charge gives (44), and the uniform gap supplies the two-eigenvalue splitting on the punctured collar. □

Figure 1. Compact-leaf concentration, the thin-core worldline, and the resolved projective link.

Figure 1. Compact-leaf concentration, the thin-core worldline, and the resolved projective link.

4.3. Moment-Resolved Source and Preservation of the Euler Grading

Let $N_{\mathsf{\Gamma }}$ be the oriented transverse normal fiber. A second-order normal source has a first moment $m\in N_{\mathsf{\Gamma }}$ and a trace-free second moment $M_0\in {Sym}_0^2\left(N_{\mathsf{\Gamma }}\right)$. The scalar trace belongs to the singlet channel and is separated before the finite support problem.

Definition 8

(Second-order moment-resolved source). A defect source is second-order moment-resolved when its associated-graded scalar-sector source up to transverse order two is represented in defect-adapted normal coordinates by a normalized core profile with first moment m and trace-free second moment $M_0$, and when the singular degree-two coefficient of the harmonic potential of its principal trace-free Codazzi representative is ${\kappa }_2{M}_0$ for a fixed $0\ne {\kappa }_2\in \mathbb{R}$.

For $\omega \in S\left(N_{\mathsf{\Gamma }}\right)$, the non-scalar link map is

$$\mathcal{M}:N_{\mathsf{\Gamma }}\oplus {Sym}_0^2\left(N_{\mathsf{\Gamma }}\right)⟶V_1\oplus V_2,\mathcal{M}(m,M_0)=\left(m·\omega ,{\omega }^T{M}_0\omega \right).$$

(185)

This is the equivariant identification of oriented normal vectors with degree-one spherical harmonics and trace-free quadratic forms with degree-two spherical harmonics.

Let ${\mathsf{N}}_{\mathrm{mom}}$ act with eigenvalue 1 on $N_{\mathsf{\Gamma }}$ and eigenvalue 2 on ${Sym}_0^2\left(N_{\mathsf{\Gamma }}\right)$. Then

$$\mathcal{M}\circ {\mathsf{N}}_{\mathrm{mom}}={\mathsf{N}}_{\partial }\circ \mathcal{M}.$$

(186)

Thus the moment map preserves the source Euler grading. Frozen normal transport preserves the same homogeneous degree on a source-free annulus. The two-channel locus $m\ne 0$, $M_0\ne 0$ is open and dense in the finite moment space.

A prescribed trace-free second moment is compatible with a positive core profile. A sufficiently large scalar trace may be added to the raw second moment; this changes only the separated scalar channel.

The associated non-scalar two-jet is

$${\left(gr{J}_{\perp }^{\le 2}\right)}_{\mathrm{ns}}=\mathcal{M}(m,M_0)\in V_1\left[1\right]\oplus V_2\left[2\right].$$

(187)

Equation (186) supplies the source-side grading required by entry A of Table 4. Together with faithful central sector reconstruction and minimal Borel-Weil support, it gives the section-space level identification of Theorem 1.

Figure 2. Preservation of the source Euler grading by the moment-resolved link map.

Figure 2. Preservation of the source Euler grading by the moment-resolved link map.

4.4. Gauss-Local Charges and Transported Boundary Data

Let $j_{\mathrm{tr}}^i$ be the principal normal reduction of (162). Extend the first moment m in the frozen normal frame by ${\nabla }^{\perp }m=0$ and put ${\mathbf{j}}_1^i=m{j}_{\mathrm{tr}}^i$. Let $A_{ij}$ be the trace-free principal Codazzi representative.

Lemma 10

(Principal current and Codazzi-Gauss charges). On a source-free principal normal annulus, ${\nabla }_i^{\perp }{\mathbf{j}}_1^i=0$. The $V_1$ current-flux charge is

$$Q_1\left(r\right)={\int }_{S_r^2}{\mathbf{j}}_1^i{n}_{i}dS\in N_{\mathsf{\Gamma }}\simeq V_1.$$

(188)

Writing $q_{\mathrm{tr}}\left(r\right)={\int }_{S_r^2}{j}_{\mathrm{tr}}^i{n}_{i}dS$, one has $Q_1\left(r\right)=m{q}_{\mathrm{tr}}\left(r\right)$ and

$$Q_1\left(r_1\right)-Q_1\left(r_0\right)={\int }_{{\mathsf{\Omega }}_{r_0,r_1}}{\nabla }_i^{\perp }{\mathbf{j}}_1^{i}dV=0.$$

(189)

The trace-free Codazzi representative has a harmonic Hessian potential,

$$A_{ij}={\partial }_i{\partial }_{j}u,\mathsf{\Delta }u=0.$$

(190)

For $M\in {Sym}_0^2\left(N_{\mathsf{\Gamma }}\right)$, let $H_M\left(x\right)=M_{ij}{x}^i{x}^j$. The $V_2$ charge is determined by

$$〈Q_2\left(r\right),M〉={\int }_{S_r^2}\left(u{\partial }_n{H}_M-H_M{\partial }_{n}u\right)dS.$$

(191)

It is independent of r, and $Q_2=0$ if and only if $M_0=0$.

Proof. 

The principal current equation gives ${\partial }_i{j}_{\mathrm{tr}}^i=0$. Parallelity of m gives ${\nabla }_i^{\perp }{\mathbf{j}}_1^i=0$, and Stokes’ theorem gives (189).

For fixed k, the principal Codazzi equation makes $A_{ik}d{x}^i$ closed. Since $H^1({\mathbb{R}}^3\setminus \left\{0\right\})=0$, one obtains $A_{ik}={\partial }_i{v}_k$. Symmetry gives $v_k={\partial }_{k}u$, and trace-freeness gives $\mathsf{\Delta }u=0$. Green’s identity gives the radius independence of (191). By Definition 8, the singular degree-two coefficient is ${\kappa }_2{M}_0$ with ${\kappa }_2\ne 0$, and the pairing on this term is a non-degenerate multiple of the standard $V_2$ inner product. □

The first moment is computed with origin on the regular component selected by Theorem 3. A normal translation which changes this component changes the defect datum. On the locus $q_{\mathrm{tr}}\ne 0$, the two charges are nonzero exactly when $m\ne 0$ and $M_0\ne 0$.

On a curved collar, $Q_1$ is compared by normal parallel transport. The $V_2$ charge is compared by the Green identity for the first-order Codazzi operator $\mathfrak{C}$ and its formal adjoint:

$${\int }_{{\mathsf{\Omega }}_{r_0,r_1}}\left(\langle \mathfrak{C}A,\mathsf{\Psi }\rangle -\langle A,{\mathfrak{C}}^{*}\mathsf{\Psi }\rangle \right)={\mathcal{B}}_{S_{r_1}}(A,\mathsf{\Psi })-{\mathcal{B}}_{S_{r_0}}(A,\mathsf{\Psi }).$$

(192)

If $\mathfrak{C}A=0$ and ${\mathfrak{C}}^{*}\mathsf{\Psi }=0$, the boundary pairing is independent of the linking radius. The transported current law is ${\mathsf{P}}_{r_0\leftarrow r_1}^{\perp }{Q}_1\left(r_1\right)=Q_1\left(r_0\right)$.

The fixed charge pair generates the boundary-charge algebra used in Definition 4. Lemma 3 then gives central type projections in every finite sector representation. Thus the current flux realizes the $V_1$ sector and the Codazzi-Gauss pairing realizes the $V_2$ sector.

4.5. Torsor-Admissible Transport and Spectral Admissibility

The charge pair must be transported compatibly with the affine finite shadow before the torsor complex of (97) is obtained. This is an additional boundary condition on the realized source data.

Definition 9

(Torsor-compatible boundary transport). An Alena-Codazzi collar is torsor-admissible when:

(i)

the $V_1$ charge (188) has a compact unitary phase calibration on linking spheres;

(ii)

the $V_2$ charge (191) is represented in a transported unitary Green-adjoint frame;

(iii)

the transported order-three normal-frame action is identified with the translation in (95);

(iv)

the relative charge transport descends to the unitaries $U_a$ in (98);

(v)

its completed low compression is compatible with (131);

(vi)

the edge transport and the low-high coupling are subcritical with respect to the separating Callias-Schur gap.

The transport is torsor-compatible when (102) vanishes. The flat subbranch additionally satisfies (103).

The first two entries fix unitary boundary normalizations. The third and fourth give the finite edge descent. The fifth controls the Schur-visible low symbol. Centrality of the Wilson defect gives a well-defined structural torsor cycle; its value remains a finite holonomy datum.

For torsor-admissible data, (189) and (192) induce the transports in (98). The scalar and mixed responses are then those of (105) and (125). The structural closure fixes centrality, while flatness selects the subbranch (103).

The analytic attachment requires a boundary-admissible normal representative with an isolated primitive cluster. The spectral projection is defined by a Riesz contour and persists under perturbations smaller than the spectral separation, in the standard sense of [8]. The open normal problem is compared with the Callias mechanism [9]; its geometric Fredholm form is represented by [10].

The primitive linking degree is locally constant under deformations which preserve the resolved transverse frame. The conditions $m\ne 0$ and $M_0\ne 0$ are open in the finite moment space. A nonzero Codazzi gap remains nonzero on a fixed compact subcollar under sufficiently small constrained perturbations. If the low-high coupling has norm below one half of the Callias-Schur separation, the rank of the isolated low projection remains fixed.

Persistence is taken inside the current-Codazzi class. An unrestricted neighborhood theorem for the multiplier would require solvability of the nonlinear map $(\phi ,B)\mapsto C\left(\phi B\right)$ together with the period condition for (170). Torsor admissibility remains a finite boundary-transport condition.

4.6. Microscopic Normal Family and Marked Schur Descent

Let ${\mathcal{R}}_{\mathrm{mic}}$ collect the differentiable current-Codazzi realization residuals retained before the torsor, completed Schur, and renormalization closures are imposed. On a local physical slice, let ${\mathsf{\Pi }}_{\mathrm{ret}}$ project to the retained microscopic tangent coordinates and let $G_{\mathrm{mic}}>0$ be the residual metric. The microscopic Jacobian and its positive normal family are

$$L_b=G_{\mathrm{mic}}^{1/2}D{\mathcal{R}}_{\mathrm{mic}}\left({\mathsf{\Phi }}_b\right){\mathsf{\Pi }}_{\mathrm{ret}},N_{+}\left(b\right)=L_b^{†}{L}_b.$$

(193)

Only realization residuals preceding the finite torsor, low Schur, and running closures enter ${\mathcal{R}}_{\mathrm{mic}}$. This ordering keeps the finite quantities read from (193) downstream of the microscopic deformation problem.

Let P be an isolated zero projection of $N_{+}\left(0\right)$ such that $L_{0}P=0$, and put $Q=\mathbf{1}-P$, $M=L_{0}Q$, and $G_H=M^{†}M$. The Callias-Schur gap is the invertibility of the positive Gram block $G_H$ on the active high subspace.

Proposition 33

(Microscopic Gram-Riesz splitting). Assume that $G_H$ is invertible. For a tangent direction X, put $A_X={\left({\partial }_{X}L\right)}_{0}P$ and define

$${\mathsf{\Pi }}_{\mathrm{ran}}=M{G}_H^{-1}{M}^{†},{\mathsf{\Pi }}_{\mathrm{cok}}=\mathbf{1}-{\mathsf{\Pi }}_{\mathrm{ran}},{\mathcal{J}}_X={\mathsf{\Pi }}_{\mathrm{cok}}{A}_X,{\mathcal{B}}_X=Q\left({\partial }_{X}P\right)P=-G_H^{-1}{M}^{†}{A}_X.$$

(194)

Then

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

(195)

The first term is the stationary Kuranishi-Schur quadratic form, while ${\mathcal{B}}_X$ is the second fundamental form of the moving Riesz bundle.

Proof. 

Differentiation of $N_{+}=L^{†}L$ at $L_{0}P=0$ gives $Q\left({\partial }_X{N}_{+}\right)P=M^{†}{A}_X$. The standard differentiated spectral-projection equation therefore gives the last identity in (194). Since ${\mathsf{\Pi }}_{\mathrm{ran}}$ is the orthogonal projection onto $RanM$, one has $A_X={\mathsf{\Pi }}_{\mathrm{cok}}{A}_X+{\mathsf{\Pi }}_{\mathrm{ran}}{A}_X={\mathcal{J}}_X-M{\mathcal{B}}_X$. Orthogonality of the two residual components gives (195). □

Theorem 4

(Equivariant metric Gram-Riesz descent). Let $g:b\to b^{\prime }$ be an admissible marking-preserving reconstruction-frame transition, and let $U_g$ and $V_g$ be the corresponding unitary identifications of the retained microscopic tangent and residual fibers. Assume

$$L_{b^{\prime }}{U}_g=V_g{L}_b.$$

(196)

Let the selected low cluster be isolated by a common Riesz contour and write $Q_b=\mathbf{1}-P_b$, $M_b=L_b{Q}_b$, and $G_H\left(b\right)=M_b^{†}{M}_b$. If $G_H\left(b\right)$ is invertible on the active high subspace, then

$$N_{+}\left(b^{\prime }\right)=U_g{N}_{+}\left(b\right)U_g^{-1},P_{b^{\prime }}=U_g{P}_b{U}_g^{-1},G_H\left(b^{\prime }\right)=U_g{G}_H\left(b\right)U_g^{-1}.$$

(197)

If derivatives are taken with Hermitian connections compatible with $U_g$ and $V_g$, the data of (194) satisfy

$${\mathcal{J}}_{g_{*}X}=V_g{\mathcal{J}}_X{U}_g^{-1},{\mathcal{B}}_{g_{*}X}=U_g{\mathcal{B}}_X{U}_g^{-1}.$$

(198)

Consequently the reduced Kuranishi-Schur quadratic form in (195) and the curvature of the projected connection (148) descend up to unitary conjugation. Their rank, spectrum, gap, trace invariants, and determinant holonomy are basic on the protected quotient.

Proof. 

Equation (196) and unitarity give the first identity in (197). Riesz functional calculus gives the covariance of $P_b$ and $Q_b$. Hence $M_{b^{\prime }}=V_g{M}_b{U}_g^{-1}$, which gives the covariance of $G_H$ and of the projections ${\mathsf{\Pi }}_{\mathrm{ran}}$ and ${\mathsf{\Pi }}_{\mathrm{cok}}$ in (194). Compatible covariant differentiation gives $A_{g_{*}X}=V_g{A}_X{U}_g^{-1}$, and substitution in (194) gives (198). The reduced quadratic form therefore transforms by $U_g$-conjugation. The projected connection has the corresponding unitary gauge transformation, so its curvature transforms by conjugation as well. □

On an irreducible active high channel, equivariance of $G_H$ makes its restriction scalar by Schur’s lemma. The positive Gram gap and the singular gap of an odd fermionic lift are distinguished below. The projected connection of (148) is determined by the ${\mathcal{B}}_X$ component, while stationary reduction retains the ${\mathcal{J}}_X$ component. Theorem 4 makes the basicness of these Gram-Riesz data a consequence of microscopic Jacobian covariance rather than an independent marked-descent condition.

Definition 10

(Common Clifford-Picard high-gap completion). Let $D_{\mathrm{ad}}:=P_{{\mathfrak{a}}_W}+2P_{{\mathfrak{a}}_C}$ be the block-label lift of (58) to ${\mathfrak{a}}_W\oplus {\mathfrak{a}}_C$. Using (96), (68), and (78), put

$${\widehat{\mathsf{\Omega }}}_{\mathrm{adj}}={\displaystyle \frac{dim{\mathcal{H}}_{\mathrm{cen}}}{{Tr}_{F_{\mathsf{\Gamma }}^{\mathrm{even}}}\left(X_{\mathrm{F}}^2\right)}}{\left(dim{F}_{\mathsf{\Gamma }}^{\mathrm{even}}\right)}^{D_{\mathrm{ad}}-1}={\displaystyle \frac{3}{80}}{P}_{{\mathfrak{a}}_W}+{\displaystyle \frac{3}{5}}{P}_{{\mathfrak{a}}_C}.$$

(199)

With $M_H>0$ the adjoint high scale, the normalized branch imposes

$${\left.G_H\right|}_{{\mathcal{H}}_{H,\mathrm{adj}}}=M_H^2{\mathbf{1}}_{\mathrm{cen}}\otimes {\mathbf{1}}_{\mathrm{par}}\otimes {\widehat{\mathsf{\Omega }}}_{\mathrm{adj}}^2,{\mathcal{D}}_{H,\mathrm{adj}}^F=M_H{\mathbf{1}}_{\mathrm{cen}}\otimes J_{\mathrm{par}}\otimes {\widehat{\mathsf{\Omega }}}_{\mathrm{adj}},$$

(200)

where $J_{\mathrm{par}}$ is the self-adjoint involution exchanging the matched parity/chirality lines of Proposition 26. Since $J_{\mathrm{par}}^2=\mathbf{1}$, one has ${\left({\mathcal{D}}_{H,\mathrm{adj}}^F\right)}^{†}{\mathcal{D}}_{H,\mathrm{adj}}^F=G_H{|}_{{\mathcal{H}}_{H,\mathrm{adj}}}$. The threshold subbranch additionally places ${\mathcal{D}}_{H,\mathrm{adj}}^F$ in a Grassmann-valued four-dimensional Dirac kinetic operator with the standard Dirac principal symbol. Under this kinetic lift the three parity pairs are three Dirac adjoint multiplets, equivalently six Weyl/Majorana adjoint multiplets for one-loop counting.

The singular gaps of (200) are therefore $M_2=3M_H/80$ on ${\mathfrak{a}}_W$ and $M_3=3M_H/5$ on ${\mathfrak{a}}_C$. The ratio $M_3/M_2=16$ is fixed by the same normalized high block. The absolute gap normalization and fermionic kinetic lift are completed data; the block scalarity is the equivariant consequence above.

Proposition 34

(Faithful finite-core Schur descent). Let $H^{\mathrm{pr}}={\left(H_{ab}\right)}_{a,b=1,2}$ be a principal source Hessian on $V_1\oplus V_2$, with $H_{11}$ invertible on the active middle channel. On the faithful finite-core images put

$${\widehat{H}}_{ab}={\iota }_{F,a}{H}_{ab}{\rho }_{F,b}.$$

(201)

If ${\rho }_{F,a}{\iota }_{F,a}={id}_{V_a}$, then on the faithful image ${\widehat{H}}_{11}^{-1}={\iota }_{F,1}{H}_{11}^{-1}{\rho }_{F,1}$ and the principal Schur complement satisfies

$${\widehat{H}}_{22}^{\mathrm{red}}={\iota }_{F,2}{H}_{22}^{\mathrm{red}}{\rho }_{F,2}.$$

(202)

Proof. 

The inverse formula follows by composition with ${\rho }_{F,1}{\iota }_{F,1}={id}_{V_1}$. Substitution of (201) into ${\widehat{H}}_{22}-{\widehat{H}}_{21}{\widehat{H}}_{11}^{-1}{\widehat{H}}_{12}$ and cancellation of the intermediate reconstruction-encoding pairs gives (202). □

Definition 11

(Marked microscopic Schur realization). A Schur-admissible microscopic family has marked Schur realization when its principal source Hessian is exported blockwise by (201). The source Euler assignment is already fixed by (186), the local fidelity condition (18), and Theorem 1; no independent Schur-factorization condition is imposed.

Proposition 35

(Marked current-Codazzi support chain). Assume marked microscopic Schur realization and the hypotheses of Proposition 30. After the fixed chirality orientation has been chosen, the principal multiplier block has finite neutral support chain

$$V_2|V_1|V_2⟼C|W_{\sigma }|C,3|2|3.$$

(203)

Here $W_{\sigma }$ is one directed weak leg; the reverse block is its Hermitian adjoint and does not define a second independent incidence. The statement fixes support and factorization pattern; the finite quadratic weights are supplied separately by the represented metric.

Proof. 

Proposition 30 fixes the source incidence $V_2\left|V_1\right|V_2$. Proposition 34 carries its principal Schur reduction to the faithful finite image. The degree assignment sends the outer $V_2$ type to $C=E_3$ and the middle $V_1$ type to $W=E_2$ by Theorem 1, giving ranks $3\left|2\right|3$. The chirality-oriented weak leg is fixed by the linear odd-channel classification of Table 8, and Hermitian adjointness fixes the reverse orientation. □

Proposition 36

(Determinant-centered weak-leg connection). Let $X={\partial }_{\vartheta }$ be the calibrated phase direction of the directed weak leg in (203). Write $a_C$ and $a_W$ for the block-scalar coefficients of the determinant-reduced projected connection along this direction and put $r_{\vartheta }=a_W-a_C$. The determinant condition (80) gives

$$(a_C,a_W)=r_{\vartheta }\left(-{\displaystyle \frac{2}{5}},{\displaystyle \frac{3}{5}}\right).$$

(204)

For the unit weak-leg lift $r_{\vartheta }=1$, the projected connection is

$${\mathcal{A}}_{\vartheta }^{\det }=i\left(P_W-{\displaystyle \frac{2}{5}}\mathbf{1}\right)d\vartheta =-{\displaystyle \frac{i}{5}}{H}_{\mathsf{\Gamma }}^{\mathrm{deg}}d\vartheta .$$

(205)

Proof. 

The unit relative weak-leg lift has raw central coefficient difference $a_W-a_C=1$. Determinant reduction projects it to $s\left(u\right(C)\oplus u(W\left)\right)$, so $3a_C+2a_W=0$ by (80). The general normalization gives (204); the unit value and (59) give (205). □

For a microscopic phase deformation, ${\mathcal{B}}_{\vartheta }$ in (194) fixes the off-diagonal derivative of the Riesz projection and hence the projected transport. The scalar $r_{\vartheta }$ in (204) is the corresponding relative phase rate after the torsor-compatible calibration. Thus $r_{\vartheta }=1$ is the microscopic target of the unit weak-leg branch, while the determinant-centered factor $-2/5$ is fixed independently of that rate.

For the normalized neutral phase branch, the oriented neutral coefficient $c_{Z{S}^2}^{\mathrm{neu}}$ is attached to the outer C transport by

$${\partial }_{\vartheta }arg{c}_{Z{S}^2}^{\mathrm{neu}}=a_C=-{\displaystyle \frac{2}{5}}{r}_{\vartheta }.$$

(206)

This is the completed Takagi phase-attachment condition; its block weight is fixed by (204).

4.7. Alena-Codazzi Realization Theorem

Definition 12

(Primitive Alena-Codazzi realization class). A current-residual Alena-Codazzi collar belongs to the primitive realization class when:

(i)

(161), (162), and (163) hold;

(ii)

the split conservation law (165) holds on the punctured collar;

(iii)

the residual scalar satisfies Proposition 29;

(iv)

the compact-leaf source has a nonzero frozen gap and a one-core thin limit;

(v)

the selected thin-core component is timelike, regular, multiplicity one, and carries the primitive linking charge;

(vi)

the moment pair satisfies $m\ne 0$, $M_0\ne 0$, and $q_{\mathrm{tr}}\ne 0$;

(vii)

the local algebra is relative Gauss-local with the charge pair fixed on the boundary.

It is Schur-admissible when the microscopic family (193) has an open Callias-Schur gap and subcritical low-high coupling, and when its microscopic Jacobian satisfies (196) under admissible reconstruction-frame transitions. Theorem 4 then supplies the basic Riesz projection, low-high blocks, projected connection, and reduced normal form. It has marked Schur realization when the blockwise finite image of Definition 11 is used.

The relation between the structural entries of Section 3 and the present realization is summarized in Table 10.

Theorem 5

(Alena-Codazzi realization of the primitive graded defect). Let $\left\{{\mathcal{C}}_{\epsilon }\right\}$ be a family of collars satisfying:

(i)

current-Codazzi closure in the sense of Proposition 29;

(ii)

the compact-leaf source construction of Proposition 31;

(iii)

the current-residual compatibility of Proposition 32;

(iv)

the thin-core hypotheses of Theorem 3;

(v)

a moment-resolved source with $m\ne 0$, $M_0\ne 0$, and $q_{\mathrm{tr}}\ne 0$;

(vi)

a relative Gauss-local algebra with the two boundary charges fixed.

The limiting collar supplies a primitive graded defect datum in the sense of Definition 5. Its source Euler grading is preserved by (186). After faithful marked visibility in entry B, minimal reconstruction gives the finite support (56), and degree fidelity follows on that support. If the collar is Schur-admissible, the selected support is represented by an isolated low bundle on the protected quotient. Its rank persists under the stated gap condition, while its Riesz projection, projected connection, and reduced normal penalty descend by Theorem 4. If it has marked Schur realization and the frozen principal normal multiplier coefficient is non-degenerate, Proposition 35 gives the finite support chain (203).

Proof. 

The current-Codazzi scalar and the exact multiplier are supplied by Proposition 29. The compact-leaf construction gives a source model with nonzero frozen gap, while Proposition 32 gives the current-residual representative.

Theorem 3 identifies the selected regular component with the limiting timelike worldline and supplies its projective link and primitive line. The nonzero moment pair gives (187), and (186) preserves its normal order. Lemma 10 gives the two nonzero charges and their invariance on the source-free annulus. Relative Gauss locality supplies their central sector labels. These are the data of Definition 5. The support statement follows from Theorem 1. Spectral persistence follows from the Callias-Schur separation, and the descent of the Gram-Riesz data follows from Theorem 4. The marked multiplier statement follows from Proposition 35 when its additional hypotheses hold. □

Corollary 12

(Realization of the finite boundary transport). Assume the hypotheses of Theorem 5 and let the limiting collar be torsor-admissible. Then its transported charge pair supplies the edge transports of (98). Closure of (102) realizes the projective-color Hodge-spectral module (99) on the boundary data. The flat subbranch is selected by (103).

Proof. 

The $V_1$ transport follows from (189) and the normal connection. The $V_2$ transport follows from (192). Definition 9 gives their descent to the finite edge transports. Closure of (102) places the Wilson element in $Z\left({\mathfrak{A}}_{\mathsf{\Gamma }}^{\mathrm{tr}}\right)$. □

The geometric realization therefore supplies the primitive link, the two graded source channels, their Gauss-local boundary charges, and the admissible finite transport. Minimal marked reconstruction gives degree fidelity, while the microscopic Gram family separates stationary obstruction from Riesz-bundle motion through (195). Under (196) and the gap condition, this Gram-Riesz packet descends by Theorem 4. Under marked Schur realization, Proposition 34 exports the principal multiplier support chain (203). The remaining finite closures are applied through Theorem 2, and the completed direct, neutral, and contact branches are evaluated on the resulting boundary cycle.

5. Schur Completion, Quantitative Branches, and Falsification

The primitive carrier, determinant-compatible Fock module, finite cocharacter shadow, and conditional family factor have been fixed in Section 3. Section 4 supplies the current-Codazzi realization and the microscopic Gram family (193). The completed low operator is read in the order fixed by (25): microscopic obstruction, chiral direct data, determinant data, Hermitian spectral response, relative family motion, neutral Schur response, and scale matching. The $B-L$ and exterior-bidegree filtration of Table 8 is kept throughout.

The distinction between generated directions and completed coefficients is retained at each step. Marked-Schur admissibility promotes the principal multiplier chain to a conditional finite support statement before the neutral trace weights are evaluated. Direct Rees valuations are separated from the normalized Rees-Picard coefficient and from the arithmetic torsor completion. In the neutral mixing sector, the finite Lie directions are exact while their microscopic reduced matrix elements remain completed data. Definition 10 specifies one common high-block normalization for the adjoint thresholds, and Proposition 36 fixes the determinant-centered phase weights. Matrix elements outside these normalized branches, the global strong determinant line, and the remaining Takagi column phases stay outside the structural carrier theorem.

5.1. Completed Low Operator and Protected Schur Descent

The low operator is the locked finite operator of (150), acting on the isolated module supplied by Proposition 28. It is taken after the structural witness descent and the penalty reduction of Lemma 1. The projected connection is (148), and the first high-sector correction is the Schur compression (153). Higher integer moments enter with the suppression stated in Proposition 27. The structural channel classes have already been fixed in Table 8. Mass readings in the completed branch are extracted from singular values, spectral gaps, or finite Hessians of the reduced operator. The microscopic splitting (195) separates the stationary obstruction component from the Riesz-bundle motion before these readings are assigned. The former supplies reduced Schur penalties, while the latter controls the relative-basis response through the projected connection.

The finite Yukawa blocks are compressed matrix elements of the zero-order part of the locked operator:

$${\left(Y_{AB}\right)}_{ij}=〈{\psi }_i^A,P_{\mathrm{low}}\left(c(\mathsf{\Phi }+{\mathsf{\Phi }}^{†})+Q_{\mathrm{hol}}+Q_C\right)P_{\mathrm{low}}{\psi }_j^B〉.$$

(207)

The exterior package fixes the possible pairs $(A,B)$; the entries in (207) are completed-branch data. In a local orthonormal low-mode frame, (148) has matrix coefficients $A_{ij}=\langle {\psi }_i,d{\psi }_j\rangle$. These are the projected gauge coefficients of the selected carrier. The determinant reduction of the projected connection is the projection to (82).

On the determinant-nondegenerate quark branch, the basic strong character is

$${\mathsf{\Xi }}_{\mathrm{str}}:=e^{i{\theta }_{\mathrm{QCD}}}{\displaystyle \frac{det\left(Y_u{Y}_d\right)}{|det(Y_u{Y}_d\left)\right|}}=e^{i\overline{\theta }}.$$

(208)

Scalar quark rephasings preserve the Hermitian direct data while the quark determinant factor and the anomalous QCD theta coordinate transform oppositely. The combination (208) is therefore retained at ${\mathfrak{E}}_{\mathrm{dir},\det }$ before the Hermitian direct quotient. Since $det\left(Y_f{Y}_f^{†}\right)={|det{Y}_f|}^2$, the later direct spectral and CKM data do not reconstruct this phase. The Rees-Picard variables introduced below are relative variables built from the difference of the up- and down-sector Hermitian basis responses. They control mixing, whereas the axial determinant phase depends on the sectorwise sum $argdet{Y}_u+argdet{Y}_d$. Any determinant-reality condition is therefore imposed on pre-Hermitian lifts of $Y_u$ and $Y_d$ separately; no strong-phase conclusion is drawn from the relative packet alone.

Let ${\mathcal{L}}_{\mathrm{str}}^{\mathrm{gen}}$ denote the strong determinant line generated by the completed quark family-index datum and its retained transgressions. An external flat translate ${\mathcal{F}}_{\alpha }$ gives the more general completion

$${\mathcal{L}}_{\mathrm{str}}^{\left(\alpha \right)}={\mathcal{F}}_{\alpha }\otimes {\mathcal{L}}_{\mathrm{str}}^{\mathrm{gen}}.$$

(209)

The autonomous reconstruction class retains ${\mathcal{L}}_{\mathrm{str}}^{\mathrm{gen}}$ and contains no independent parameter $\alpha$. Its holonomy on a primitive color winding generator is the reconstructed strong phase. The general character (208) is retained before Hermitianization so that an externally specified flat theta completion, when present, is treated in the same invariant line.

Remark 4

(Differential-index reading of minimal reconstruction). At determinant depth, the completed operator family determines a differential family-index class in $\widehat{K}$, as represented by [14]. The generated strong line and its canonical transgressions are read from this class. Tensoring by an independent flat class as in (209) changes the reconstruction class. The holonomy generated by the family itself remains part of the completed determinant problem.

Remark 5

(Real generated strong line). If ${\mathcal{L}}_{\mathrm{str}}^{\mathrm{gen}}$ admits a connection-preserving real structure, its holonomy is reduced from $U\left(1\right)$ to $\{\pm 1\}$. The remaining sign is the orientation class of the real generated determinant line along the primitive color cycle and may be formulated as a mod-two determinant/Pfaffian spectral-flow problem. The neutral Pfaffian construction below provides the corresponding finite algebraic pattern but does not compute this color-family sign. The chirality orientation of the finite Clifford module does not by itself establish the required global determinant orientation.

If ${\mathcal{L}}_{\mathrm{str}}^{\mathrm{gen}}$ admits a gauge-invariant parallel trivialization, Proposition 12 gives $\overline{\theta }=0(mod2\pi )$ on the autonomous branch. The construction of the global transgression and the proof of this trivialization remain part of the completed determinant problem.

After the central factor (96) has been isolated, a direct Hermitian family block $H_f=Y_f{Y}_f^{†}$ carrying retained edge transport has a Schur-visible first family-motion part $H_f^{\left(1\right)}$. Its finite Hodge-Schur bookkeeping is

$$H_f^{\left(1\right)}=H_f^{\mathrm{circ}}+{\mathsf{\Gamma }}_f,{\mathsf{\Gamma }}_f={\mathsf{\Pi }}_{\mathrm{noncirc}}\left(H_f^{\left(1\right)}\right).$$

(210)

Here $H_f^{\mathrm{circ}}$ is the transported degree-zero part, while $H_f-H_f^{\left(1\right)}$ contains higher Schur terms not used in the first family-motion test. The projection is the one in (123). Thus only ${\mathsf{\Gamma }}_f$ is tested by the mixed torsor curvature (125) at first order. The vertex-resolved charged spectral representative is reduced before this transport decomposition.

The completed branch is represented by a finite Schur-Kuranishi map. After gauge, diffeomorphism, and unitary redundancies have been removed, let b denote the remaining branch coordinates and put

$${\mathcal{F}}_{\mathrm{fin}}\left(b\right)=P_{\mathrm{obs}}\left(K_{\mathrm{eff}}\left(b\right)\right),{\mathcal{F}}_{\mathrm{fin}}\left(b\right)=0.$$

(211)

Here $P_{\mathrm{obs}}$ denotes the projection to the finite observable obstruction coordinates of the completed low branch. Different sectors of ${\mathcal{F}}_{\mathrm{fin}}$ are filtered by $B-L$ and by exterior bidegree. The Schur entry is refined according to (25). Let ${\mathcal{S}}_{\mathrm{dir}}\left(b\right)$ collect the ordered direct-sector singular values and Hodge coordinates retained at the direct spectral depth. The intermediate protected interface is

$${\mathcal{P}}_{\mathrm{spec}}\left(b\right)=\left(j_b^{r_{\mathrm{spec}}}{\mathcal{F}}_{\mathrm{fin}},{\mathcal{M}}_{\mathrm{pre}-\mathrm{Schur}},{\mathcal{S}}_{\mathrm{dir}}\left(b\right),{\mathsf{\Delta }}_{\mathrm{low}}\left(b\right)\right).$$

(212)

An independent family orientation is not retained at this depth. At relative-basis depth the interface is enlarged to

$${\mathcal{P}}_{\mathrm{Sch}}\left(b\right)=\left(j_b^{r_{\mathrm{Sch}}}{\mathcal{F}}_{\mathrm{fin}},{\mathcal{M}}_{\mathrm{pre}-\mathrm{Schur}},[P_{\mathrm{low}}\left(b\right),{\nabla }_{\mathrm{low}}\left(b\right)],{\mathsf{\Delta }}_{\mathrm{low}}\left(b\right)\right),$$

(213)

where $j_b^{r_{\mathrm{Sch}}}{\mathcal{F}}_{\mathrm{fin}}$ is the jet retained by the chosen Schur order, ${\mathcal{M}}_{\mathrm{pre}-\mathrm{Schur}}$ collects the boundary-sector projections, the $B-L$ and exterior-bidegree filtrations, and the finite-shadow and torsor labels, and $[P_{\mathrm{low}},{\nabla }_{\mathrm{low}}]$ denotes the marking-preserving unitary class of the isolated low bundle with its projected connection. The scalar ${\mathsf{\Delta }}_{\mathrm{low}}$ is the isolating Riesz gap.

Proposition 37

(Protected Schur descent). Let b be a local zero of (211) in a fixed reconstruction class, after the entries preceding ${\mathfrak{E}}_{\mathrm{Schur}}$ in (13) have been closed. Every finite channel generated by the retained source transport or by (153) and required by the protected Schur interface belongs to the initial reconstructed object at this depth. A presentation coordinate absent from this object is not retained in the finite reconstruction core.

Proof. 

The required Schur terms belong to the admissible data before (22) is formed; presentation data outside the least protected object are removed by Proposition 4. □

At the earlier spectral depth, an independent family orientation is retained only when generated by a source transport or required by a protected downstream observable. Gap-mediated terms from (153) and the higher-moment terms of Proposition 27 are retained before auxiliary high coordinates are eliminated.

Figure 3. Completed Schur architecture on the structurally fixed low carrier. The common adjoint high block is normalized by Definition 10.

Figure 3. Completed Schur architecture on the structurally fixed low carrier. The common adjoint high block is normalized by Definition 10.

The reduced completed branch contains the channels forced by the retained observables, the finite filtration, and the gap-mediated compression; auxiliary coordinates outside the least protected Schur interface are removed by Proposition 37.

5.2. Marked Neutral Hessian and Determinant Rees Coordinate

The finite neutral quadratic form is identified with the Gram Hessian (8) followed by the stationary reduction (24). On the multiplicity-free Clifford response spaces, the retained metric is the normalized trace form (69). Its restrictions to the weak-angular reservoir ${\mathcal{R}}_{\angle }=C\oplus W\oplus W^{*}$ and the radial reservoir ${\mathcal{R}}_r=C\oplus W$ are proportional to ${\mathbf{1}}_7$ and ${\mathbf{1}}_5$, respectively. Let ${\tau }_{\angle }={Tr}_{{\mathcal{R}}_{\angle }}/7$ and ${\tau }_r={Tr}_{{\mathcal{R}}_r}/5$.

Under marked Schur realization, Proposition 35 fixes the source support of the principal multiplier Hessian before the trace weights are evaluated. The corresponding finite projector incidence is

$${\mathfrak{I}}_{\mathrm{neu}}=\left(P_C^{\angle },P_C^{\angle }\otimes P_{W_{\sigma }}^{\angle },P_C^{\angle }\otimes P_C^r\right).$$

(214)

The ranks of these three legs are those of (203). The weak orientation is fixed after the chirality convention; its reverse block is the Hermitian adjoint and does not define an additional incidence coefficient. If the blockwise marked Schur realization is not available on the isolated low bundle, (214) retains the status of a finite completion.

The canonical trace gives

$$A_0={\tau }_{\angle }\left(P_C^{\angle }\right)={\displaystyle \frac{3}{7}},B_0={\tau }_{\angle }\left(P_C^{\angle }\right){\tau }_{\angle }\left(P_{W_{\sigma }}^{\angle }\right)={\displaystyle \frac{6}{49}},C_0={\tau }_{\angle }\left(P_C^{\angle }\right){\tau }_r\left(P_C^r\right)={\displaystyle \frac{9}{35}}.$$

(215)

Thus the three leading coefficients are conditional consequences of marked Schur realization and the finite trace metric, rather than independent continuous parameters.

The scalar trace separated before (45) reappears in the scalar completed block. Its compression is block-scalar and carries no torsor clock degree. Tangency to the split determinant condition leaves one first-order direction:

$$\delta S_{\mathrm{sc}}=u{P}_C-{\displaystyle \frac{3}{2}}u{P}_W.$$

(216)

By (59), this tangent is $(u/2)H_{\mathsf{\Gamma }}^{\mathrm{deg}}$. Equation (60) therefore identifies the same line as the infinitesimal determinant-normalized Rees direction on the selected carrier. The primitive scalar unit used in the minimal branch is

$$u_{\mathrm{prim}}={\displaystyle \frac{1}{16S_{\mathrm{prim}}}},S_{\mathrm{prim}}=4{\pi }^2.$$

(217)

The value $S_{\mathrm{prim}}=4{\pi }^2$ is the Bohr-Sommerfeld closed-cell normalization used for the primitive link phase cell. The resolved ${\mathbb{CP}}_{\mathsf{\Gamma }}^1$ link has one complex, equivalently two real, phase directions, and the closed primitive cell carries the two $2\pi$ phase periods. The factor 16 is the dimension of the even package (68). The finite central trace part of the corresponding projective-color reading is the torsor coefficient (135).

In the minimal determinant-tangent branch, (216) leaves one scalar u. With the primitive normalization (217), the neutral determinant seed and the projective-color central seed are two readings of the same finite determinant-tangent unit, the latter using the torsor trace (135).

Remark 6

(Normalization status of the primitive unit). The top-form constraint and the selected carrier fix the one-dimensional direction (216). The absolute normalization in (217) depends on the high-block normalization in the Schur compression (153). In particular, replacing $K_{HH}$ by $c{K}_{HH}$ rescales the second term in (153) by $c^{-1}$. Thus (217) is the minimal-branch normalization of the determinant-tangent unit.

The first determinant-tangent Schur correction is read additively in the same reduced neutral Hessian:

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

(218)

Because (216) is scalar on the family factor and carries no torsor clock degree, a first-order family-shape response requires a retained noncentral direct component. In the subbranch where no such linear component is source-generated, family-shape coordinates begin at quadratic Schur order. This is a conditional selection rule of the completed descent and is tested below.

The rank-split determinant contribution used in this branch is

$${\left({\displaystyle \frac{dimW}{dimC}}\right)}^2{u}_{\mathrm{prim}}={\displaystyle \frac{1}{144{\pi }^2}}.$$

(219)

It is a one-loop-sized completed coefficient obtained from the chosen primitive unit and the rank ratio. The scale-free determinant-shadow angle is

$$s_{\det }={\displaystyle \frac{B}{A+B}}.$$

(220)

The same completed branch assigns the primitive determinant scale

$$v_{\mathrm{EW}}^{\mathrm{prim}}=2\sqrt{2}{M}_{\mathrm{Pl}}exp\left(-S_{\mathrm{prim}}-{\displaystyle \frac{1}{144{\pi }^2}}\right)\sqrt{{\displaystyle \frac{C_0}{C_0+u_{\mathrm{prim}}}}}.$$

(221)

At the same zero-remainder scale, the neutral-cell masses are read by

$$\begin{array}{cccc}\hfill m_{\gamma }^{\mathrm{prim}}& =0,\hfill & \hfill m_W^{\mathrm{prim}}& ={\displaystyle \frac{v_{\mathrm{EW}}^{\mathrm{prim}}}{2}}\sqrt{A},\hfill \\ \hfill m_Z^{\mathrm{prim}}& ={\displaystyle \frac{v_{\mathrm{EW}}^{\mathrm{prim}}}{2}}\sqrt{A+B},\hfill & \hfill m_h^{\mathrm{prim}}& =v_{\mathrm{EW}}^{\mathrm{prim}}\sqrt{C}.\hfill \end{array}$$

(222)

Eliminating $u_{\mathrm{prim}}$ and $v_{\mathrm{EW}}^{\mathrm{prim}}$ from (218) and (222) gives

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

(223)

Equation (223) is parameter-eliminated within marked neutral descent, the tracial metric, and the common determinant-tangent completion.

The same neutral-cell mass reading fixes the tree-level electromagnetic combination. With $G_F^{\mathrm{prim}}:={\left[\sqrt{2}{\left(v_{\mathrm{EW}}^{\mathrm{prim}}\right)}^2\right]}^{-1}$,

$${\alpha }_{\mathrm{cell}}={\displaystyle \frac{AB}{4\pi (A+B)}}={\displaystyle \frac{\sqrt{2}{G}_F^{\mathrm{prim}}{\left(m_W^{\mathrm{prim}}\right)}^2}{\pi }}\left(1-{\displaystyle \frac{{\left(m_W^{\mathrm{prim}}\right)}^2}{{\left(m_Z^{\mathrm{prim}}\right)}^2}}\right).$$

(224)

Thus ${\alpha }_{\mathrm{cell}}$ is the $G_F$-scheme mass combination of the same neutral cell. The corresponding electroweak scheme convention is summarized in [75]; conversion to the Thomson-limit $\alpha \left(0\right)$ belongs to the radiative matching layer.

The Planck scale in (221) supplies the dimensional reference for the determinant suppression. The coefficients A, B, and C and the determinant shadow (220) are read at the zero-remainder neutral-cell matching scale, denoted by ${\mu }_{\mathrm{cell}}$. The numerical branch below takes ${\mu }_{\mathrm{cell}}$ in the electroweak region, with ${\mu }_{\mathrm{cell}}\simeq m_Z^{\mathrm{prim}}$. Accordingly, $B/A$ is used as a neutral-cell Hessian ratio at ${\mu }_{\mathrm{cell}}$; a Planck-scale boundary condition on the running ratio $g^{\prime 2}/g^2$ is not part of this branch. Gauge-coupling evolution is assigned to the common matching and threshold prescription below.

5.3. Direct Spectral Response, Rees-Picard Paths, and Family Polarization

5.3.1. Direct Hodge Rank Ladder

By (70) and $Q=T_3+Y$, the weak one-particle factor splits as $W=W_0\oplus W_{+}$ with charges 0 and 1, while the color factor has charge $-1/3$. Let $T_{+}:W_0\to W_{+}$ be normalized as a partial isometry and put $T_{-}=T_{+}^{†}$, so that $T_{-}{T}_{+}=P_{W_0}$ and $T_{+}{T}_{-}=P_{W_{+}}$. The nested weak-incidence kernels are represented by

$$F_0=\mathbf{1},F_1=\mathbf{1}-T_{-}{T}_{+},F_2=\mathbf{1}-T_{-}{T}_{+}-T_{+}{T}_{-},(rank{F}_0,rank{F}_1,rank{F}_2)=(5,4,3).$$

(225)

The normalized one-particle trace therefore gives ${\rho }_k={tr}_V\left(F_k\right)=(5-k)/5$. If the direct Hodge residual identifies $e^{{\mathsf{\Sigma }}_f}={\rho }_{k_f}$, Lemma 7 gives

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

(226)

A compact bookkeeping of the motivated sector readout uses a color-support bit ${\chi }_C$ and a directed weak-bridge bit ${\chi }_W$, with $k_f={\chi }_C\left(f\right)+{\chi }_W\left(f\right)$. The assignments ${({\chi }_C,{\chi }_W)}_{\ell }=(0,0)$, ${({\chi }_C,{\chi }_W)}_d=(1,0)$, and ${({\chi }_C,{\chi }_W)}_u=(1,1)$ give $k_{\ell }=0$, $k_d=1$, $k_u=2$, and hence $Q_{\ell }=2/3$, $Q_d=3/4$, and $Q_u=8/9$. The weak orientation is read after the chiral convention and $Q=T_3+Y$ have been fixed. The Hodge readout and this sector assignment remain a motivated direct-sector completion; no identification with signed color triality is made.

Lemma 11

(Primitive Hodge-Eisenstein incidence packet). For the first adjacent incidence step in (225), put $p_H=rank{F}_0-rank{F}_1$, $q_H=rank{F}_1$, and $r_H=rank{F}_0$. Then

$$(p_H,q_H,r_H;n_H)=(1,4,5;7).$$

(227)

The second adjacent step gives the packet $(1,3,4)$ and a nonintegral value $13/3$. Hence the first step is the unique adjacent incidence packet of the Hodge ladder compatible with the integral commensurate condition of (116).

Proof. 

The ranks are fixed by (225). Substitution in (116) gives (227); the same substitution for the second adjacent step gives $13/3$. □

5.3.2. Source-Generated Quark Transport and Schur-Berry Motion

The projective-color carrier and its mixed-curvature detector were fixed in SubSection 3.6. Equation (132) places the common degree-zero image in ${\mathcal{A}}_0=\mathbb{C}\left[S\right]$, while the marked positive degrees lie in ${\mathcal{A}}_1\oplus {\mathcal{A}}_2$. In the source-generated quark branch used below, the leading up- and down-type blocks share the degree-zero transport polarization. Their relative family motion is therefore read from the first retained non-circulant degrees. The Picard translation S has degree zero by Lemma 8; positive direct valuation is assigned by the source Rees degree.

Accordingly, the direct flavor problem is treated as the relative response of two sectoral compressions of the same graded source class. The source valuation controls the order at which a non-circulant correction can enter, while the sectoral spectra and low compression control its coefficient. The CKM hierarchy can therefore be tested without an independent entry-by-entry Yukawa ansatz. The mixed-curvature seminorm retains the non-circulant norm but not the commensurate singular-value ratios of $d_{\theta }$, since its scalar term drops from the commutator. Any Hodge-Eisenstein normalization of the sectoral jets must therefore descend through the protected spectral interface before this norm quotient.

The positive normal representative is the microscopic family (193). Let $P_b$ be its isolated low projection and put $Q_b=\mathbf{1}-P_b$. For a branch parameter X, the Schur-visible detector is

$$V_X=Q_b\left({\partial }_X{N}_{+}\right)P_b.$$

(228)

The inverse Sylvester map transfers such a component to the motion of the low projection and hence to the finite Berry-Wilczek-Zee connection [63]. For two branch directions $X,Y$, the curvature of the projected low bundle is read in the standard projected form

$${\mathsf{\Omega }}_{XY}^{\mathrm{BWZ}}=P_b[{\partial }_X{P}_b,{\partial }_Y{P}_b]P_b.$$

(229)

Thus a Schur-visible detector contributes to family mixing through the induced motion of the isolated low projection. Proposition 33 identifies this bundle motion with the high-compensable component of the microscopic derivative, while the complementary component contributes to the stationary Schur penalty. By Proposition 22, the finite test for the central family-motion part is the non-circulant projection (123). Relative $V_1/V_2$ charge transport supplies such a source when its low symbol in (131) is nonzero and has positive degree.

A CP-sensitive central invariant is obtained after the two direct Dirac sectors have been compared. Let ${\mathsf{\Gamma }}_u$ and ${\mathsf{\Gamma }}_d$ denote the first non-circulant Schur-visible Hermitian corrections in the up and down blocks, after the common circulant baseline has been removed. The finite CP-odd diagnostic is

$$J_{\mathrm{cen}}=\Im Tr\left({[{\mathsf{\Gamma }}_u,{\mathsf{\Gamma }}_d]}^3\right).$$

(230)

It vanishes when the two corrections are contained in a common commuting family algebra. The normalization relating (230) to the physical quark invariant is a completed Schur-Berry datum, with the standard comparison convention given by [76].

The Hermitian detector pair (127) is used with $\vartheta ={\vartheta }_{\mathsf{\Gamma }}$. The completed Schur-Berry edge phase is identified in the one-phase lift with the torsor edge phase of (98), subject to the finite-shadow coherence (92). Its finite central loop has the invariant

$${\mathcal{J}}_{\mathrm{cen}}\left({\vartheta }_{\mathsf{\Gamma }}\right)=\Im Tr\left({[K_Z,H_S\left({\vartheta }_{\mathsf{\Gamma }}\right)]}^3\right)={\displaystyle \frac{9\sqrt{3}}{2}}sin\left(3{\vartheta }_{\mathsf{\Gamma }}\right).$$

(231)

Thus the invariant depends on the total phase around the central three-cycle. In the one-phase direct-Dirac reduction, the CP-sensitive closure is

$$J_q=s_{12}{s}_{23}{s}_{13}{c}_{12}{c}_{23}{c}_{13}^{2}sin\left(3{\vartheta }_{\mathsf{\Gamma }}\right).$$

(232)

Here $3{\vartheta }_{\mathsf{\Gamma }}$ is the rephasing-invariant CKM phase in the one-phase reduction. Equation (232) relates the physical Jarlskog invariant to the torsor loop phase after the three CKM magnitudes have been fixed. In the primitive commensurate subbranch of Proposition 21, the torsor spectrum fixes this phase and $J_q$ is then read from the same closure. The carrier, torsor complex, and non-circulant detector remain fixed under this phase closure. The one-phase reduction uses the central torsor closure (102); flatness is the separate subbranch (103). The primitive direct two-jet criterion for this phase identification is isolated below in Proposition 43.

Proposition 38

(Central spectral-CP identity). Assume the scalar one-phase lift, with ${\theta }_{\mathsf{\Gamma }}={\vartheta }_{\mathsf{\Gamma }}$ and cycle-trivial coefficient transport. Then

$${Disc}_{\mu }{\chi }_{{\mathsf{\Delta }}_{\theta }}=108{sin}^2\left(3{\vartheta }_{\mathsf{\Gamma }}\right)={\displaystyle \frac{16}{9}}{\mathcal{J}}_{\mathrm{cen}}{\left({\vartheta }_{\mathsf{\Gamma }}\right)}^2.$$

(233)

Consequently, the central CP loop is nonzero precisely when the scalar torsor spectrum is simple.

Proof. 

Under the stated lift, (101) gives $w_{\mathsf{\Gamma }}=4{sin}^2(3{\vartheta }_{\mathsf{\Gamma }}/2)$. Substitution in (114), followed by (231), gives (233). □

Proposition 39

(Cubic clock-circulant factorization). Let $A=Zf\left(S\right)\in {\mathcal{A}}_1$, put $\mathsf{\Gamma }=A+A^{†}$, and let $H_{\alpha }=h_0\mathbf{1}+\alpha S+\overline{\alpha }{S}^2$ be Hermitian circulant. Then

$$\Im Tr\left({[\mathsf{\Gamma },H_{\alpha }]}^3\right)=-36\sqrt{3}\Im \left({\alpha }^3\right)\Im detA.$$

(234)

Moreover,

$$\left|\Im Tr\left({[\mathsf{\Gamma },H_{\alpha }]}^3\right)\right|\le 3\sqrt{2}{\parallel \mathsf{\Gamma }\parallel }_{\mathrm{F}}^3|\Im \left({\alpha }^3\right)|,$$

(235)

For $\Im \left({\alpha }^3\right)\ne 0$, equality holds precisely when $A^{†}A$ is scalar and $detA$ is purely imaginary. The choice $A=-iZ/2$ and $\alpha =e^{-i{\vartheta }_{\mathsf{\Gamma }}}$ reproduces (231).

Proof. 

The Weyl relation reduces the cubic trace to the determinant character of the degree-one component and gives (234). Since $A^2\in {\mathcal{A}}_2$, one has ${\parallel \mathsf{\Gamma }\parallel }_{\mathrm{F}}^2=2Tr\left(A^{†}A\right)$. The determinant bound $|detA|\le {(Tr\left(A^{†}A\right)/3)}^{3/2}$ gives (235); equality requires equal singular values and $|\Im detA|=|detA|$. □

A Schur-visible central detector contained in $\mathbb{C}\left[S\right]$ is simultaneously diagonalized with the circulant family baseline and has vanishing nonabelian family curvature; equivalently, (123) vanishes in that subcase. A detector pair containing the two degrees in (127) has nonzero commutator for generic ${\vartheta }_{\mathsf{\Gamma }}$ and gives the central loop invariant (231). Relative $V_1/V_2$ charge transport supplies such a clock degree when its Schur-visible low symbol has a nonzero component in ${\mathcal{A}}_1\oplus {\mathcal{A}}_2$.

Let $H_f^{\left(0\right)}$ be the leading circulant Hermitian family block in a direct Dirac sector and let ${\mathsf{\Gamma }}_f$ be the first non-circulant Schur-visible correction. In the Fourier basis of $H_f^{\left(0\right)}$, the first basis change is

$${\left({\mathsf{\Theta }}_f\right)}_{ij}={\displaystyle \frac{{\left({\mathsf{\Gamma }}_f\right)}_{ij}}{h_{f,j}^{\left(0\right)}-h_{f,i}^{\left(0\right)}}},i\ne j,{\left({\mathsf{\Theta }}_f\right)}_{ii}=0.$$

(236)

Lemma 12

(Covariant second family-basis jet). Let $H_f\left(ϵ\right)=H_{f,0}+ϵH_{f,1}+{ϵ}^2{H}_{f,2}+O\left({ϵ}^3\right)$ with simple-spectrum $H_{f,0}\in {\mathcal{A}}_0$, and let $U_f\left(ϵ\right)=exp(ϵK_{f,1}+{ϵ}^2{K}_{f,2}+O\left({ϵ}^3\right))$ diagonalize $H_f\left(ϵ\right)$ in off-diagonal gauge. Let ${\mathcal{S}}_f$ denote the inverse Sylvester map represented by (236). Then

$$\begin{array}{cc}\hfill K_{f,1}& ={\mathcal{S}}_f\left(H_{f,1}^{\mathrm{off}}\right),\hfill \\ \hfill K_{f,2}& ={\mathcal{S}}_f{\left(H_{f,2}+[H_{f,1}^{\mathrm{diag}},K_{f,1}]+{\displaystyle \frac{1}{2}}[H_{f,1}^{\mathrm{off}},K_{f,1}]\right)}^{\mathrm{off}}.\hfill \end{array}$$

(237)

If $b\left(ϵ\right)=b_0+ϵx_1+{ϵ}^2{x}_2+O\left({ϵ}^3\right)$ is a source path, the component linear in the independent second source jet is

$$K_{f,2}^{\mathrm{prim}}={\mathcal{S}}_f\left(D{H}_f{\left[x_2\right]}^{\mathrm{off}}\right).$$

(238)

Proof. 

Expansion of $U_f{\left(ϵ\right)}^{†}{H}_f\left(ϵ\right)U_f\left(ϵ\right)$ to second order and cancellation of its off-diagonal part gives (237). In $H_{f,2}=D{H}_f\left[x_2\right]+{\displaystyle \frac{1}{2}}{D}^2{H}_f[x_1,x_1]$, only the first term is linear in the independent source jet $x_2$, which gives (238). □

Corollary 13

(Determinant-neutral two-jet family transport). Under the hypotheses of Lemma 12, the off-diagonal gauge gives

$$Tr{K}_{f,1}=Tr{K}_{f,2}=0,det{U}_f\left(ϵ\right)=1+O\left({ϵ}^3\right).$$

(239)

Thus the retained two-jet family-basis motion is special-unitary to this order and carries no central determinant phase.

Proof. 

Both generators in (237) are taken in off-diagonal gauge and therefore have zero trace. The determinant identity for the exponential gives (239). □

This statement concerns the Hermitian family-basis transport and does not fix the central chiral determinant lift retained in (208).

Corollary 14

(Low-symbol grading of oriented basis jets). Under the hypotheses of Lemma 12, the inverse Sylvester map commutes with ${Ad}_S$ and

$${\mathcal{S}}_f\left({\mathcal{A}}_r\right)\subset {\mathcal{A}}_r,[{\mathcal{A}}_1,{\mathcal{A}}_1]\subset {\mathcal{A}}_2.$$

(240)

For a family matrix T, write $T^{\left[r\right]}$ for its ${\mathcal{A}}_r$ component in the Weyl expansion (122). Put $G_{f,1}={\left(H_{f,1}^{\mathrm{off}}\right)}^{\left[1\right]}$ and $G_{f,2}^{\mathrm{prim}}={\left(D{H}_f{\left[x_2\right]}^{\mathrm{off}}\right)}^{\left[2\right]}$. Then $B_{f,1}:={\mathcal{S}}_f\left(G_{f,1}\right)\in {\mathcal{A}}_1$ and $C_{f,2}:={\mathcal{S}}_f\left(G_{f,2}^{\mathrm{prim}}\right)\in {\mathcal{A}}_2$. The Hermitian and anti-Hermitian jets are recovered by adjoint completion, so $H_{f,1}^{\mathrm{off}}=G_{f,1}+G_{f,1}^{†}$, $K_{f,1}=B_{f,1}-B_{f,1}^{†}$, and $K_{f,2}^{\mathrm{prim}}=C_{f,2}-C_{f,2}^{†}$. If $B_{u,1}=a_{u}X$ and $B_{d,1}=a_{d}X$ on one retained degree-one line with unitary X, the first-order physical generators commute, and the $x_2$-linear primitive second relative response is obtained from $C_{d,2}-C_{u,2}$ and its adjoint.

Proof. 

Since $H_{f,0}\in {\mathcal{A}}_0=\mathbb{C}\left[S\right]$, its Sylvester denominators are invariant under conjugation by S, which gives the first inclusion in (240). The second follows from the Weyl grading. Hermitian conjugation exchanges ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$, while the inverse Sylvester map sends the corresponding Hermitian off-diagonal pair to its anti-Hermitian pair. The relative statement follows from the second-order Baker-Campbell-Hausdorff expansion. If the oriented first components share the unitary line $\mathbb{C}X$, the completed first-order generators commute because X commutes with $X^{†}$. The oriented degree-two part of ${\displaystyle \frac{1}{2}}{ϵ}^2{K}_{f,1}^2$ produces the generated square line $\mathbb{C}{X}^2$ kinematically. The inverse Sylvester map preserves the graded subspace but need not act scalarly on an individual unitary Weyl line when the simple-spectrum gaps are unequal. □

Lemma 13

(Spectral-gap phase preservation). Let $H_0=diag(h_0,h_1,h_2)$ in the Fourier basis and let $B\in {\mathcal{A}}_1$ be an invertible oriented degree-one component. Put $A_B=[B,H_0]$. With the cyclic ordering fixed by ${\mathcal{A}}_1$,

$$det{A}_B=\mathsf{\Delta }\left(H_0\right)detB,\mathsf{\Delta }\left(H_0\right)=(h_1-h_0)(h_2-h_1)(h_0-h_2)\in \mathbb{R}.$$

(241)

For a simple spectrum, $argdet{A}_B=argdetB(mod\pi )$.

Proof. 

A degree-one Weyl component has one nonzero entry on each edge of the oriented three-cycle. The commutator with $H_0$ multiplies these entries by the three real cyclic spectral differences in (241). Their product gives the determinant factor. Thus the Sylvester spectral gaps change the cubic magnitude and may reverse its orientation sign, but introduce no continuous phase. □

Thus

$$V_{\mathrm{CKM}}=1+{\mathsf{\Theta }}_d-{\mathsf{\Theta }}_u+O\left({\mathsf{\Gamma }}^2\right).$$

(242)

The standard CKM conventions are those of [77,78], and [79]. The CP-sensitive normalization is the one associated with (230).

Proposition 40

(Direct Dirac CKM mechanism). Assume that the leading up- and down-type Hermitian family blocks are simple-spectrum degree-zero low symbols of the same transported $V_1/V_2$ source class. Then they belong to $\mathbb{C}\left[S\right]$, and in the common Fourier phase convention CKM mixing is generated to first order by the relative degree-one and degree-two Schur-visible corrections. It is given by (242), and the finite test for the relevant correction is (125).

Proof. 

Equation (132) places the common degree-zero image in $\mathbb{C}\left[S\right]$, so the leading blocks have a common Fourier basis. A degree-zero correction remains diagonal in that basis and gives no off-diagonal term in (236). Hence only the degree-one and degree-two parts of the Schur-visible correction contribute to the first basis motion. Proposition 22 identifies this part with the mixed torsor curvature norm (125). Taking the relative basis change between the down and up sectors gives (242). □

Corollary 15

(CKM torsor-curvature bound). Assume that the leading circulant Hermitian blocks in the up and down direct Dirac sectors have simple spectra. Let ${\delta }_f={min}_{i\ne j}|h_{f,j}^{\left(0\right)}-h_{f,i}^{\left(0\right)}|$ for $f=u,d$, and let ${\parallel ·\parallel }_{\mathrm{F}}$ be the Frobenius norm on the central block. Then the first-order CKM displacement satisfies

$$\parallel V_{\mathrm{CKM}}{-\mathbf{1}\parallel }_{\mathrm{F}}\le {\displaystyle \frac{1}{\sqrt{3}}}\left({\displaystyle \frac{\parallel [d_{\theta },{\mathsf{\Gamma }}_d]{\parallel }_{\mathrm{F}}}{{\delta }_d}}+{\displaystyle \frac{\parallel [d_{\theta },{\mathsf{\Gamma }}_u]{\parallel }_{\mathrm{F}}}{{\delta }_u}}\right)+O_{{\delta }_u,{\delta }_d}\left((\parallel {\mathsf{\Gamma }}_u{\parallel }_{\mathrm{F}}+\parallel {\mathsf{\Gamma }}_d{{\parallel }_{\mathrm{F}})}^2\right).$$

(243)

Proof. 

Equation (236) gives the first-order basis motion with denominators bounded below by ${\delta }_f$. Taking the Frobenius norm in (242) gives the corresponding sum of up and down contributions. The identity (125) gives $\parallel {\mathsf{\Pi }}_{\mathrm{noncirc}}{\mathsf{\Gamma }}_f{\parallel }_{\mathrm{F}}=3^{-1/2}{\parallel [d_{\theta },{\mathsf{\Gamma }}_f]\parallel }_{\mathrm{F}}$ for the uniform-phase torsor differential (104). The second-order remainder is controlled by the same simple-spectrum gaps. Substitution gives (243). □

Definition 13

(Minimal non-circulant Hodge-Schur direct branch). After the carrier, finite shadow, projective-color torsor, and $B-L$ filtration have been fixed, let ${\mathcal{P}}_{\mathrm{dir}}$ be a reduced source-generated class of Schur-visible direct paths, including their positive valuations and retained path lines. A direct quark branch is called minimal non-circulant Hodge-Schur relative to ${\mathcal{P}}_{\mathrm{dir}}$ if the common circulant baseline is removed as in (210), the branch is non-flat in the seminorm (126), and, at fixed nonzero direct-block normalization $\parallel {\mathsf{\Gamma }}_u{\parallel }_{\mathrm{F}}^2+{\parallel {\mathsf{\Gamma }}_d\parallel }_{\mathrm{F}}^2={\mathsf{\Lambda }}_{\mathrm{dir}}^2$, the relative up-down correction minimizes the residual

$${\mathcal{K}}_{\mathrm{HS}}(u,d)={\displaystyle \frac{{\mathcal{L}}_{\mathsf{\Gamma }}{\left({\mathsf{\Gamma }}_d\right)}^2}{{\delta }_d^2}}+{\displaystyle \frac{{\mathcal{L}}_{\mathsf{\Gamma }}{\left({\mathsf{\Gamma }}_u\right)}^2}{{\delta }_u^2}}$$

(244)

among corrections supported on ${\mathcal{P}}_{\mathrm{dir}}$ whose relative $V_1/V_2$ low symbol has a nonzero component in ${\mathcal{A}}_1\oplus {\mathcal{A}}_2$. Here ${\delta }_f$ is the direct-sector gap used in Corollary 15, while ${\mathsf{\Lambda }}_{\mathrm{dir}}>0$ is completed-branch data expressed in the primitive units of Remark 6. Reduced closure excludes an independent lower-valuation direct $1\leftrightarrow 3$ edge when it is not sourced by the fixed Gauss-local data.

The definition is used only after the structural entries of Table 1 have been fixed. The class ${\mathcal{P}}_{\mathrm{dir}}$ is fixed before the minimization in (244); its reduced line content is specified by the source/path branch used below. On the non-circulant correction space, (125) gives ${\mathcal{L}}_{\mathsf{\Gamma }}{\left({\mathsf{\Gamma }}_f\right)}^2=3{\parallel {\mathsf{\Gamma }}_f\parallel }_{\mathrm{F}}^2$. The residual therefore weights the admissible amplitudes by the direct gaps without distinguishing Weyl directions carrying the same path data. In particular, the extremal central representative $K_Z$ identified by Proposition 39 is not imposed as the shape of a physical Hermitian direct jet. The fixed direct-block normalization prevents continuous rescaling from replacing the minimum by the zero infimum, and the least gap-weighted representative is selected within ${\mathcal{P}}_{\mathrm{dir}}$.

For the first family-motion block in (210), let

$${\rho }_f\left(\beta \right)={\displaystyle \frac{e^{-\beta H_f^{\left(1\right)}}}{{Tr}_{\mathrm{cen}}\left(e^{-\beta H_f^{\left(1\right)}}\right)}},{\mathcal{C}}_f\left(\beta \right)={\mathcal{C}}_{\mathsf{\Gamma }}\left({\rho }_f\left(\beta \right)\right),$$

(245)

where $\beta >0$ is a spectral probe parameter.

Proposition 41

(Entropic family susceptibility). For $\beta \to 0$,

$${\mathcal{C}}_f\left(\beta \right)={\displaystyle \frac{{\beta }^2}{18}}{\mathcal{L}}_{\mathsf{\Gamma }}{\left({\mathsf{\Gamma }}_f\right)}^2+O({\beta }^3\parallel H_f^{\left(1\right)}{\parallel }_{\mathrm{F}}^3).$$

(246)

Consequently, the entropic susceptibility

$${\mathcal{X}}_f^{\mathrm{ent}}:=\underset{\beta \downarrow 0}{lim}{\displaystyle \frac{\sqrt{{\mathcal{C}}_f\left(\beta \right)}}{\beta }}={\displaystyle \frac{1}{3\sqrt{2}}}{\mathcal{L}}_{\mathsf{\Gamma }}\left({\mathsf{\Gamma }}_f\right)$$

(247)

exists, and (243) becomes

$$\parallel V_{\mathrm{CKM}}{-\mathbf{1}\parallel }_{\mathrm{F}}\le \sqrt{6}\left({\displaystyle \frac{{\mathcal{X}}_d^{\mathrm{ent}}}{{\delta }_d}}+{\displaystyle \frac{{\mathcal{X}}_u^{\mathrm{ent}}}{{\delta }_u}}\right)+O_{{\delta }_u,{\delta }_d}\left((\parallel {\mathsf{\Gamma }}_u{\parallel }_{\mathrm{F}}+\parallel {\mathsf{\Gamma }}_d{{\parallel }_{\mathrm{F}})}^2\right).$$

(248)

Moreover,

$${\mathcal{K}}_{\mathrm{HS}}(u,d)=18\left({\displaystyle \frac{{\left({\mathcal{X}}_d^{\mathrm{ent}}\right)}^2}{{\delta }_d^2}}+{\displaystyle \frac{{\left({\mathcal{X}}_u^{\mathrm{ent}}\right)}^2}{{\delta }_u^2}}\right).$$

(249)

Proof. 

At $\beta =0$, the state in (245) is $\mathbf{1}/3$, and its first non-circulant variation is $-\beta {\mathsf{\Gamma }}_f/3$. Equation (246) follows from (141). Equations (247)-(249) follow from (126), (243), and (244). □

Thus the minimal non-circulant Hodge-Schur branch minimizes the weighted entropic susceptibility within the fixed source-generated class ${\mathcal{P}}_{\mathrm{dir}}$. Entanglement-based flavor selection in scattering gives an independent comparison [80]; here the entropy is fixed by the projective-color expectation (138), while the admissible Weyl directions and their valuations are fixed before the gap-weighted minimization.

5.3.3. Rees-Picard Path Hierarchy and Arithmetic CP Completion

The positive direct valuations inherit the two source orders. With no independently sourced lower-valuation $1\leftrightarrow 3$ edge, the connected relative-basis paths have

$$v_{12}=1,v_{23}=2,v_{13}=3.$$

(250)

For direct-sector denominators bounded away from zero in (236), and without an imposed up-down cancellation, the corresponding orders are

$$s_{12}=O\left(ϵ\right),s_{23}=O\left({ϵ}^2\right),s_{13}=O\left({ϵ}^3\right),J_q=O\left({ϵ}^6\right)$$

(251)

for a central loop phase of order one. The valuation counts positive Rees degree. Insertions of the protected Picard translation $S^b$ carry degree zero by Lemma 8.

Lemma 14

(Normalized Weyl path coefficient). Put ${\widehat{W}}_{ab}=3^{-1/2}{Z}^a{S}^b$ for $a,b\in {\mathbb{Z}}_3$, so that $\parallel {\widehat{W}}_{ab}{\parallel }_{\mathrm{F}}=1$. A product containing $v\ge 1$ normalized positive-degree Weyl insertions, with arbitrary unitary degree-zero Picard factors between them, has the form

$${\widehat{W}}_{a_1{b}_1}\dots {\widehat{W}}_{a_v{b}_v}=3^{-(v-1)/2}{e}^{i\varphi }{\widehat{W}}_{ab}$$

(252)

for some $a,b\in {\mathbb{Z}}_3$ and a cubic-root phase $e^{i\varphi }$. If m equal-weight path operators of positive valuation v are pairwise orthogonal in the Hilbert-Schmidt product, the norm coefficient is

$$c_{v,m}=\sqrt{m}{3}^{-(v-1)/2}.$$

(253)

Proof. 

Repeated use of $ZS=\omega SZ$ reduces the positive-degree product to a phase times one Weyl monomial. Degree-zero factors $S^b$ are unitary and do not change the valuation normalization. Hilbert-Schmidt orthogonality gives (253). □

Definition 14

(Primitive Rees-Picard angular data). Let $R_1=B_{d,1}-B_{u,1}=a_{1}X$ be a nonzero oriented degree-one relative response with unitary $X\in {\mathcal{A}}_1$, and let $Y=C_{d,2}-C_{u,2}\in {\mathcal{A}}_2$ be the oriented source-marked primitive second response associated with the $x_2$-linear jet in (238) and Corollary 14. Its component orthogonal to the generated square is

$$Y_{\perp }=Y-{\displaystyle \frac{1}{3}}Tr\left(X^{-2}Y\right)X^2.$$

(254)

Put ${\mathsf{\Delta }}_{\mathrm{prim}}={\parallel Y_{\perp }\parallel }_{\mathrm{F}}^2$. For ${\mathsf{\Delta }}_{\mathrm{prim}}>0$, define

$${\rho }_{\mathrm{RP}}={\displaystyle \frac{\sqrt{3}{\parallel Y_{\perp }\parallel }_{\mathrm{F}}}{\parallel R_1{\parallel }_{\mathrm{F}}^2}}.$$

(255)

Since $X^{-2}{Y}_{\perp }\in \mathbb{C}\left[S\right]$ and has zero trace, it is written uniquely as

$$Y_{\perp }=X^2\left({\beta }_{+}S+{\beta }_{-}{S}^2\right).$$

(256)

The Picard polarization defect is

$${\mathsf{\Delta }}_{\mathrm{pol}}=1-{\displaystyle \frac{|{\langle Y_{\perp },{Ad}_X\left(Y_{\perp }\right)\rangle }_{\mathrm{HS}}{|}^2}{\parallel Y_{\perp }{\parallel }_{\mathrm{F}}^4}}.$$

(257)

If $p=|{\beta }_{+}{|}^2/\left(\right|{\beta }_{+}{|}^2+|{\beta }_{-}{|}^2)$, then ${\mathsf{\Delta }}_{\mathrm{pol}}=3p(1-p)$.

Lemma 15

(Norm Rees-Picard reduction). For the data of Definition 14 with ${\mathsf{\Delta }}_{\mathrm{prim}}>0$, the normalized valuation coefficients are

$$c_1=1,c_2=\sqrt{{\displaystyle \frac{1+{\rho }_{\mathrm{RP}}^2}{3}}},c_3={\displaystyle \frac{{\rho }_{\mathrm{RP}}}{3}}.$$

(258)

The corresponding effective Hilbert-Schmidt ranks satisfy

$$m_{23}^{\mathrm{eff}}=1+{\rho }_{\mathrm{RP}}^2,m_{13}^{\mathrm{eff}}={\rho }_{\mathrm{RP}}^2,m_{23}^{\mathrm{eff}}-m_{13}^{\mathrm{eff}}=1.$$

(259)

Writing the normalized Rees tangent coordinates as ${\sigma }_{12}=ϵc_1$, ${\sigma }_{23}={ϵ}^2{c}_2$, and ${\sigma }_{13}={ϵ}^3{c}_3$, elimination of the common Rees seed and of ${\rho }_{\mathrm{RP}}$ gives

$${\left({\displaystyle \frac{{\sigma }_{23}}{{\sigma }_{12}^2}}\right)}^2-3{\left({\displaystyle \frac{{\sigma }_{13}}{{\sigma }_{12}^3}}\right)}^2={\displaystyle \frac{1}{3}}.$$

(260)

No Picard polarization condition is used in these norm relations. The ${\sigma }_{ij}$ are graded tangent/path coordinates; their identification with exact CKM angles requires the matrix completion below.

Proof. 

The generated square $X^2$ is Hilbert-Schmidt orthogonal to $Y_{\perp }$ by (254). For each of the two components in (256), the Heisenberg relation (133) gives a factor $1+{\omega }^{-b}$ in $X{Y}_{\perp }+Y_{\perp }X$, with $b=1,2$ and unit modulus. The two Picard components remain Hilbert-Schmidt orthogonal, so $\parallel X{Y}_{\perp }+Y_{\perp }{X\parallel }_{\mathrm{F}}={\parallel Y_{\perp }\parallel }_{\mathrm{F}}$. With the normalization of Lemma 14, this gives (258) and (259). Equation (260) follows by eliminating ${\rho }_{\mathrm{RP}}$ between the last two valuation coefficients. □

The two Picard directions in (256) are distinct ${Ad}_X$ eigenlines. Consequently, the initial invariant extension is one-dimensional only when the generated response itself lies in one of these eigenlines. If both coefficients are nonzero, the invariant extension is two-dimensional. The condition ${\mathsf{\Delta }}_{\mathrm{pol}}=0$ therefore remains a restricted generated-line subbranch until a microscopic source-line statement is established.

Definition 15

(Primitive marked Rees-Picard direct branch). For the data of Definition 14 with ${\mathsf{\Delta }}_{\mathrm{prim}}>0$, put ${\ell }_2=\mathbb{C}{Y}_{\perp }$. The data are called marked Rees-Picard compatible when ${\ell }_2$ is invariant under ${Ad}_X$. Equivalently, ${\mathsf{\Delta }}_{\mathrm{pol}}=0$. This is the strictly Picard-polarized subbranch of the generic angular data.

Lemma 16

(Rees-Picard path reduction). On the branch of Definition 15, there is an orientation $\epsilon =\pm 1$ such that

$${\ell }_2=\mathbb{C}{X}^2{S}^{\epsilon }.$$

(261)

After the central cubic word has been removed from relative family motion, the literal retained path-line ranks are $(1,2,1)$. Their norm coefficients and effective ranks are those of (258) and (259).

Proof. 

Write the primitive response as in (256). The two nonzero Picard directions are the two distinct ${Ad}_X$ eigenlines in the traceless part of $\mathbb{C}\left[S\right]$. Hence ${\mathsf{\Delta }}_{\mathrm{pol}}=0$ selects one of them and gives (261). At valuation three the two mixed orderings lie on the same line and differ by a cubic-root phase, giving the literal $(1,2,1)$ line pattern. The norm coefficients are already fixed by Lemma 15. □

Definition 16

(Hodge-Eisenstein matrix-completion branch). Let $L_{ij}=E_{ij}-E_{ji}$ in the ordered family basis, and let $c_2,c_3$ be the unit-normalized coefficients of (258). The Hodge-Eisenstein jet bridge identifies the common-to-relative first-jet ratio with $\mathfrak{q}=q_H/p_H=4$ from (227). The minimal magnitude representatives are

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

(262)

No independent common cubic $L_{13}$ jet is retained. Put $V_{\mathrm{HE}}\left(ϵ\right)=exp(-A_u\left(ϵ\right))exp\left(A_d\left(ϵ\right)\right)$. The one-phase torsor closure is attached after its three standard mixing magnitudes have been extracted.

The identification $\mathfrak{q}=q_H/p_H$ is a completed Schur-Riesz bridge from the protected Hodge incidence data to sectoral reduced matrix elements. It is not implied by the mixed-curvature seminorm alone.

Proposition 42

(Minimal Hodge-Eisenstein unitary completion). On the branch of Definition 16, require the leading 13 entry of $V_{\mathrm{HE}}$ to retain the Rees coefficient $c_3$ and retain no independent connected order-four correction in the 23 entry. Then

$$\mathfrak{t}={\displaystyle \frac{3}{2}}+{\displaystyle \frac{5\sqrt{6}}{12}},\mathfrak{h}=-{\displaystyle \frac{5}{12}}+{\displaystyle \frac{7\sqrt{6}}{24}}.$$

(263)

If the first standard mixing angle is calibrated by $s_{12}={\lambda }_{\mathrm{cen}}$, the physical magnitudes satisfy

$$\begin{array}{cc}\hfill s_{23}& =c_2{\lambda }_{\mathrm{cen}}^2\left(1+{\displaystyle \frac{1}{3}}{\lambda }_{\mathrm{cen}}^2+O\left({\lambda }_{\mathrm{cen}}^4\right)\right),\hfill \\ \hfill s_{13}& =c_3{\lambda }_{\mathrm{cen}}^3\left(1+{\kappa }_{13}^{\mathrm{HE}}{\lambda }_{\mathrm{cen}}^2+O\left({\lambda }_{\mathrm{cen}}^4\right)\right),\hfill \end{array}$$

(264)

where

$${\kappa }_{13}^{\mathrm{HE}}={\displaystyle \frac{185}{192}}-{\displaystyle \frac{203\sqrt{6}}{384}}=-0.33137088\dots .$$

(265)

Thus the $+1/3$ relative correction in the 23 magnitude is fixed by the unitary seed calibration, while the 13 correction is predicted by the same completion.

Proof. 

Expansion of (262) gives

$$\begin{array}{cc}\hfill {\left(V_{\mathrm{HE}}\right)}_{12}& =ϵ-{\displaystyle \frac{1}{6}}{ϵ}^3+O\left({ϵ}^5\right),\hfill \\ \hfill {\left(V_{\mathrm{HE}}\right)}_{23}& =c_2{ϵ}^2+C_{23}{ϵ}^4+O\left({ϵ}^6\right),\hfill \\ \hfill {\left(V_{\mathrm{HE}}\right)}_{13}& =B_3{ϵ}^3+C_{13}{ϵ}^5+O\left({ϵ}^7\right),\hfill \end{array}$$

(266)

with $B_3=\mathfrak{h}+\mathfrak{t}/2-3c_2/2$, $C_{23}=3\mathfrak{h}/2+5\mathfrak{t}/12-11c_2/6$, and $C_{13}=19c_2/12-11\mathfrak{h}/6-13\mathfrak{t}/32$. The conditions $B_3=c_3$ and $C_{23}=0$ give (263). The first line gives $ϵ={\lambda }_{\mathrm{cen}}+{\lambda }_{\mathrm{cen}}^3/6+O\left({\lambda }_{\mathrm{cen}}^5\right)$. Substitution in the last two lines gives (264); the remaining fifth-order coefficient gives (265). □

For unitary $X\in {\mathcal{A}}_1$, the operator $Z^{†}X$ belongs to $\mathbb{C}\left[S\right]$. A common unitary circulant rephasing therefore reduces the retained line $\mathbb{C}X$ to $\mathbb{C}Z$ while leaving the circulant baseline fixed. This frame is used in the following phase diagnostic.

Proposition 43

(Primitive cubic phase factorization). In the frame $X=Z$, write (256) as $Y_{\perp }=Z^2({\beta }_{+}S+{\beta }_{-}{S}^2)$ and put $H_Y=Y_{\perp }+Y_{\perp }^{†}$. Then

$${\mathcal{J}}_{2\mathrm{j}}:=\Im Tr\left({[K_Z,H_Y]}^3\right)={\displaystyle \frac{9\sqrt{3}}{2}}\Im \left({\beta }_{-}^3-{\beta }_{+}^3\right).$$

(267)

Write ${\beta }_{+}=r_{+}{e}^{-i{\vartheta }_{\mathsf{\Gamma }}}{\zeta }_{+}$ and ${\beta }_{-}=r_{-}{e}^{i{\vartheta }_{\mathsf{\Gamma }}}{\zeta }_{-}$ with $r_{\pm }\ge 0$ and $|{\zeta }_{\pm }|=1$. If the forward and reverse primitive Picard components descend as adjoint-dual orientations of one unitary coefficient line, define its residual holonomy by ${\zeta }_{+}^3=e^{i{\psi }_{\mathrm{prim}}}$ and ${\zeta }_{-}^3=e^{-i{\psi }_{\mathrm{prim}}}$. Then

$${\mathcal{J}}_{2\mathrm{j}}={\displaystyle \frac{9\sqrt{3}}{2}}\left(r_{+}^3+r_{-}^3\right)sin\left(3{\vartheta }_{\mathsf{\Gamma }}-{\psi }_{\mathrm{prim}}\right).$$

(268)

If the induced primitive coefficient transport is cycle-trivial, ${\psi }_{\mathrm{prim}}=0(mod2\pi )$ and the phase in (268) is the torsor loop phase.

Proof. 

Direct Weyl multiplication using $ZS=\omega SZ$ gives (267). Under the adjoint-dual holonomy condition the two cubic terms have the common phase $3{\vartheta }_{\mathsf{\Gamma }}-{\psi }_{\mathrm{prim}}$, which gives (268). □

Under the adjoint-dual condition of Proposition 43, the Picard shape enters the cubic CP diagnostic through the real factor $r_{+}^3+r_{-}^3$, while the phase is $3{\vartheta }_{\mathsf{\Gamma }}-{\psi }_{\mathrm{prim}}$. The cycle-trivial primitive coefficient condition is a direct completed transport condition. Definition 9 controls the boundary charge transports $U_a$ and their low compression, but does not by itself identify the completed primitive direct coefficient line. Thus ${\psi }_{\mathrm{prim}}=0$ is kept separate from the Picard polarization condition ${\mathsf{\Delta }}_{\mathrm{pol}}=0$. The normalization from this central two-jet diagnostic to the physical invariant remains the completed Schur-Berry datum specified after (230).

Lemma 17

(Weyl determinant cubic split). In the frame $X=Z$, let $M=aZ+Z^2({\beta }_{+}S+{\beta }_{-}{S}^2)$. Then $TrM=Tr{M}^2=0$ and

$$det(\mathbf{1}+M)=1+a^3+{\beta }_{+}^3+{\beta }_{-}^3.$$

(269)

Thus the determinant-sensitive cubic depends on the symmetric combination ${\beta }_{+}^3+{\beta }_{-}^3$, whereas (267) depends on the antisymmetric combination ${\beta }_{-}^3-{\beta }_{+}^3$.

Proof. 

Weyl orthogonality gives the two vanishing traces. Using $ZS=\omega SZ$ and $Z^3=S^3=\mathbf{1}$ gives $Tr{M}^3=3(a^3+{\beta }_{+}^3+{\beta }_{-}^3)$. The $3\times 3$ Newton identity then gives (269). □

Corollary 16

(Orientation-real balanced Weyl packet). Let ${\mathfrak{c}}_{\mathrm{Pic}}$ be the canonical real structure of the regular ${\mathbb{Z}}_3$ coefficient module. In the vertex basis it may be written as ${\mathfrak{c}}_{\mathrm{Pic}}=RK$, where K is coefficientwise complex conjugation and $R{e}_a=e_{-a}$; in the $X=Z$ coefficient frame it satisfies ${\mathfrak{c}}_{\mathrm{Pic}}Z{\mathfrak{c}}_{\mathrm{Pic}}^{-1}=Z$ and ${\mathfrak{c}}_{\mathrm{Pic}}S{\mathfrak{c}}_{\mathrm{Pic}}^{-1}=S^{-1}$. If the Weyl packet of Lemma 17 is fixed by ${\mathfrak{c}}_{\mathrm{Pic}}$, then $a\in \mathbb{R}$ and ${\beta }_{-}=\overline{{\beta }_{+}}$. Hence the symmetric cubic in (269) is real, while the antisymmetric cubic in (267) is generically nonzero. The fixed locus has $p=1/2$ and ${\mathsf{\Delta }}_{\mathrm{pol}}=3/4$.

Proof. 

Antilinearity and the stated action on Z and S give $a=\overline{a}$ and exchange the two degree-two coefficients by conjugation. The two cubic combinations then have the stated reality properties, and the value of ${\mathsf{\Delta }}_{\mathrm{pol}}$ follows from its definition. □

The involution in Corollary 16 acts on the abstract Picard coefficient module. It does not impose a common real structure on the physical up- and down-sector family spaces: their coefficient identifications may differ by the retained torsor transport. The relative torsor phase entering (268) is therefore not removed by coefficient-module reality.

Proposition 44

(Sectorwise chiral determinant reality). Suppose that, for each $f=u,d$, the pre-Hermitian direct map admits on a determinant-nondegenerate stratum a factorization $Y_f=Y_f^{\left(0\right)}(\mathbf{1}+M_f)$ whose sectoral coefficient identification sends $M_f$ to the Weyl normal form of Lemma 17. If the pulled-back baseline and coefficient packet are fixed by ${\mathfrak{c}}_{\mathrm{Pic}}$, then $det{Y}_f\in \mathbb{R}$. On a connected component containing a positive reference determinant, $argdet\left(Y_u{Y}_d\right)=0$.

Proof. 

The real structure makes each pulled-back chiral map conjugate to itself, so its determinant is real. Determinant non-degeneracy fixes its sign on each connected component; positivity at the reference point gives the stated phase of the product. □

The coefficients $B_{f,1}$, $C_{f,2}$, and ${\beta }_{\pm }$ in Definition 14 are relative Hermitian data obtained after inverse Sylvester reduction. Proposition 44 therefore requires a separate pre-Hermitian lift in each quark sector and cannot be inferred from the relative balanced condition alone. The canonical candidate ${\mathfrak{c}}_{\mathrm{Pic}}$ is already fixed by the regular ${\mathbb{Z}}_3$ module; the remaining determinant question is whether the direct bridge and boundary transport preserve its sectoral lifts.

The polarized branch ${\mathsf{\Delta }}_{\mathrm{pol}}=0$ and the balanced fixed locus ${\mathsf{\Delta }}_{\mathrm{pol}}=3/4$ are distinct for ${\mathsf{\Delta }}_{\mathrm{prim}}>0$. The first is the boundary specialization $p\in \{0,1\}$ and gives the literal one-line path interpretation of Lemma 16. The balanced locus is the unique interior stationary point of ${\mathsf{\Delta }}_{\mathrm{pol}}=3p(1-p)$ and is fixed by ${\mathfrak{c}}_{\mathrm{Pic}}$. The norm relations of Lemma 15 do not constrain p; no numerical reading used below presently selects either polarization subbranch.

The normalized direct branch takes ${\rho }_{\mathrm{RP}}=1$. By (255), this is the microscopic norm condition $\sqrt{3}\parallel Y_{\perp }{\parallel }_{\mathrm{F}}={\parallel R_1\parallel }_{\mathrm{F}}^2$. It is independent of Picard polarization; the literal path-line interpretation additionally uses Lemma 16. Under the integral reconstruction refinement of Remark 1, the unit condition becomes a consequence if the scalar coefficient is an automorphism of the same canonically identified primitive rank-one $\mathbb{Z}\left[\omega \right]$ coefficient lattice. It then lies in $\mathbb{Z}{\left[\omega \right]}^{\times }={\mu }_6$ and has unit modulus. A nonzero morphism between distinct fractional-ideal coefficient lines does not give this conclusion. Without the identified-lattice automorphism statement, ${\rho }_{\mathrm{RP}}=1$ remains a normalized completed condition. The determinant coordinate supplying the common dimensionless seed is obtained from the projective-color central sector. The finite trace coefficient (135) gives

$${\kappa }_{\mathrm{SB}}=1+{tr}_{\mathrm{cen}}\left({\mathsf{\Delta }}_{\theta }\right){Tr}_{{\mathsf{\Lambda }}^{\mathrm{even}}V}\left(Y^2\right)u_{\mathrm{prim}}=1+{\displaystyle \frac{20}{3}}{u}_{\mathrm{prim}},{\lambda }_{\mathrm{cen}}={\displaystyle \frac{{\kappa }_{\mathrm{SB}}B}{A+{\kappa }_{\mathrm{SB}}B}}.$$

(270)

The factor $20/3$ is the finite torsor trace contribution from (135). The variable u in (216) is an infinitesimal determinant-tangent coordinate, whereas ${\lambda }_{\mathrm{cen}}$ is a finite projective Schur coordinate. Before the primitive normalization is fixed, the same tangent branch gives ${\lambda }_{\mathrm{cen}}\left(u\right)=(12+80u)/(54-67u)$, so ${\lambda }_{\mathrm{cen}}\left(0\right)=2/9$ and $d{\lambda }_{\mathrm{cen}}{/du|}_0=427/243$. Its identification with the normalized direct Rees amplitude is the one-seed calibration of the minimal direct branch. Eliminating $u_{\mathrm{prim}}$ between (220) and (270) gives

$${\lambda }_{\mathrm{cen}}={\displaystyle \frac{507s_{\det }-80}{67+360s_{\det }}}.$$

(271)

This relation inherits marked neutral descent, the common determinant-tangent unit, and the finite trace normalization, but contains no fitted coefficient.

The coefficients of Lemma 15 fix the tangent hierarchy. Exact CKM magnitudes on the Hodge-Eisenstein matrix branch are obtained from $V_{\mathrm{HE}}$ rather than by direct identification of the Rees coordinates with the standard angles. For (309), solving the standard extraction condition $s_{12}={\lambda }_{\mathrm{cen}}$ gives ${ϵ}_{\mathrm{HE}}=0.22702747$ and exact exponentiation of (262) gives

$$s_{12}=0.2250096,s_{23}=0.04203605,s_{13}=0.00373527.$$

(272)

The asymptotic physical magnitude relations are those of (264). The exact tangent identity (260) remains independent of ${\rho }_{\mathrm{RP}}=1$ and Picard polarization, while the physical matrix ratios include the unitary corrections fixed by Proposition 42. No second continuous mixing seed is introduced.

The commensurate-spectrum condition of Proposition 21 remains an arithmetic completion unless the microscopic edge transport is shown to preserve the commensurability class of the integral finite-shadow lattice. The least simple packet is the same $(1,4,5;7)$ packet isolated independently by Lemma 11. Under the integral reconstruction refinement, and with simple spectrum required by the direct Sylvester problem, the least-index admissible commensurate enlargement is then (120). It gives $w_{\mathsf{\Gamma }}=400/343$. Under the scalar one-phase lift, (101) gives $cos{\delta }_{\mathrm{CKM}}=143/343$. With the positive CP orientation,

$$sin{\delta }_{\mathrm{CKM}}^{\min }={\displaystyle \frac{180\sqrt{3}}{343}},{\delta }_{\mathrm{CKM}}^{\min }=65.{360}^{\circ },{\vartheta }_{\mathsf{\Gamma }}^{\min }=21.{787}^{\circ }.$$

(273)

The opposite orientation has ${\delta }_{\mathrm{CKM}}\equiv -65.{360}^{\circ }(mod{360}^{\circ })$. Substitution of (272) and (273) into (232) gives

$$J_q=3.1262\times {10}^{-5}.$$

(274)

The standard CKM convention is the one of [77,78], and [79]; the CP-sensitive comparison uses [76]. The normalized Rees-Picard condition, the Hodge-Eisenstein jet bridge, the cycle-trivial primitive coefficient condition, and microscopic preservation of the commensurate arithmetic class remain separate completions. Their tangent, matrix, or discrete consequences are (260), Proposition 42, and (273).

5.3.4. Charged-Lepton Vertex Polarization and Hodge Response

The charged-lepton block is the color-singlet direct channel $L_L\leftrightarrow e_L^c$ in Table 8. At the direct spectral depth (212), the minimal sourced branch retains the vertex-resolved spectrum and no independent charged edge-transport coefficient. Its spectral representative is selected by

$$R_{\mathrm{vert}}^{\ell }=H_{\ell }-{\mathbb{E}}_{\mathrm{vert}}\left(H_{\ell }\right),R_{\mathrm{vert}}^{\ell }=0,$$

(275)

so that $H_{\ell ,0}\in \mathbb{C}\left[Z\right]$ after an auxiliary torsor origin has been chosen. This is the minimal charged-direct completion at the spectral depth; if a later PMNS map fails to be basic on this quotient, the protected interface must be enlarged.

Let $x_{\ell }$ denote the positive charged-lepton amplitude vector on the three-sector torsor carrier,

$$x_{\ell ,r}=\sqrt{m_{\ell ,r}},r\in {\mathbb{Z}}_3.$$

(276)

The invariant-line and torsor-complement decomposition of Lemma 7 is used on $x_{\ell }$. In the reference normalization ${\theta }_{\mathsf{\Gamma }}=0$, the invariant line is harmonic. If the torsor component is nonzero, then after removal of the overall scale a real positive vector on the three vertices may be written as

$$x_{\ell ,r}=C_{\ell }\left(a_{\ell }+cos\left({\varphi }_{\ell }+{\displaystyle \frac{2\pi r}{3}}\right)\right),r\in {\mathbb{Z}}_3.$$

(277)

This is the real coordinate form of the vertex-carrier decomposition; a mass prediction additionally requires the completed direct block. At ${\varphi }_{\ell }=0$ two vertex amplitudes are degenerate. The generator G in (127) gives $x_{\ell }\left(\varphi \right)=e^{-\varphi G}{x}_{\ell }\left(0\right)$ in the real torsor representation, with the invariant line fixed. Thus ${\varphi }_{\ell }$ is the finite angular coordinate resolving the torsor doublet.

Put ${\mathfrak{X}}_0={\sum }_r{x}_{\ell ,r}$ and ${\mathfrak{X}}_1={\sum }_r{x}_{\ell ,r}{\omega }^{-r}$. In the coordinate convention (277), ${\mathfrak{X}}_1=(3/2)C_{\ell }{e}^{i{\varphi }_{\ell }}$. The balance and clock coordinates are the real and imaginary parts of one local complex response,

$${\mathcal{Z}}_{\ell }=log{\displaystyle \frac{{\mathfrak{X}}_0}{\sqrt{2}{\mathfrak{X}}_1}},{\mathfrak{B}}_{\ell }=2Re{\mathcal{Z}}_{\ell },{\varphi }_{\ell }=-Im{\mathcal{Z}}_{\ell }.$$

(278)

The scale-free charged-lepton balance residual is

$$R_{\mathrm{bal}}^{\ell }={\mathfrak{B}}_{\ell }-{\mathsf{\Sigma }}_{\ell }.$$

(279)

The scalar ${\mathsf{\Sigma }}_{\ell }$ records the finite Schur correction to the norm balance between the invariant line and the torsor complement. The zero-correction case ${\mathsf{\Sigma }}_{\ell }=0$ is the leading balance reading. In a completed charged-lepton branch, ${\mathsf{\Sigma }}_{\ell }$ would be supplied by the finite Schur tensor of the direct Dirac block.

Lemma 18

(Charged-lepton balance form). For the torsor coordinate form (277), the residual (279) closes if and only if

$$2a_{\ell }^2=e^{{\mathsf{\Sigma }}_{\ell }}.$$

(280)

In particular, the zero-correction balance gives $a_{\ell }=1/\sqrt{2}$.

Proof. 

The singlet component has norm $3C_{\ell }^2{a}_{\ell }^2$. The torsor component has norm ${\displaystyle \frac{3}{2}}{C}_{\ell }^2$, by the standard trigonometric orthogonality on the three-cycle. Substitution in (279) gives (280). □

Applying (106) to (276) identifies the charged-lepton quotient $Q_{\ell }:={\sum }_r{m}_{\ell ,r}/{\left({\sum }_r\sqrt{m_{\ell ,r}}\right)}^2$ with the torsor Hodge quotient. Thus the Koide value $Q_{\ell }=2/3$ [81] is the zero-correction Hodge-balance limit of (279). The central experimental residual provides a diagnostic for the completed charged-lepton Schur correction, with its significance assessed after propagation of the lepton-mass uncertainties. Near the zero-correction branch, writing $a_{\ell }=1/\sqrt{2}+\delta a_{\ell }$ gives ${\mathsf{\Sigma }}_{\ell }=2\sqrt{2}\delta a_{\ell }+O\left(\delta a_{\ell }^2\right)$ from (280). Hence the logarithmic balance defect and the displacement of the torsor-shape parameter are two coordinates on the same residual.

The differential of (278) gives the paired balance and clock responses.

Proposition 45

(Linearized charged-lepton balance response). Let $H_{\ell }=Y_{\ell }{Y}_{\ell }^{†}$ be positive with simple spectrum $h_r>0$ and normalized eigenvectors $u_r$, in a fixed local ordering. Put $x_{\ell }=\mathfrak{s}\left(H_{\ell }\right)$, where

$${\left(D{\mathfrak{s}}_{H_{\ell }}\left[\delta H_{\ell }\right]\right)}_r={\displaystyle \frac{〈u_r,\delta H_{\ell }{u}_r〉}{4h_r^{3/4}}},\mathfrak{s}{\left(H\right)}_r={\lambda }_r{\left(H\right)}^{1/4}.$$

(281)

The linearized balance functional on Hermitian direct-block variations is

$$\begin{array}{cc}\hfill {Tr}_{\mathrm{bal}}^{\left(H_{\ell }\right)}\left(\delta H_{\ell }\right)& =2Re{\displaystyle \frac{〈P_{\mathbf{1}}{x}_{\ell },P_{\mathbf{1}}D{\mathfrak{s}}_{H_{\ell }}\left[\delta H_{\ell }\right]〉}{\parallel P_{\mathbf{1}}{x}_{\ell }{\parallel }^2}}\hfill \\ \hfill & -2Re{\displaystyle \frac{〈P_{\mathrm{T}}{x}_{\ell },P_{\mathrm{T}}D{\mathfrak{s}}_{H_{\ell }}\left[\delta H_{\ell }\right]〉}{\parallel P_{\mathrm{T}}{x}_{\ell }{\parallel }^2}}.\hfill \end{array}$$

(282)

If $D{\mathfrak{s}}_{H_{\ell }}\left[\delta H_{\ell }\right]=(a{P}_{\mathbf{1}}+b{P}_{\mathrm{T}})x_{\ell }$, then ${Tr}_{\mathrm{bal}}^{\left(H_{\ell }\right)}\left(\delta H_{\ell }\right)=2(a-b)$. The conjugate clock functional is

$${Tr}_{\mathrm{clk}}^{\left(H_{\ell }\right)}\left(\delta H_{\ell }\right)=-{\displaystyle \frac{2}{3C_{\ell }}}\sum _{r\in {\mathbb{Z}}_3}sin\left({\varphi }_{\ell }+{\displaystyle \frac{2\pi r}{3}}\right){\left(D{\mathfrak{s}}_{H_{\ell }}\left[\delta H_{\ell }\right]\right)}_r,$$

(283)

and equals $D{\varphi }_{\ell }\left[\delta H_{\ell }\right]$ in the local ordering of (277).

Let a Hermitian perturbation $T_3$ of type $V_3$ satisfy the assumptions of Proposition 27, and assume that its Schur compression is represented in the charged-lepton Hermitian block. At the zero-correction reference ${\mathsf{\Sigma }}_{\ell }\left(0\right)=0$ and a fixed clock reference ${\varphi }_{\ell ,0}$, one has

$$\begin{array}{cc}\hfill {\mathsf{\Sigma }}_{\ell }\left(\epsilon \right)& =-{\epsilon }^2{Tr}_{\mathrm{bal}}^{\left(H_{\ell ,0}\right)}\left(P{T}_{3}Q{\left(K_{HH}^{\left(0\right)}\right)}^{-1}Q{T}_{3}P\right)+O\left({\epsilon }^3\right),\hfill \\ \hfill {\varphi }_{\ell }\left(\epsilon \right)-{\varphi }_{\ell ,0}& =-{\epsilon }^2{Tr}_{\mathrm{clk}}^{\left(H_{\ell ,0}\right)}\left(P{T}_{3}Q{\left(K_{HH}^{\left(0\right)}\right)}^{-1}Q{T}_{3}P\right)+O\left({\epsilon }^3\right).\hfill \end{array}$$

(284)

Proof. 

Equation (281) is obtained from first-order perturbation of the simple eigenvalues of $H_{\ell }$ and the relation $x_{\ell ,r}=h_r^{1/4}$. Differentiation of the logarithmic norm ratio in (279) gives (282). Substitution of (154) gives (284). □

Writing ${\mathcal{S}}_{\ell }^H$ for the Hermitian component of the completed charged-lepton Schur data, the corresponding readings are

$${\mathsf{\Sigma }}_{\ell }={Tr}_{\mathrm{bal}}^{\left(H_{\ell ,0}\right)}\left({\mathcal{S}}_{\ell }^H\right)+O\left(\parallel {\mathcal{S}}_{\ell }^H{\parallel }^2\right),{\varphi }_{\ell }-{\varphi }_{\ell ,0}={Tr}_{\mathrm{clk}}^{\left(H_{\ell ,0}\right)}\left({\mathcal{S}}_{\ell }^H\right)+O\left(\parallel {\mathcal{S}}_{\ell }^H{\parallel }^2\right).$$

(285)

The branch below uses ${\varphi }_{\ell ,0}=2/9$ as a clock-lift completion. Under marked neutral descent, the value $2/9$ is already the common zero-order neutral and central determinant weight; its identification with the charged clock coordinate remains a direct-sector completion. The balance condition does not determine this reference angle. The two linear functionals in (285) are the paired components of the same complex response (278), while the overall scale $C_{\ell }$ remains independent.

5.4. Neutral Determinant, Schur Response, and PMNS

The neutral sector is the $\left|\mathsf{\Delta }\right(B-L\left)\right|=2$ Schur problem of Table 8. Let $D_{\nu }$ be the neutral Dirac bridge, $R_N$ the heavy singlet family denominator, $M_R$ its scale, and $A_L$ the active Majorana correction.

$$K_N=\left(\begin{array}{cc}{A}_L& D_{\nu }^T\\ D_{\nu }& M_R{R}_N\end{array}\right).$$

(286)

The Majorana entries are taken complex symmetric, $A_L^T=A_L$ and $R_N^T=R_N$, so that $K_N^T=K_N$. Their physical neutral bases are defined by Takagi factorization. The heavy scale is separated from its family shape by the convention $|det{R}_N|=1$, so that $M_R={|det\left(M_R{R}_N\right)|}^{1/3}$. This convention removes the rescaling redundancy between $M_R$ and $R_N$ and carries no additional spectral prediction.

The quadratic half-flux normalization used in the minimal neutral branch is

$$S_M={\displaystyle \frac{1}{4}}{S}_{\mathrm{prim}},M_R=M_{\mathrm{Pl}}{e}^{-S_M}.$$

(287)

The unreduced Planck mass convention $M_{\mathrm{Pl}}=1.22089\times {10}^{19}\mathrm{GeV}$ is used for the completed numerical reading. The fraction in (287) is a completed geometric scale assumption beyond the carrier theorem. It does not define a holomorphic square root of $L_{\mathsf{\Gamma }}$: a line M with $M^{\otimes 2}\simeq L_{\mathsf{\Gamma }}$ would imply $2c_1\left(M\right)=1$ in $H^2({\mathbb{CP}}^1,\mathbb{Z})$, which has no integral solution.

The Pfaffian structure is attached to the fermionic quadratic form rather than imposed as an additional skew condition on the symmetric family matrix. Let

$$E_{\mathrm{sp}}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),{\mathbb{A}}_N=E_{\mathrm{sp}}\otimes K_N,{\mathbb{A}}_H=E_{\mathrm{sp}}\otimes \left(M_R{R}_N\right).$$

(288)

Then ${\mathbb{A}}_N^T=-{\mathbb{A}}_N$ and ${\mathbb{A}}_H^T=-{\mathbb{A}}_H$ follow from the symmetry of the corresponding family matrices. In a fixed product orientation,

$$Pf\left({\mathbb{A}}_N\right)=-det{K}_N,Pf\left({\mathbb{A}}_H\right)=-det\left(M_R{R}_N\right).$$

(289)

The paired singular values of the skew kernels in (288) belong to the two-dimensional spinor factor and do not constrain the Takagi singular values of $K_N$.

The seesaw comparison is the usual heavy-denominator comparison [82]. If $R_N$ is invertible, the light active block is the Schur complement

$$K_{\nu }^{\mathrm{eff}}=A_L-D_{\nu }^T{\left(M_R{R}_N\right)}^{-1}{D}_{\nu }.$$

(290)

The block determinant identity and (289) give

$$det{K}_{\nu }^{\mathrm{eff}}={\displaystyle \frac{Pf\left({\mathbb{A}}_N\right)}{Pf\left({\mathbb{A}}_H\right)}},m_1{m}_2{m}_3=\left|{\displaystyle \frac{Pf\left({\mathbb{A}}_N\right)}{Pf\left({\mathbb{A}}_H\right)}}\right|,$$

(291)

where $m_i$ are the Takagi singular values of $K_{\nu }^{\mathrm{eff}}$. In the pure type-I subbranch $A_L=0$, the determinant identity reduces to

$$m_1{m}_2{m}_3={\displaystyle \frac{|det{D}_{\nu }{|}^2}{M_R^3}}\left(|det{R}_N|=1\right).$$

(292)

Thus the half-flux scale fixes the denominator scale, while an absolute light-neutrino prediction additionally requires the determinant shape of the neutral Dirac bridge.

The normalized determinant-suppressed type-I branch used below takes $R_N=\mathbf{1}$ and $A_L=0$. Put ${\mathsf{\Sigma }}_{\nu }=diag(B_0,A,1+16u_{\mathrm{prim}})$. In the charged-vertex/heavy-singlet basis its Takagi-compatible Dirac orientation is fixed by

$$D_{\nu }=m_D{\mathsf{\Sigma }}_{\nu }{U}_{\nu }^{†},r_{\nu }:={\displaystyle \frac{|det{D}_{\nu }|}{m_D^3}}=B_{0}A(1+16u_{\mathrm{prim}})=0.05351.$$

(293)

The entries A, $B_0$, and $u_{\mathrm{prim}}$ are those of (215)-(218). Substitution in (290) gives

$$K_{\nu }^{\mathrm{eff}}=-{\displaystyle \frac{m_D^2}{M_R}}{U}_{\nu }^{*}{\mathsf{\Sigma }}_{\nu }^2{U}_{\nu }^{†},m_i={\displaystyle \frac{m_D^2}{M_R}}{\left({\mathsf{\Sigma }}_{\nu }\right)}_{ii}^2.$$

(294)

The common sign in (294) is absorbed by a common Takagi column phase and does not affect the oscillation quotient. Equation (293) is a normalized neutral-shape and orientation completion; no rank reduction is imposed.

The charged-lepton Hermitian block determines a unitary basis $U_e$, while the complex symmetric effective neutral block (290) determines its Takagi basis $U_{\nu }$. The lepton mixing matrix is the relative basis

$$U_{\mathrm{PMNS}}=U_e^{†}{U}_{\nu }.$$

(295)

Thus PMNS is read from the comparison between a $B-L$ preserving direct Dirac basis and a $B-L$ breaking neutral Schur basis. In the minimal sourced charged branch, (275) fixes the leading charged basis to the torsor vertices up to diagonal phases and ordering. The neutral denominator may instead carry the transport polarization and its degeneracy structure. Large PMNS angles can therefore arise from the relative vertex/transport polarization and from the resolved neutral gaps.

At oscillation depth, right multiplication $U_{\nu }\mapsto U_{\nu }{D}_{\eta }$ by a diagonal phase matrix leaves the oscillation probabilities invariant. The neutrinoless-double-beta effective mass,

$$m_{\beta \beta }=\left|\sum _i{m}_i{\left(U_{\mathrm{PMNS}}\right)}_{ei}^2\right|,$$

(296)

depends on the relative column phases. These phases are therefore not basic on the oscillation-only mixing quotient and must be retained when (296) is protected downstream.

A useful diagnostic denominator on the central carrier is the circulant Majorana shape

$$R_N^{\mathrm{circ}}=m\mathbf{1}-r(S+S^{†}).$$

(297)

Its eigenvalues are $m-2r,m+r,m+r$. If $m=2r+\delta$ with $0<\delta \ll r$, then

$${\left(R_N^{\mathrm{circ}}\right)}^{-1}={\displaystyle \frac{1}{3r+\delta }}\mathbf{1}+{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{1}{\delta }}-{\displaystyle \frac{1}{3r+\delta }}\right)J,J_{ab}=1.$$

(298)

The scale-shape convention gives $\delta {(3r+\delta )}^2=1$ for this positive circulant representative. With $x=\delta /r$, the two parameters reduce to the single dimensionless softness x, with $r={\left[x{(3+x)}^2\right]}^{-1/3}$. Thus a soft singlet denominator gives a democratic neutral enhancement. If the charged-lepton block and all neutral blocks are contained in the same circulant algebra $\mathbb{C}\left[S\right]$, the same Fourier basis diagonalizes the family part. Large PMNS angles in this branch require a non-circulant neutral component, a non-circulant charged-lepton component, or a degeneracy resolution outside the common circulant algebra.

A minimal degeneracy-resolution test is obtained by adding a small clock-even perturbation to (297). In the Fourier basis of the circulant denominator, the two heavy eigenvalues $m+r,m+r$ are degenerate. The clock-even perturbation $Z+Z^{†}$ restricts on this degenerate subspace to an off-diagonal two-by-two block. Its eigenvectors are the symmetric and antisymmetric combinations of the two nontrivial Fourier modes. In the torsor basis this gives the leading neutral basis

$${\displaystyle \frac{1}{\sqrt{3}}}(1,1,1),{\displaystyle \frac{1}{\sqrt{6}}}(2,-1,-1),{\displaystyle \frac{1}{\sqrt{2}}}(0,1,-1).$$

(299)

Thus the soft circulant denominator with a clock-even perturbation gives a tri-bimaximal-type leading neutral basis. For the physical column ordering, let $U_{\mathrm{TBM}}$ be obtained from (299) by placing the democratic vector in the second column, and put $L_{ij}=E_{ij}-E_{ji}$ in this ordered basis.

Lemma 19

(TBM torsor Lie closure). Put $H_0=H_S\left(0\right)$. The operators in (127) satisfy

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

(300)

Hence G and $[K_Z,H_0]$ generate $\mathfrak{so}\left(3\right)$ on the real TBM basis.

Proof. 

The identities follow by substitution of (127) in the ordered basis defined by (299). After the second channel has been normalized to unit angular speed, the last identity follows from $-[L_{23},-L_{13}]=L_{12}$. □

Since G fixes the invariant line $P_{\mathbf{1}}{\mathcal{H}}_{\mathrm{cen}}$, a neutral correction contained in the circulant algebra preserves the democratic PMNS column in the minimal charged vertex branch. With the democratic vector assigned to the second column, this gives

$$|{\left(U_{\mathrm{PMNS}}\right)}_{e2}{|}^2={\displaystyle \frac{1}{3}},{sin}^2{\theta }_{12}={\displaystyle \frac{1}{3{cos}^2{\theta }_{13}}}.$$

(301)

The restricted pure-circulant repair therefore does not reproduce the central solar and reactor values of [83]. The required non-circulant real rotation channels are fixed by Lemma 19 without adjoining a new family generator: G supplies the 13 direction, the normalized detector commutator supplies the 23 direction, and their commutator supplies the 12 direction.

The finite Lie closure fixes the available real 13, 23, and 12 directions. Their microscopic reduced matrix elements remain completed data, so the neutral response is normalized separately from the direct Rees-Picard path counting.

Lemma 20

(Transpose grading of the neutral Schur contraction). For the Weyl grading (132), transposition preserves each low-symbol space: ${\mathcal{A}}_r^{\mathsf{T}}={\mathcal{A}}_r$. Hence, if a fixed heavy inverse lies in ${\mathcal{A}}_0$ and degree-pure neutral couplings satisfy $D_1\in {\mathcal{A}}_1$ and $D_2\in {\mathcal{A}}_2$, then

$$D_1^{\mathsf{T}}{H}^{-1}{D}_2+D_2^{\mathsf{T}}{H}^{-1}{D}_1\in {\mathcal{A}}_0.$$

(302)

Thus a non-circulant cubic 12 response is not fixed by low-symbol degree alone.

Proof. 

Transposing $ST{S}^{-1}={\omega }^{-r}T$ and using $S^{\mathsf{T}}=S^{-1}$ gives $S{T}^{\mathsf{T}}{S}^{-1}={\omega }^{-r}{T}^{\mathsf{T}}$. The product rule ${\mathcal{A}}_r{\mathcal{A}}_s\subset {\mathcal{A}}_{r+s}$ then places each term in (302) in ${\mathcal{A}}_0$. □

Definition 17

(Normalized neutral three-channel branch). The normalized neutral branch uses the three generated directions in (300). The adjacent inverse-Sylvester factor in the 23 channel is retained, while the reduced numerators are fixed in the numerical subbranch by

$${\mathsf{\Xi }}_{23}=\sqrt{{\displaystyle \frac{3}{8}}},{\mathsf{\Xi }}_{13}={\displaystyle \frac{13}{7}},{\mathsf{\Xi }}_{12}={\displaystyle \frac{7}{5}}.$$

(303)

These three coefficients are normalized reduced-matrix-element data. The phase rate $r_{\vartheta }$ is the projected-connection diagnostic of (204); the unit weak-leg subbranch takes $r_{\vartheta }=1$.

Writing $R_{12}$ and $R_{13}$ for real column rotations and $R_{23}(\alpha ,\varphi )$ for the complex column rotation with ${\left(R_{23}\right)}_{23}=sin\alpha e^{-i\varphi }$ and ${\left(R_{23}\right)}_{32}=-sin\alpha e^{i\varphi }$, the retained neutral basis is

$$\begin{array}{cc}\hfill U_{\nu }& =U_{\mathrm{TBM}}{R}_{12}\left({\beta }_{\nu }\right)R_{13}\left({\gamma }_{\nu }\right)R_{23}({\alpha }_{\nu },{\varphi }_{\nu }),\hfill \\ \hfill {\alpha }_{\nu }& ={\displaystyle \frac{2}{\sqrt{3}}}{\mathsf{\Xi }}_{23}{\lambda }_{\mathrm{cen}}={\displaystyle \frac{{\lambda }_{\mathrm{cen}}}{\sqrt{2}}},{\gamma }_{\nu }=-{\mathsf{\Xi }}_{13}{\lambda }_{\mathrm{cen}}^2,{\beta }_{\nu }=-{\mathsf{\Xi }}_{12}{\lambda }_{\mathrm{cen}}^3,\hfill \\ \hfill {\varphi }_{\nu }& ={\displaystyle \frac{4\pi }{3}}-{\displaystyle \frac{2}{5}}{r}_{\vartheta }{\vartheta }_{\mathsf{\Gamma }}.\hfill \end{array}$$

(304)

The $4\pi /3$ offset is the ${\omega }^2$ phase of the oriented $Z{S}^2$ coefficient in (128). The second term in ${\varphi }_{\nu }$ is the integrated outer-C attachment (206). The directions in (300) and this determinant-centered phase weight are fixed before the three reduced coefficients in (303) are chosen. Lemma 20 keeps the cubic response at completed Jacobian depth. No additional continuous mixing seed is introduced in this normalized subbranch. The CP invariant is read from the standard rephasing-invariant product $J_{\mathrm{PMNS}}=\Im \left(U_{e1}{U}_{\mu 2}{U}_{e2}^{*}{U}_{\mu 1}^{*}\right)$.

The different sizes of CKM and PMNS mixing have a corresponding reduced-Hessian interpretation. For a simple Hermitian block, (236) makes the quadratic cost of a small basis rotation proportional to the relevant spectral gap squared. The charged-lepton gaps are nonzero, while the circulant neutral denominator has an exactly degenerate doublet before its clock-even resolution. In that doublet a basis choice has vanishing leading gap cost. Under the minimal completed descent, a required leading lepton rotation is therefore assigned to the neutral degenerate subspace before an additional charged transport direction is retained. The quark blocks instead share the degree-zero transport polarization and have simple gaps, so their relative basis motion starts with the non-circulant corrections. This argument is conditional on the sourced-polarization and soft-neutral completions used in this subsection.

A symmetric perturbation $\delta K_N$ induces the skew perturbation $E_{\mathrm{sp}}\otimes \delta K_N$ of (288). Preservation of the neutral quadratic-form class therefore requires no additional family-space skew constraint; the remaining restrictions are the $B-L$ filtration, the retained gaps, and the chosen determinant orientation.

The projective-color expectation (138) may also be applied to a Hermitian representative of the effective neutral block, for example $K_{\nu }^{\mathrm{eff}}{\left(K_{\nu }^{\mathrm{eff}}\right)}^{†}$. The resulting entropy measures the non-circulant component of this Hermitian representative, while the basis response is controlled by the resolved neutral gaps. Hence the direct and neutral sectors use the same sectoral entropy with different spectral susceptibilities: large direct gaps suppress CKM motion through (248), whereas the soft denominator in (298) can amplify PMNS motion. Outside the normalized type-I shape (293), the shapes of $D_{\nu }$, $R_N$, and $A_L$ remain completed data. The relative-basis reading of the normalized three-channel branch is fixed by (304).

5.5. Radial, Contact, and Common Scale Completion

5.5.1. Radial and Decay Diagnostics

The neutral bosonic cell gives a separate radial test. Let $\delta A:={\partial }_{logv}A$, $\delta B:={\partial }_{logv}B$, and $\delta C:={\partial }_{logv}C$ at the reference point. If the finite neutral cell is radially locked, these derivatives vanish and the standard $hWW$ and $hZZ$ normalizations are recovered. A trace-neutral leakage direction satisfies $\delta A+\delta B+\delta C=0$. The corresponding first radial coupling modifiers are

$${\kappa }_W-1={\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta A}{A}},{\kappa }_Z-1={\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta A+\delta B}{A+B}}.$$

(305)

Thus a one-parameter logarithmic leakage direction gives a correlated $hWW/hZZ$ test. The resulting relation is a completed-cell test. In the reduced branch, an additional first-order scalar direction which changes this correlation but is invisible to $P_{\mathrm{obs}}$ and to the filtered sector labels is removed by Proposition 37. A projected-curvature stress reading may also be compared with the Alena-compatible Rainich branch response; that comparison belongs to the completed curvature sector and does not modify the carrier.

Fermion masses and Yukawa hierarchies remain singular values of the compressed weak-bridge blocks (207). A vortex-Yukawa relation, if imposed, is therefore a sectoral branch condition on those matrix elements. In a locked scalar island one may record the branch condition as $y_f=cosh{\phi }_f-1$ and $g_{hff}=m_f/v$ at zero radial leakage. A nonzero ${\partial }_{logv}{\phi }_f$ is a sectoral leakage diagnostic; the exterior channel remains fixed.

Decay channels are read as allowed punctures of the Riesz gap. For a parent island i and a final channel F, the kinematic excess is

$${\mathsf{\Delta }}_{i\to F}=m_i-\sum _{a\in F}{m}_a.$$

(306)

A channel is open only when (306) is non-negative and the corresponding exterior, weak-bridge, or contact matrix element is nonzero. Standard Higgs decay formulae and loop form factors are used only as comparison readings of the bridge-resolvent calculation [84].

5.5.2. Doubled Top-Form Contact and Scale Conversion

The contact sector is the doubled top-form class in Table 8. Its baryon-violating component obeys (160); the remaining contractions preserve B and L separately. The contact coefficient is a completed Schur-Kuranishi datum separated from the one-Higgs bridge and from the Majorana denominator. The separation is preserved by Proposition 37.

A direct tunnelling assignment $M_X=M_{\mathrm{Pl}}exp(-S_{\mathrm{prim}})$ would put the baryon-violating contact near the electroweak scale and is therefore excluded as a proton-decay reading. The proton-decay comparison convention is the standard GUT effective-operator one [85]. The doubled top-form assignment places the contact at the tower scale. For an order-one finite coefficient in its baryon-violating component and $M_X\simeq M_{\mathrm{Pl}}$, the standard dimension-six estimate gives ${\tau }_p\sim {10}^{48}\mathrm{yr}$. Under the alternative completed-branch assignment $M_X=M_{\mathrm{Pl}}/{\left(2\pi \right)}^2\simeq 3\times {10}^{17}\mathrm{GeV}$, the same estimate gives ${\tau }_p\sim {10}^{41}\mathrm{yr}$. An observed $p\to e^{+}{\pi }^0$ signal near ${10}^{35}\mathrm{yr}$ would rule out the minimal doubled-top-form contact assignment.

The trace normalizations (76) and (77) fix the relative finite normalization of the three gauge factors. A common coupling prediction additionally requires a matching scale, the kinetic normalization of the completed gauge block, and the heavy threshold spectrum generated by the same boundary-cycle completion. The $SU\left(3\right)$ coupling therefore belongs to the completed scale test together with the electroweak couplings, and its low-energy value is left to this common matching problem.

Threshold conversion and running are applied after the finite branch and its heavy denominators have been fixed. The matching map is evaluated on the same Schur-compressed operator, with heavy-field elimination organized by the covariant derivative expansion of [86]. One-loop matching across separated mass scales is represented by [87]. Entries generated by one heavy boundary-cycle completion inherit one matching prescription within the same reconstruction class; their masses and threshold positions need not coincide. Independent sector-by-sector threshold functions constitute an additional completed input.

A threshold-free common-kinetic diagnostic may be evaluated before the heavy spectrum is specified. With one common gauge-block kinetic coefficient, (76) identifies the $SU\left(2\right)$ and $SU\left(3\right)$ matching conditions, while ${\alpha }_1$ is taken in the canonical $k_Y=5/3$ normalization of (77). Using the one-loop Standard-Model coefficients $b_i=(41/10,-19/6,-7)$ and the reference electroweak and strong-coupling inputs of [75], the central crossings are ${\mu }_{23}=9.7\times {10}^{16}\mathrm{GeV}$ from ${\alpha }_2={\alpha }_3$ and ${\mu }_{12}=1.0\times {10}^{13}\mathrm{GeV}$ from ${\alpha }_1={\alpha }_2$. At ${\mu }_{23}$ one finds ${\alpha }_1^{-1}-{\alpha }_2^{-1}=-10.6$, about $23\%$ of ${\alpha }_2^{-1}$. The threshold-free one-scale subbranch is therefore excluded at this order.

In the fermionic kinetic subbranch of Definition 10, Proposition 26 supplies three Dirac-form adjoint pairs and the kinetic lift assigns them the Dirac one-loop weight. Their contributions and singular thresholds are

$$(\mathsf{\Delta }{b}_1,\mathsf{\Delta }{b}_2,\mathsf{\Delta }{b}_3)=(0,8,12),M_2={\displaystyle \frac{3}{80}}{M}_H,M_3={\displaystyle \frac{3}{5}}{M}_H.$$

(307)

At the Standard-Model crossing ${\mu }_{23}$, preservation of ${\alpha }_2={\alpha }_3$ under these thresholds gives

$${\displaystyle \frac{{\mu }_{23}}{M_3}}={16}^2,{\displaystyle \frac{{\mu }_{23}}{M_2}}={16}^3,{\mathsf{\Delta }}_{12}^{\mathrm{th}}={\displaystyle \frac{48log2}{\pi }}=10.5905.$$

(308)

Thus the one-loop threshold shift cancels the quoted $-10.6$ mismatch to the precision of the input crossing. The same identity gives $M_H^{\mathrm{RG}}={\mu }_{23}/153.6=6.3151\times {10}^{14}\mathrm{GeV}$. Independently, the half-flux completion (287) gives $M_R=6.3148\times {10}^{14}\mathrm{GeV}$ with the stated Planck convention. The common-scale specialization sets $M_H=M_R$; the numerical comparison tests this identification at the precision of the one-loop crossing. Two-loop running and finite matching corrections are not fixed by (308). This scale test remains separate from the zero-remainder neutral-cell reading at ${\mu }_{\mathrm{cell}}$.

5.6. Quantitative Status and Numerical Readings

The quantitative relations below are evaluated after the spectral-before-mixing descent and the reduced Schur closure. Numerical reference values are taken from [75]. The oscillation-fit convention follows the published NuFIT 6.0 analysis [83], while the numerical PMNS comparison is updated to NuFIT 6.1 [88]. Direct reactor and long-baseline comparisons are stated separately where used. Table 11 separates exact finite-algebra consequences, conditional reconstruction consequences, normalized branch conditions, arithmetic completions, absolute normalizations, and unresolved analytic data. Parameter-eliminated and conditional no-refit tests are listed together with normalized benchmarks in Table 12 before their dependent numerical readings.

5.6.1. Quantitative Tests and Normalized Benchmarks

Table 12 separates parameter-eliminated relations, conditional no-refit tests, restricted subbranch tests, and normalized benchmarks. The normalized entries are displayed as completed benchmarks and are not counted as independent predictions.

For the direct Hodge completion, a reference common-scale conversion of the PDG quark-mass inputs [75] gives $Q_d^{\mathrm{obs}}=0.74803\pm 0.00280$ and $Q_u^{\mathrm{obs}}=0.88763\pm 0.00232$, compared with $3/4$ and $8/9$. The deviations are $0.70\sigma$ and $0.54\sigma$ in magnitude. These values depend on the common renormalization convention used for the quark masses; no exact renormalization-group invariance is assumed.

Using the observed CKM angles as a leading proxy for the Rees tangent coordinates, the left-hand side of (260) is $0.36065\pm 0.02858$, compared with $1/3$, an offset of $0.96\sigma$. The uncertainty is a first-order diagnostic obtained by symmetrizing the quoted asymmetric one-sigma errors and neglecting correlations. After the Hodge-Eisenstein unitary completion the corresponding physical combination is $0.36683$, which is $0.22\sigma$ from the same observed combination. The tangent relation remains the test of source valuations; the second comparison tests its matrix completion.

At leading Rees order the two higher-angle ratios reconstruct the remaining norm coordinate separately. The ratio $s_{23}/s_{12}^2$ gives ${\rho }_{\mathrm{RP}}^{\left(23\right)}=1.024\pm 0.037$, while $s_{13}/s_{12}^3$ gives ${\rho }_{\mathrm{RP}}^{\left(13\right)}=0.983\pm 0.025$. The two diagnostics share the $s_{12}$ input and are not combined as statistically independent measurements. They are consistent with the unit tangent normalization; once Proposition 42 is imposed, these leading inversions are not exact reconstructions of ${\rho }_{\mathrm{RP}}$.

5.6.2. Dependent Neutral-Cell Readings

Evaluation of (220), (221), and (222) gives Table 13. The link angle is compared with the on-shell value $1-M_W^2/M_Z^2$, while the determinant scale is compared with $v_F={\left(\sqrt{2}{G}_F\right)}^{-1/2}$.

5.6.3. Dependent Central-Family and CKM Readings

With (217), the central seed (270) gives

$${\lambda }_{\mathrm{cen}}=0.2250096.$$

(309)

Solving (270) directly for the PDG central value of $s_{12}$ gives ${\kappa }_{\mathrm{CKM}}=1.0105566$, while the completed branch gives ${\kappa }_{\mathrm{SB}}=1.0105543$. The difference is $2.3\times {10}^{-6}$ at the level of the central scaling factor. This is a dependent consistency check of the common primitive unit and the torsor trace.

The primitive commensurate solutions with $n\le 49$ are listed in Table 14. The list is exhaustive in this denominator range. The $n=1$ entry is the nonsimple endpoint, while $n=7$ is the least-denominator simple branch selected in Proposition 21. The distances use the phase reference in Table 15. All other simple entries shown are more than $11\sigma$ away. This separation is a low-denominator comparison and is not used as a lower bound on the spacing of the unrestricted tower.

Evaluation of (272), (273), and (274) gives Table 15. For (309), one has $J_q/{\lambda }_{\mathrm{cen}}^6=0.240882$. The entropic susceptibility remains CP-even and leaves the orientation of the central loop.

The same matrix and arithmetic choices determine the unitarity-triangle geometry. In the standard Wolfenstein reading they give $A=0.830271$ and $R_b=0.385015$. The dependent triangle readings use the same PDG compilation, with the direct $\gamma$ comparison updated to the HFLAV Summer 2025 average [89], in Table 16. The matrix completion changes the higher-angle ratios while leaving the commensurate phase fixed.

The values in Table 15 and Table 16 are dependent readings of the same completed direct branch. The valuations (250) and Lemma 15 fix the tangent hierarchy. The choice ${\rho }_{\mathrm{RP}}=1$ fixes the primitive norm ratio, while (227) and the Hodge-Eisenstein jet bridge fix the common-to-relative first-jet ratio. Proposition 42 then converts the retained tangent data to a matrix-level magnitude representative without introducing a second continuous quark-mixing scale. The strict Picard polarization of Lemma 16 is needed only for the literal one-line path interpretation.

Proposition 43 isolates the residual primitive coefficient phase. Cycle-trivial coefficient transport is the phase-minimal completed subbranch, after which Proposition 21 and (232) give the discrete commensurate CP reading. The same integral packet $(1,4,5;7)$ therefore enters the magnitude completion through the Hodge bridge and the phase through the torsor spectrum, while the microscopic identification of these two appearances remains part of the completed transport problem. Proposition 37 excludes an independent observable-invisible lower-valuation $1\leftrightarrow 3$ repair.

The direct identification of the Rees path coefficients with exact CKM angles would give $sin2\beta =0.73012$ at the phase (273). The exact Hodge-Eisenstein unitary completion gives $0.71030$ with the same phase. Against the value $0.709\pm 0.011$ used in Table 16, the matrix-completed reading is $0.12\sigma$. The shift is therefore generated at the magnitude-completion step.

The same branch also provides a discriminator for the inclusive-exclusive $|V_{cb}|$ difference. The 2025 review [75] quotes $|V_{cb}|=(42.0\pm 0.5)\times {10}^{-3}$ from inclusive decays and $(39.5\pm 0.5)\times {10}^{-3}$ from exclusive decays. The branch value $|V_{cb}|=0.0420361$ lies $+0.07\sigma$ from the inclusive and $+5.07\sigma$ from the exclusive determination. This comparison favors the inclusive central value within the completed matrix branch and is a dependent discriminator of the same $s_{23}$ prediction.

5.6.4. Charged-Lepton Balance Benchmark

The charged-lepton diagnostic evaluates the scale-free residual (279) on the direct block $L_L\leftrightarrow e_L^c$. It is independent of the quark relative-basis problem and of the neutral Majorana denominator. Using $m_e=0.51099895069\mathrm{MeV}$, $m_{\mu }=105.6583755\mathrm{MeV}$, and $m_{\tau }=1776.93\pm 0.09\mathrm{MeV}$ from the numerical data compilation [75], with the torsor ordering $(\tau ,e,\mu )$, the central coordinate extraction in (277) gives

$$\begin{array}{cccc}\hfill a_{\ell }& =0.70710912,\hfill & \hfill {\varphi }_{\ell }& =0.22222476,\hfill \\ \hfill C_{\ell }& =25.05390{\mathrm{MeV}}^{1/2}.\hfill \end{array}$$

(310)

The corresponding central balance reading is

$${\mathsf{\Sigma }}_{\ell }^{\mathrm{obs}}=6.61\times {10}^{-6},Q_{\ell }^{\mathrm{obs}}=0.66666446.$$

(311)

The central displacements from the zero-correction balance and the common determinant baseline are

$$a_{\ell }-{\displaystyle \frac{1}{\sqrt{2}}}=2.34\times {10}^{-6},{\varphi }_{\ell }-{\displaystyle \frac{2}{9}}=2.54\times {10}^{-6}.$$

(312)

Propagating the quoted tau-mass uncertainty through the same coordinate extraction, with the electron and muon contributions negligible at this precision, gives

$${\mathsf{\Sigma }}_{\ell }^{\mathrm{obs}}=(0.66\pm 1.52)\times {10}^{-5},{\varphi }_{\ell }-{\displaystyle \frac{2}{9}}=(2.54\pm 6.26)\times {10}^{-6}.$$

(313)

The balance and clock displacements are therefore $0.43\sigma$ and $0.41\sigma$ from their zero-correction values, respectively. The first displacement is controlled by (280) once ${\mathsf{\Sigma }}_{\ell }$ is supplied. The second probes the clock-lift completion through the conjugate response (283).

The determinant-tangent branch also gives an internal response-order diagnostic. At $u=u_{\mathrm{prim}}$, the carrier quantities have $(A/A_0-1)/u=-3.500$, $(C/C_0-1)/u=3.889$, $(s_{\det }-2/9)/u=0.608$, and $({\lambda }_{\mathrm{cen}}-2/9)/u=1.761$. The observed central family residuals give ${\mathsf{\Sigma }}_{\ell }/u^2=2.64$ and $({\varphi }_{\ell }-2/9)/u^2=1.01$. This separation is consistent with the conditional linear-versus-quadratic selection rule following (218). The present charged-lepton uncertainties do not by themselves establish the power law.

The zero-correction balance ${\mathsf{\Sigma }}_{\ell }=0$, combined with $m_e$ and $m_{\mu }$ as inputs, gives $m_{\tau }^{\mathrm{bal}}=1776.9690\mathrm{MeV}$. With the clock-lift reference ${\varphi }_{\ell }=2/9$, the second reading gives $m_{\tau }^{\mathrm{clk}}=1776.9665\mathrm{MeV}$ from the same two mass inputs. Thus

$$m_{\tau }^{\mathrm{bal}}-m_{\tau }^{\mathrm{clk}}=2.5\mathrm{keV}.$$

(314)

Relative to the PDG value used above, the balance and clock readings lie at $+0.43\sigma$ and $+0.41\sigma$, respectively. Their $2.5\mathrm{keV}$ separation is a factor 36 smaller than the present $90\mathrm{keV}$ one-sigma uncertainty on $m_{\tau }$ and supplies a keV-scale internal overdetermination of the completed charged-lepton branch. Simultaneous exact closure is not imposed at the present completed order. A threshold study for a proposed super tau-charm facility reaches a projected $1\mathrm{keV}$ precision on $m_{\tau }$ [90], so that the scale of (314) is experimentally resolvable in such a programme. Evaluating the complementary residual on each zero-order solution gives ${\mathsf{\Sigma }}_{\ell }\left(m_{\tau }^{\mathrm{clk}}\right)=4.26\times {10}^{-7}$ and ${\varphi }_{\ell }\left(m_{\tau }^{\mathrm{bal}}\right)-2/9=-1.75\times {10}^{-7}$. These cross-residuals quantify the overdetermination without assigning a statistical weight to either closure. The diagnostic is summarized in Table 17.

5.6.5. Neutral Determinant-Scale and PMNS Diagnostics

The half-flux scale completion (287) gives $M_R=6.31\times {10}^{14}\mathrm{GeV}$. In the unit-determinant pure type-I subbranch of (292), $|det{D}_{\nu }{|}^{1/3}=v_F/\sqrt{2}=174.10\mathrm{GeV}$ gives ${\left(m_1{m}_2{m}_3\right)}^{1/3}=0.0480\mathrm{eV}$. Combining this prefactor with the normal-ordering splittings of [88] gives $m_1\simeq 0.0409\mathrm{eV}$ and ${\sum }_i{m}_i\simeq 0.1474\mathrm{eV}$.

For flat $\mathsf{\Lambda }$CDM, the DESI+CMB analysis [91] reports the $95\%$ limits $\sum m_{\nu }<0.072\mathrm{eV}$ for the prior $\sum m_{\nu }>0$ and $\sum m_{\nu }<0.113\mathrm{eV}$ for the hierarchy-motivated prior $\sum m_{\nu }>0.059\mathrm{eV}$. The unit-determinant subbranch lies above both limits. With the same oscillation splittings, the two limits translate through (292) to $|det{D}_{\nu }|/m_D^3\lesssim 0.22$ and $\lesssim 0.64$, respectively, where $m_D=v_F/\sqrt{2}$.

The normalized branch (293) gives $r_{\nu }=0.05351$. With $m_D=174.10\mathrm{GeV}$ and the half-flux scale, the Takagi factorization (294) gives

$$m_1=0.720\mathrm{meV},m_2=8.72\mathrm{meV},m_3=50.46\mathrm{meV},\sum _i{m}_i=0.05990\mathrm{eV}.$$

(315)

The corresponding squared differences are $7.55\times {10}^{-5}{\mathrm{eV}}^2$ and $2.55\times {10}^{-3}{\mathrm{eV}}^2$. The adjoint threshold identity (308) independently fixes $M_H^{\mathrm{RG}}$ in the same scale region, while (287) fixes $M_R$. The normalized Dirac orientation remains completed neutral data; the pure type-I determinant identity itself is (292). The seesaw comparison is represented by [82].

For the relative basis, (304) is evaluated with (309), (273), and the unit projected-connection rate $r_{\vartheta }=1$. Its PMNS reading is compared with the normal-ordering NuFIT 6.1 fit [88] in Table 18.

The branch value has ${sin}^2{\theta }_{23}=0.47033$ and selects the lower octant. The joint T2K+NOvA analysis [92] gives ${sin}^2{\theta }_{23}=0.{56}_{-0.05}^{+0.03}$ and weakly favors the upper octant with Bayes factor $3.5$; after removal of the reactor constraint, the Bayes factor is $1.2$ for the lower octant relative to the upper one. The present octant comparison is therefore read from the likelihood structure rather than from a Gaussian pull between the two octant minima.

The oscillation branch leaves the Majorana column phases described after (295) unresolved. In the additional no-extra-Takagi-phase subbranch these relative phases are set to zero. Evaluation of (296) with (315) gives Table 19.

5.7. Output Status and Falsification

The structural outputs were fixed in Theorem 2. Table 20 records the additional closures and the corresponding falsifiers for the realized and completed layers.

Table 12 separates parameter-eliminated and conditional tests from normalized benchmarks. The values in Table 13 and Table 15, Table 16, Table 17, Table 18 and Table 19 give the stated tests or benchmark readings according to their recorded status. Table 11 records the marked-descent, normalization, arithmetic, and scale assumptions inherited by those readings. The carrier, carrier group, Fock module, finite cocharacter shadow, and structural channel filtration retain the statuses given in Table 9.

6. Discussion and Conclusions

6.1. Results and Structural Interpretation

Filtered minimal reconstruction is defined by the least element of the admissible protected-extension poset at each fixed stratum. The explicit ambient envelopes are listed in Table 3; enlarging the downstream protected data enlarges the reconstructed object monotonically. Finite-core multiplicity reduction, Borel-Weil witness descent, the full finite-shadow orbit, and protected Schur descent are instances of this closure property. The operator algebra ${\mathcal{N}}_X$ gives the corresponding state-level initial subsystem, with ${\mathbb{E}}_X$ its trace-preserving completely positive retraction.

For the primitive codimension-three defect, entries A and B of Table 4 give the initial marked carrier (56) and degree fidelity. Entry C gives the determinant-compatible group, whose faithful effectivization is (64), together with the exterior package and canonical Abelian gradings. The protected Picard translation generates the full ${\mu }_6$ orbit, and parity lock gives the three-dimensional corner. Physical family attachment remains conditional on finite-shadow covariance and Callias-Schur isolation. Corollaries 11 and 10 record the corresponding absence of unsourced weak-doublet and light-family multiplicities on their fixed strata.

Two standard structures which are usually posited at the outset appear here as outputs of the same descent. The gauge field is not adjoined to the carrier: it is the projected connection (148) on the isolated low bundle of Proposition 28, whose determinant reduction takes values in the local kernel (82), so that its structure algebra is the same $s\left(u\left(C\right)\oplus u\left(W\right)\right)$ already fixed by the top-form condition (80). Gauge transformations are correspondingly not an added redundancy but the admissible reconstruction-frame transitions of (196); Theorem 4 shows that the Riesz projection, the projected connection, and its curvature transform by unitary conjugation, so that rank, spectrum, gap, and determinant holonomy are basic on the protected quotient. In a local low-mode frame the projected coefficients are $\langle {\psi }_i,d{\psi }_j\rangle$, and the relative normalizations of the three factors are the trace indices (76) and (77). What is not generated at this depth is the kinetic normalization of the completed gauge block; it enters only with the common matching problem of SubSection 5.5 and is recorded as completed data in Table 11.

The Dirac structure enters at three separate depths and is constrained at each. Algebraically, the exterior residual (66) closes to a graded Clifford representation, and commutant closure forces multiplicity one (Proposition 9), so the spinorial package ${\mathsf{\Lambda }}^{•}V$ and its ${\mathbb{Z}}_2$ grading are consequences of finite-core minimality rather than inputs; the CAR normalization (69) then supplies the finite metric used throughout the completed branch. Operationally, the parity lock (149) makes the completed low operator odd, which is the finite form (150) organized as a superconnection. Index-theoretically, the local anomaly coefficients of Table 6 vanish as consequences of the centered determinant degree rather than as imposed constraints, the global $SU\left(2\right)$ obstruction vanishes because the weak-doublet multiplicity is $3+1=4$, and the open normal problem carries a stable Callias-Fredholm class. If the compatible determinant trivialization assumed in Proposition 12 is supplied, its local vertical obstruction and gauge-loop holonomy vanish. Thus the centered integral degree $H_{\mathsf{\Gamma }}^{\mathrm{deg}}$ fixes the carrier group, hypercharge, finite cocharacter shadow, connection algebra, canonical block weights, and anomaly bookkeeping, while the projected connection values remain Gram-Riesz data. The remaining global determinant problem includes construction and trivialization of the generated differential-index line and, after a compatible real structure is established, its mod-two orientation class.

At direct depth, the generic Rees-Picard norm relations remain independent of Picard polarization and present numerical readings do not select ${\mathsf{\Delta }}_{\mathrm{pol}}$. The one-line branch ${\mathsf{\Delta }}_{\mathrm{pol}}=0$ is the literal path specialization. The first Hodge incidence step gives the integral packet (227); on the completed Hodge-Eisenstein jet bridge, Proposition 42 converts the Rees tangent hierarchy to matrix-level CKM magnitudes and removes the leading $sin2\beta$ residual without changing the arithmetic phase. The bridge from Hodge support to sectoral reduced matrix elements remains conditional. Corollary 16 identifies the balanced fixed locus of the canonical real structure on the relative coefficient module and Lemma 17 separates its symmetric and CP-odd antisymmetric cubics. Since the relative packet is built from the difference of the up- and down-sector Hermitian responses, strong determinant reality requires the separate pre-Hermitian sectoral lifts of Proposition 44.

The determinant subdepth precedes Hermitianization. The autonomous reconstruction class retains the generated differential-index strong line and contains no independent flat translate; an external factor in (209) defines a different completion class. Sectorwise coefficient reality reduces the chiral determinant problem to the generated global line. A real structure on that line leaves a ${\mathbb{Z}}_2$ orientation class, whose mod-two determinant/Pfaffian index on the primitive color cycle remains to be computed. In the completed high branch, Proposition 26 fixes the Dirac-form parity multiplicity, while Definition 10 supplies the adjoint gap normalization and the fermionic kinetic lift required for threshold counting. Proposition 36 fixes the determinant-centered phase weights, and (206) specifies their neutral Takagi attachment. The neutral reduced coefficients in (303), the additional Takagi column phases, the arithmetic torsor completion, and higher-order common matching retain their completed statuses. Table 11 and Table 20 record these dependencies before the numerical readings.

6.2. Decisive Completed-Branch Tests

The numerical readings below test the stated completed branches. Their exclusion removes the corresponding completion while leaving the structural carrier theorem at its stated status. Present comparisons and measurements capable of resolving the branch are summarized in Table 21. The values inherit the assumptions recorded in Table 11 and Table 20.

The no-extra-Takagi-phase, determinant-suppressed subbranch gives $m_{\beta \beta }=3.89\mathrm{meV}$ at the mass scale (315). The minimal isolated neutral branch contains exactly three light active family modes under the stated heavy-gap hypothesis. The common adjoint threshold identity (308) fixes $M_H^{\mathrm{RG}}$ independently of the neutral half-flux scale $M_R$ and provides the one-loop common-scale comparison.

6.3. Relation to Geometric and Spectral Constructions

The curvature reconstruction used here is closest in spirit to the algebraic program initiated by Rainich [97]. Its aligned Einstein-Maxwell implementation is represented by [98], while broader stress-energy classifications are discussed in [99]. The present use of a trace-adjusted Codazzi coefficient adds a graded boundary source and a finite reconstruction step to that geometric setting. Codazzi and conformal-tensor comparisons are supplied by [100] and by the recent structural analysis of [101]; the differential-geometric conventions follow [102] and the spacetime convention of [103].

A complementary comparison is provided by first-order and connection-based formulations. The Plebanski separation of metric data [104] and the pure-connection formulations of [105] and [106] show how gravitational variables may be reorganized before matter is attached. Spinorial and differential-form extensions are developed in [107,108], and [109]. The self-dual-form background is represented by the Urbantke construction [110], the self-dual complex of [111], and the related form analysis of [112]. In the present setting, these comparisons concern the geometric realization layer; the finite internal package is selected by the boundary grading and its reconstruction closures.

The projective link also places the construction in the spin and twistor setting of [113], the gauge-theoretic twistor discussion of [114], and the spin-geometric framework of [115]. The isolated normal block is naturally compared with the index theorem of [116], magnetic zero-mode counting in [117], and homogeneous spherical analysis in [118]. These results provide the analytic and representation-theoretic background for the link spectrum; the rank-five support additionally uses faithful marked-sector reconstruction and finite-core minimality.

In almost-commutative geometry, the internal algebra is introduced as spectral data, as in [119], and its dynamics is organized by the spectral action of [120]. Recent structural refinements are represented by [121]. Lorentzian extensions are studied in [122], while the no-doubling and electroweak-theta variants are represented by [123] and [124]. The broader spectral framework is reviewed in [125]. Division-algebraic organization of the Standard-Model and family data is represented by [126,127], and [128]. The distinction in the present construction is that the finite algebra and its integral grading are outputs of the primitive boundary reconstruction.

Geometric internal-space models give a further comparison. Dynamical principal bundles are treated in [129]; Kaluza-Klein internal symmetries are developed in [130] and [131], while relational internal spaces are considered in [132]. The gauge convention is consistent with [133], and the coupled Yang-Mills-Higgs-Dirac setting is represented by [134]. These approaches begin with an extended geometric or bundle structure. The reconstruction considered here asks which finite structure is forced after only the resolved defect observables and their grading have been retained.

6.4. Further Directions

The first open problem is analytic completion of the realization theorem. The admissible reconstruction-frame groupoid should be constructed and the microscopic Jacobian covariance (196) proved on this groupoid. Theorem 4 then supplies the basicness of the normal family, Riesz projection, projected connection, Kuranishi component, and reduced penalty. The same analysis should realize the blockwise faithful principal Hessian image (201) on the selected isolated low bundle; Proposition 34 then supplies its Schur factorization. A classification of primitive Alena-Codazzi collars would also require control of the nonlinear multiplier equation, including the period obstruction for the multiplier one-form, together with compactness and regularity of the thin-core limit. The relevant boundary estimates begin with [135] and [136]; nonhomogeneous boundary data are treated in [137]. The pseudodifferential and conic tools needed for a global collar problem are represented by [138,139], and [140].

The second analytic problem is to derive the finite torsor and the isolated family window from a microscopic boundary deformation complex. APS boundary theory is represented by [141]; the Dirac-boundary and Cauchy-data formulations are developed in [142] and [143]. Callias boundary refinements are represented by [144], perturbations on noncompact ends by [145], and the pseudodifferential Callias class by [146]. The generalized Dirac-Schrödinger comparison of [147] and the qualitative PDE framework of [148] provide natural tools for proving persistence of the low projection under geometric deformations. A successful construction should recover the edge transports, the Wilson defect, and the covariance residual from the same normal operator family. Proposition 21 identifies the commensurate scalar spectra with Eisenstein norm data; the remaining question is whether the microscopic edge transport selects this arithmetic subbranch from the integral finite-shadow lattice.

A third direction concerns global and structural generalizations. Higher normal order, other primitive positive classes, and different normal codimensions may produce other graded supports and compact Levi factors. Global consistency must then include the allowed line-operator lattice, for which [149] gives the relevant gauge-theoretic comparison. The determinant, Pfaffian, and finite-family data are organized at completed depth by the differential-K reading of Remark 4. The remaining global strong problem is to construct the generated differential-index representative on the reconstructed gauge groupoid and calculate its real determinant holonomy on a primitive color winding cycle. Once a real structure is established, the remaining sign should be computed as the mod-two orientation or spectral-flow class of the corresponding real determinant/Pfaffian line. The neutral Pfaffians in (289) give a finite algebraic analogue of this orientation problem but are not themselves the strong color invariant.

The direct-flavor microscopic programme can be narrowed to the source-to-family response needed to test the completed branch. The first target is to determine whether the $x_1$ and $x_2$ deformations produce a nonzero relative first response $R_1$ and a nonzero primitive second response $Y_{\perp }$ on the selected low bundle. In the present notation this requires the ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ components of $D{H}_u\left[x_1\right]$, $D{H}_d\left[x_1\right]$, $D{H}_u\left[x_2\right]$, and $D{H}_d\left[x_2\right]$ after the sectoral inverse Sylvester maps; these relative data govern the mixing packet. Once ${\mathsf{\Delta }}_{\mathrm{prim}}>0$ is established, (255) tests whether the unit normalization ${\rho }_{\mathrm{RP}}=1$ is a microscopic reduced-matrix-element identity. The additional matrix-level test is sharper: before the mixed-curvature norm quotient removes the scalar torsor spectrum, the protected sectoral Schur-Riesz data should map the first Hodge incidence packet (227) to the common/relative first-jet ratio $\mathfrak{q}=q_H/p_H$. Establishing this identity would derive the value 4 used in Definition 16; Proposition 42 then fixes the remaining matrix coefficients. The polarization defect (257) separately distinguishes the literal one-line branch from the orientation-real fixed locus of Corollary 16.

For the strong determinant, the calculation is sectorwise and pre-Hermitian. After factoring the generated chiral baseline, $Y_u$ and $Y_d$ should each be tested for a Weyl coefficient lift of the type used in Proposition 44. The canonical candidate real structure is already supplied by the regular ${\mathbb{Z}}_3$ coefficient module: in the vertex basis it is $RK$, equivalently ordinary conjugation after Fourier exchange of the shift and clock descriptions. The remaining microscopic check is whether the direct bridge and boundary transport preserve the two sectoral lifts. Their embeddings need not coincide, so the retained relative torsor phase may still contribute to (268). The axial diagnostic is $\Im Tr(Y_u^{-1}D{Y}_u\left[x\right]+Y_d^{-1}D{Y}_d\left[x\right])$. Failure of either sectoral lift or the appearance of an additional generated central axial component would exclude the determinant-real mechanism.

The next direct target is the transport of the primitive Picard coefficient module and its compatibility with the Hodge-Eisenstein bridge. A microscopic calculation should first determine whether the generated response occupies one ${Ad}_X$ eigenline or the full two-line invariant extension. On a generated one-line branch, preservation of the same canonically identified primitive Eisenstein coefficient lattice on both sides of the coefficient map would turn ${\rho }_{\mathrm{RP}}=1$ into a consequence of the integral reconstruction refinement. Independently, the sectoral inverse-Sylvester denominators and low compressions should be tested for the surviving-to-lost incidence normalization $q_H/p_H$ before the mixed-curvature seminorm is taken. Adjoint-dual forward/reverse holonomy as in Proposition 43 then reduces the remaining phase data to ${\psi }_{\mathrm{prim}}$. The commensurate arithmetic branch is selected by the integral reconstruction only if the microscopic edge transport preserves the integral commensurability class; its least simple packet is already the packet in (227). The source filtration should also determine the sector index in (226) and whether the clock-lift reference ${\varphi }_{\ell ,0}=2/9$ follows from the same reduced connection weight. Quark analogues of the clock coordinate remain required for the other sectoral scales $C_u:C_d:C_{\ell }$. The quadratic family response in (284) should be compared with a microscopic direct Schur tensor.

The high-block microscopic programme is reduced to four direct tests. First, the explicit Jacobian in (193) should be restricted to (152) and tested against the $M_H$-normalized Gram block of (200); failure of this relation would remove the normalized threshold spectrum while leaving the parity-paired odd multiplicity intact. Second, the odd square root should be shown to enter the fermionic kinetic class specified in Definition 10; without this lift the Dirac one-loop threshold weight is not retained. Third, the phase deformation $X={\partial }_{\vartheta }$ should be inserted in (194), the relative projected phase rate $r_{\vartheta }$ extracted from ${\mathcal{B}}_{\vartheta }$, and the outer-C attachment (206) checked on the neutral coefficient. Fourth, the common adjoint thresholds should be carried through two-loop running and finite matching with one boundary-cycle prescription.

The neutral tensor should test the normalized Dirac shape and Takagi-compatible orientation in (293)-(294), the shapes of $R_N$ and $A_L$ away from the pure type-I branch, and the three reduced coefficients in (303). Lemma 20 keeps the cubic response at completed Jacobian depth. Protection of (296) additionally requires the relative Takagi column phases, which are absent from the oscillation-only quotient. The microscopic torsor transport should also determine whether the commensurate arithmetic branch selects the minimal $n=7$ phase. Non-Abelian Wilson-loop measurements such as [150] provide a finite-holonomy comparison, while geometric soliton models such as [151] give a distinct realization test for matter-like localized sectors.

Finally, the state-level reconstruction parameter should be related to microscopic dynamics. The completely positive semigroup is fixed by the conditional expectation once the finite algebra has been selected, but its rate and its relation to Lorentzian proper time remain dynamical data. A derivation from the normal evolution would connect the algebraic reconstruction depth, the torsor Dirichlet form, and the physical relaxation of unresolved observables. This would also determine whether the entropy diagnostics used in the direct and neutral sectors admit an operational preparation and readout.

Experimental discrimination of the completed branches is correspondingly overdetermined. Table 21 collects the direct oscillation and flavor observables that can be sharpened without altering the structural reconstruction. The CKM completion is tested jointly by its phase and unitarity-triangle geometry, while the charged-lepton closure has an independent threshold-scale test. The neutral completion is constrained simultaneously by the three oscillation angles, the leptonic CP phase, the absolute spectrum from (315), and, once the relative Takagi phases are fixed, the neutrinoless-double-beta amplitude (296). The common high-scale completion is independently sharpened by two-loop running and finite matching of (307) against the neutral half-flux scale. Completion of the real generated strong determinant line would reduce the remaining strong-CP test to the holonomy in (208), placing that branch under electric-dipole-moment bounds. Agreement is therefore required across correlated observables within each completed branch; one incompatible member excludes that branch while leaving the structural carrier theorem unchanged.

The resulting statements separate into structural minimal-reconstruction consequences, conditional analytic attachment, and completed coefficient, determinant, arithmetic, and scale data. The structural theorem fixes the primitive carrier, faithful global group, exterior package, Abelian grading plane, and generated finite-shadow orbit. The microscopic realization layer conditionally attaches the family window and the finite principal Hessian support through covariance, the Callias-Schur gap, and faithful marked Schur realization. The remaining tasks are analytic realization of these conditional statements, derivation of the Hodge-to-Sylvester jet bridge underlying $\mathfrak{q}=4$, primitive integral control of the direct coefficient module, derivation of separate pre-Hermitian chiral Weyl lifts for $Y_u$ and $Y_d$ preserving the canonical coefficient real structure, microscopic evaluation of the $M_H$-normalized $G_H$, the fermionic kinetic lift, $r_{\vartheta }$, and the outer-C Takagi attachment, derivation of the remaining neutral reduced coefficients and Dirac orientation, computation of the mod-two orientation and holonomy of the generated strong determinant line, and higher-order common matching of (307). None of these later tasks changes the structural carrier theorem.