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

1. Introduction

The geometrization of field interactions often proceeds by enlarging the geometric structure. In Kaluza-Klein-type mechanisms, gauge variables are represented by higher-dimensional or bundle-metric data [1]. In Eisenhart-Duval-type constructions, forced dynamics is rewritten as geodesic dynamics on an enlarged space [2]. In Randers and Finsler descriptions, charged motion is encoded by an effective geometry of trajectories [3]. Finite-resolution curvature-defect mechanisms provide a related local comparison class [4]. The problem considered here is more specific. A local codimension-three defect in a four-dimensional Lorentzian branch is used to reconstruct a finite boundary cycle from its resolved link, before a finite internal algebra or a gauge group is inserted as an independent input. At this level a branch is used only through the resolved link and the Gauss-local boundary charges. The Alena Tensor stress reading [5] and its current-residual extension [6] enter below as a variational source class for these charges; the finite carrier is selected before the realization class is specialized.

The organizing principle is that physical configurations may be treated as fixed points of a self-reconstruction loop of observables. A configuration $\Phi$ determines a closed observable algebra ${\mathcal{A}}_{\Phi }$, a dynamics ${\delta }_{\Phi }$, isolated stable Schur sectors, and the natural connection data on the corresponding sector bundles. These data reconstruct a configuration $\widehat{\Phi }$:

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

(1)

The equality is understood on the physical quotient, after gauge, diffeomorphism, and unitary equivalences have been removed. The local closure defect is measured by

$$D[\Phi ]={dist}_{\mathfrak{M}}^2\left([\Phi ],\left[\widehat{\Phi }\right]\right),$$

(2)

where $\mathfrak{M}$ denotes the corresponding quotient configuration space. In Section 2, this defect is refined to a filtered residual system. Its lexicographic critical equations are the Euler-reconstruction generators. Standard action, Einstein, curvature-compatibility, Gauss, Fredholm, determinant, torsor, and Schur closures are then treated as residual entries of the same reconstruction scheme.

The local branch starts from a Gauss-local worldline datum. In four dimensions, a worldline defect has a two-sphere link. In the optical Codazzi setting this sphere is read as a projective spinor line, hence as $S_{\Gamma }^2\simeq {\mathbb{CP}}_{\Gamma }^1$ in the standard spinor language of [7]. The twistor comparison class is represented by [8], while twistor-space Standard-Model comparisons give a separate representation-theoretic reference [9]. A positive transverse class gives a positive line on the projective link; the degree-one line $L_{\Gamma }\simeq 𝒪\left(1\right)$ is recovered below by reduced visibility. The positive link modes are then the Borel-Weil spaces on ${\mathbb{CP}}^1$, in the usual homogeneous-bundle sense of [10]. Equivalently, the same finite spaces are read as monopole zero modes on $S^2$ [11] or as the rank-one case of standard $SU\left(2\right)$ representation theory [12].

The source hypothesis is local. A scalar-sector normal source of transverse order at most two has, after the scalar singlet has been separated, only two principal non-scalar $SU\left(2\right)$ types: the order-one phase-current type $V_1$ and the trace-free order-two Codazzi-gap type $V_2$. In a Gauss-local collar these coefficients are boundary charges. The observable algebra is taken relative to the fixed boundary-charge algebra, and the corresponding sectors are central in finite sector representations, in the usual boundary-charge reading of local observables [13]. In the reconstruction action used below, this separation is the faithful Gauss-sector entry: independent Gauss-local residuals must be reconstructed by independent finite charge labels. The ${\mathbb{CP}}^1$ Berezin-Toeplitz cutoff [14] then makes $V_1$ first visible on $E_2$ and $V_2$ first visible on $E_3$. Fuzzy-sphere and finite-matrix comparison models give the same finite-mode interpretation [15]. Faithfulness selects the separated support $E_3\oplus E_2$, while minimal finite visibility removes unsourced principal integer types and fixes the degree-one sector.

After the carrier has been selected, the split determinant reading is applied to the same finite object. The two selected blocks have dimensions 3 and 2, and the split top-form condition gives the compact carrier group $S\left(U\right(3)\times U(2\left)\right)$, with global Standard-Model form (33). The determinant-compatible even exterior package (35) is the local one-generation representation package. This exterior organization is familiar from the $SU\left(5\right)$ and $Spin\left(10\right)$ comparison class [16]; related Clifford-ideal constructions give a separate finite-algebraic comparison [17]. In the present construction the exterior package is applied after the carrier has been reconstructed from the link.

The determinant reading also gives a finite central layer. The primitive class used integrally in the Borel-Weil tower has the finite-coefficient shadow (46) after the global form (33) has been fixed. Its ${\mathbb{Z}}_3$ projection is the projective-color shadow, while its ${\mathbb{Z}}_2$ projection is compatible with the weak parity part of the chirality/exterior-parity lock. The projective-color shadow defines the torsor ${\mathcal{T}}_{\Gamma }$ in (52). Its vertex space is the central response carrier (53), and its finite edge resolution is (54). Algebraic family models based on finite permutation or Clifford structures give useful comparison points [18]. The same torsor complex is packaged below as a projective-color finite Hodge-spectral module: its Wilson defect, vertex Laplacian, Schur-visible non-circulant detector (72), and finite trace factor are different readings of one finite edge complex. In the Alena-Codazzi realization, this finite module is read as the quantized projective-color boundary shadow of the vorticity response. The physical family multiplicity and sectoral splittings still require the locked low representative and the Callias-Schur gap.

The Alena-Codazzi sector is used only after the abstract boundary cycle has been fixed. The variational Alena Tensor family supplies the branch-stress reading [5], while the residual current and vorticity terms used in the collar are those of [6]. The continuum, variational, and Higgs-like branch-potential inputs are those of [19,20,21]. In the present paper only the collar data needed by Definition 2 are used: a current-residual scalar, a conserved translational current, a trace-adjusted Codazzi multiplier, a compact-leaf source mechanism, and the two principal boundary charges. This realization supplies the continuum vorticity source, while the finite projective-color module keeps only its quantized boundary shadow. Under torsor-admissible boundary transport, the transported charge pair also realizes the finite edge transports of (55); flat torsor closure is imposed by (59).

The analytic completion is formulated on the selected boundary cycle. A boundary-admissible Dirac-Callias representative isolates the low sector, and the Riesz projection gives a finite-rank low-sector bundle. The Callias mechanism [22] and its geometric form [23] provide the Fredholm comparison. Perturbation theory is used in the sense of [24]. The projected low-sector connection is the Berry-Wilczek-Zee connection [25,26]. The determinant-line and family-index comparisons belong to the standard regular-family language of Dirac-type operators [27].

The completed low representative is organized by the $B-L$-filtered Schur module. The weak bridge is the unique color-singlet $B-L$ preserving odd tangent to the chirality/exterior lock and gives the direct Dirac/Yukawa blocks. For a reduced primitive normal representative satisfying the locked central factorization condition, the projective-color factor gives a rank-three torsor low cluster. CKM mixing tests Schur-visible non-circulant motion in the direct Dirac sector. The mixed torsor curvature (72) is the finite detector for that motion, while relative $V_1/V_2$ charge transport gives a natural source of the clock degree. The neutral sector is different: light neutrino masses are obtained from the $B-L$ breaking Majorana Schur complement, and large PMNS angles may be denominator-driven by soft neutral gaps and degeneracy resolution.

The quantitative part of the completed branch is kept separate from the structural theorem. Yukawa matrices, fermion masses, CKM and PMNS data, scalar determinant normalizations, running-coupling matching, Pfaffian Majorana data, CP phases, and contact-class coefficients are finite Schur-Kuranishi data on the selected boundary cycle. The standard quark-sector comparison uses the usual CKM language of [28,29,30]. CP-sensitive invariants are compared with the Jarlskog invariant [31]. Numerical reference values, when used, are taken from the standard particle-data compilation [32]. In the present organization these quantities are treated as diagnostics, conditional branch predictions, or falsification tests of the completed Schur-Berry layer. The reduced completed layer uses the same minimality after low-sector projection: a finite Schur channel which is invisible to the finite observable residual and to the already closed sector labels is absent from the reduced representative. This reduction belongs to the completed layer and is not an input to the carrier theorem. One additional scale-free diagnostic is recorded for the charged-lepton direct Dirac block. It identifies the Koide relation [33] as the zero-correction Hodge-balance on the reconstructed family carrier; its nonzero observed defect is kept as a Schur-layer datum rather than as an input to the carrier theorem.

The paper is organized lexicographically. Section 2 formulates the filtered self-reconstruction principle and its faithful minimal reconstruction action. The reconstructive minimality, faithful Gauss-sector reconstruction, no-phantom visibility, finite-shadow conservation, and low-isolation readings used later are read as entries or consequences of this ordered action. Section 3 applies these generators to the primitive Gauss-local defect, proves the separated support theorem, recovers the degree-one line from reduced positive-degree visibility, records the determinant-compatible structural package, and constructs the projective-color finite Hodge-spectral module attached to the selected carrier. Section 4 gives a compact-leaf Alena-Codazzi realization criterion for the primitive boundary data and separates torsor-admissible boundary transport from flat torsor closure. Section 5 treats the locked low cycle and the $B-L$-filtered Schur completion: projected gauge fields, weak-bridge Dirac blocks, the direct CKM detector, the neutral Majorana Schur complement, Pfaffian orientation, and doubled top-form contact classes are separated by finite channel. Section 6 collects the minimal-branch numerical diagnostics and falsification tests, including the determinant-shadow cell, central family seed, charged-lepton balance defect, Pfaffian seesaw scale, radial leakage, and contact-scale estimates. The final sections collect the structural conclusions and discuss the geometric, representation-theoretic, noncommutative, K-theoretic, and spectral comparison frameworks.

2. Filtered Self-Reconstruction and Generated Closure Laws

The self-reconstruction loop (1) is used below in a filtered form. The point of the filtration is to separate closure conditions which enter at different logical depths. The geometric link, the Gauss-local boundary charges, the Toeplitz support, the determinant carrier, the finite shadow, the torsor response, and the Schur-completed low operator are not varied at the same level. The closure defect (2) is therefore refined to an ordered residual system.

The construction in this section is local on the physical quotient. A slice, a Hilbert chart, or a Kuranishi chart is fixed whenever the quotient by gauge, diffeomorphism, and unitary equivalence is used. All gradients and projections below are understood in such a local representative. The spectral projections used later are stable in the standard perturbative sense [24]; the Callias comparison for the low-high split is the one used for Fredholm representatives on open spaces [22,23].

Let $\mathfrak{C}$ denote the local configuration space and let $\mathcal{G}$ denote the equivalence group which has already been removed in (2). A reconstructed configuration determines a residual section over $\mathfrak{C}/\mathcal{G}$. In the filtered version, this residual is written as

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

(3)

where $R_i(\Phi )$ takes values in a Hilbert obstruction bundle ${\mathcal{E}}_i\to \mathfrak{C}/\mathcal{G}$. The index i records reconstruction depth, not perturbative order.

For the primitive branch considered in this paper, the residual components are ordered as

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

(4)

The first entries fix the local geometric and boundary data. The middle entries fix the finite carrier, exterior package, Abelian channel filtration, finite shadow, parity factor, and torsor cycle. The last entries belong to the isolated spectral representative and to the comparison of different resolutions. Within this order, the finite and parity entries contain the compatibility of the full ${\mathbb{Z}}_6$ shadow with its ${\mathbb{Z}}_3$ and ${\mathbb{Z}}_2$ projections, while the Schur entry contains the power counting of higher source moments, the central-loop CP phase, the Pfaffian orientation of the neutral branch, and possible scale-free shape subresiduals of the completed direct Dirac blocks. The determinant-line and differential-K comparison used in the determinant entries is standard [27,34]. The odd finite insertions used later are compared with the superconnection language of [35].

For each component of (3) put $J_i(\Phi )={\displaystyle \frac{1}{2}}{\parallel R_i(\Phi )\parallel }_{{\mathcal{E}}_i}^2$. The filtered closure functional is the ordered value

$${\mathcal{S}}_{\mathrm{rec}}^{\mathrm{lex}}[\Phi ]=\left(J_0[\Phi ],J_1[\Phi ],\dots ,J_N[\Phi ]\right)\in {\mathbb{R}}_{\ge 0,\mathrm{lex}}^{N+1}.$$

(5)

The order on the target is lexicographic. Thus the i-th residual is minimized only after the previous residuals have been closed. Equivalently, the critical sets are defined recursively by

$$Z_{-1}:=\mathfrak{C}/\mathcal{G},Z_i:=Crit\left(J_i{|}_{Z_{i-1}}\right),0\le i\le N.$$

(6)

A configuration is called lexicographically self-reconstructing to depth N if $[\Phi ]\in Z_N$.

The same ordering can be approximated by a one-parameter scalar functional

$${\mathcal{S}}_{\eta }[\Phi ]={\displaystyle \sum _{i=0}^N}{\eta }^{2i}{J}_i[\Phi ],0<\eta \ll 1,$$

(7)

provided the limit $\eta \to 0^{+}$ is taken after the critical equations have been ordered. This representation is used only as a local analytic device. The structural object is the ordered functional (5).

Assume that $Z_{i-1}$ is represented near $\Phi$ by a clean critical submanifold or by a Kuranishi chart. Let ${\Pi }_i$ be the orthogonal projection onto $T_{\Phi }{Z}_{i-1}$ in the chosen slice. The i-th projected gradient is

$${\mathfrak{E}}_i(\Phi )={\Pi }_i\left(D{R}_i{(\Phi )}^{*}{R}_i(\Phi )\right).$$

(8)

The equations ${\mathfrak{E}}_i(\Phi )=0$ are the Euler-reconstruction generators associated with the filtered residual.

The following observation is used only to fix notation.

Under the cleanness assumption above, the recursive criticality condition is simply

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

(9)

This is the constrained first-variation condition for $J_i$ on $Z_{i-1}$, with tangential gradient given by (8).

Thus the single closure condition (2) is replaced, at filtered depth, by the ordered system (9). The entries of this system are the generated closure laws used in the rest of the construction.

2.1. Faithful minimal reconstruction action

The filtered residual may be read as a generalized action on a space of descriptions rather than as a classical action on fields alone. A local description contains the branch representative together with the finite data reconstructed from it:

$$X=\left(\Phi ,{\mathcal{A}}_{\Phi },F,{\rho }_F,L_{\Gamma },\Omega ,{\mathcal{H}}_{\mathrm{fin}},{\mathcal{T}}_{\Gamma },U,P_{\mathrm{low}},K_{\mathrm{low}}\right).$$

(10)

Here F is the finite positive support, ${\rho }_F$ is the finite charge-reconstruction map, $\Omega$ is the determinant line datum, ${\mathcal{H}}_{\mathrm{fin}}$ is the finite shadow carrier, ${\mathcal{T}}_{\Gamma }$ is the finite torsor, U denotes its edge transports, and $P_{\mathrm{low}}$ is the low-sector projection when it is defined. The reconstruction steps in (4) act as transformations of such descriptions. Their residuals are the reconstruction displacements. Thus the role played by a velocity in an ordinary action is played here by the failure of a reconstruction step to close.

Let ${\mathcal{Q}}_{\partial }^{\mathrm{pr}}\subset {\mathcal{Q}}_{\partial }$ denote the finite vector space spanned by the principal boundary charges which enter $R_{\mathrm{Gauss}}$. In the two-channel branch it is spanned by the two principal charge classes. Let $Z_F$ denote the center of the finite support algebra at the Gauss level. The charge-reconstruction map is

$${\rho }_F:Z_F⟶{\mathcal{Q}}_{\partial }^{\mathrm{pr}}.$$

(11)

The finite Gauss description is faithful when ${\rho }_F$ has full rank on ${\mathcal{Q}}_{\partial }^{\mathrm{pr}}$. The corresponding defect is

$${\delta }_{\mathrm{fid}}\left(F\right)=dim{\mathcal{Q}}_{\partial }^{\mathrm{pr}}-rank{\rho }_F\ge 0.$$

(12)

No representation type is prescribed in (12); only faithful recovery of independent principal boundary charges is required.

A finite description is measured by a monotone complexity. Only its ordering class is used. In the support entries it is enough to use

$$\mathcal{C}\left(F\right)=rankF+\alpha dim{Vis}_{\mathrm{pr}}\left(F\right),\alpha >0.$$

(13)

The rank term is sufficient for the positive-degree comparison used below. The second term records that additional principal visibility is not a free datum of a minimal finite description. Any monotone representative which strictly decreases when an unused finite summand or a removable unsourced principal visible channel is removed gives the same primitive support selection.

The faithful minimal reconstruction action is the ordered functional

$${\mathcal{S}}_{\mathrm{FMR}}\left[X\right]={\left(J_0,\dots ,J_{\mathrm{Gauss}},{\delta }_{\mathrm{fid}}^2,J_{\mathrm{vis}},\mathcal{C},J_{\det },\dots ,J_N\right)}_{\mathrm{lex}}.$$

(14)

The named entries are those of (4). The fidelity entry is evaluated after the Gauss residual and before visibility. The support-complexity entry is evaluated after visibility and before the determinant and later finite entries. Thus (14) contains closure, faithful recovery, and minimal finite description, but it contains no preferred support, carrier group, or finite torsor.

The following statement records the canonicality content used by the finite entries. It separates the fixed-point condition from the reduced finite description condition.

The canonical finite reconstruction content used below is the following. If the self-reconstruction loop (1) is required to be a local identity on the physical quotient, principal charge classes occurring in $R_{\mathrm{Gauss}}$ must be recovered faithfully by (11). A finite summand, or a principal visible channel, which is undetected by the residuals already closed and does not change this rank is removed in the reduced representative. Thus minimal closed representatives are selected within each reconstruction class by a monotone complexity of the form (13), removable unsourced visibility satisfies (15), and the order in (14) is a linear extension of the corresponding definability order.

Proposition 1

(Faithful minimal reconstruction). Let X be a local zero-closure minimizer of (14) in a fixed reconstruction class. Then the filtered closure equations (9) hold. If ${\delta }_{\mathrm{fid}}\left(F\right)=0$, the principal Gauss-local charges are faithfully reconstructed by finite central labels. In particular, independent principal charge classes require independent finite charge sectors. If a principal visible channel can be removed without changing the closed residuals or the fidelity rank and with strictly smaller complexity (13), it is absent in a minimal representative.

Proof. 

The closure equations are (9). The condition ${\delta }_{\mathrm{fid}}\left(F\right)=0$ says that the finite sector labels reconstruct ${\mathcal{Q}}_{\partial }^{\mathrm{pr}}$ with full rank. If independent principal charge classes were forced through a smaller set of central labels, the rank of (11) would drop. The last assertion is the lexicographic minimization of $\mathcal{C}$ in (14). □

In the finite support entries, the last statement will be used in the form

$${Vis}_{\mathrm{pr}}\left(F_i\right)\subseteq {Src}_{\le i},$$

(15)

whenever the extra principal visibility is removable in the above sense. Condition (15) is called no-phantom visibility. The finite-sector consequence of ${\delta }_{\mathrm{fid}}=0$ is faithful Gauss-sector reconstruction. The corresponding reduced-description rule is used later for the finite Schur-Kuranishi layer, after the low window, the $B-L$ filtration, the exterior bidegree, and the torsor labels have been fixed. In that layer, a reduced representative is chosen by minimizing a monotone finite Schur-coordinate complexity on the constrained zero set of the already closed residuals. The Schur reduction removes only channels which leave the finite observable obstruction coordinates and the already closed sector labels unchanged. Gap-mediated Schur terms are not removed by this rule.

The finite-coefficient entry is read with full finite-shadow conservation. Once a topological coefficient class has been reduced by the determinant global form, the full finite coefficient class is retained by $R_{\mathrm{fin}}$ before any coprime projection is selected for a later low-sector factor. For the primitive branch this is the ${\mathbb{Z}}_6$ class whose ${\mathbb{Z}}_3$ and ${\mathbb{Z}}_2$ projections are used in the torsor and parity entries.

In particular, an ordinary action principle is obtained when one residual component is the first variation of a classical action on the chosen slice. The reconstruction principle does not require this component to be primitive. It requires only that the corresponding displacement be one of the closure defects of the reconstructed description. In this sense the Euler-Lagrange, Einstein, and curvature-compatibility equations are recovered as special projected closure equations, while the primitive branch uses the link, Gauss, visibility, determinant, torsor, and Schur entries. The familiar variational cases are recovered by choosing the corresponding residual component. This is the sense in which the reconstruction principle extends the usual minimal-action reading.

The Riemannian curvature conventions are the standard ones used in Einstein and Codazzi comparison geometry [36]. The boundary-charge entry is the local-observable superselection reading of [13]. The determinant entry is the finite part of the regular-family determinant reading [27].

Table 1 is used only as a comparison table. In the primitive branch, the relevant residuals are the link, Gauss-local, Toeplitz, determinant, torsor, and Schur entries listed in (4).

The primitive Codazzi branch is obtained by applying the generated closure laws to the ordered residuals in (4). The entries in Table 2 record the role of each generator. The proofs of the structural entries are given in the sections where the corresponding data are introduced.

The Toeplitz entry is the ${\mathbb{CP}}^1$ Berezin-Toeplitz cutoff used in (24); the general comparison is represented by [14] and by the review [37]. The determinant and exterior entries are compared with the usual finite $SU\left(5\right)$ and $Spin\left(10\right)$ organization [16]. The torsor entry is finite and is treated by the cochain complex (54). In the Alena-Codazzi realization it is the finite boundary shadow of the vorticity response. Thus the same ordered residual list may be read in geometric, topological, Gauss-local, quantized-link, determinant, finite-torsor, and spectral languages.

The table separates structural closure from completed-branch data. The support, determinant group, exterior package, finite shadow, and torsor complex are fixed before a numerical low operator is chosen. Yukawa singular values, CKM and PMNS angles, CP phases, scalar normalizations, and contact coefficients enter only through the later Schur-completed representative.

The generated laws in Table 2 have different logical status. The first block is structural: it fixes the projective link, the Gauss-local charge sectors, the Toeplitz support, the primitive positive degree, the determinant carrier, and the exterior package. The second block is finite: it fixes the $B-L$ channel check, the full ${\mathbb{Z}}_6$ shadow, the weak-parity factor, and the projective-color finite Hodge-spectral module. The last block is spectral: it fixes the low-window hypotheses, the finite family-motion tests, the direct-block shape diagnostics, the Schur completion, and the scale comparison.

This status is summarized in Table 3.

The entries in Table 3 will be called structural when they are fixed before the low spectral representative is chosen. In this sense the carrier, determinant group, exterior package, full finite shadow, and projective-color torsor complex are structural outputs of the primitive reconstruction. The later spectral entries are conditional branch outputs: they depend on the completed low representative, but they act on the boundary cycle already fixed by the first line of (16).

The spectral entries may be refined into channel subresiduals only after the low window and the $B-L$ filtration have been fixed. The charged-lepton balance subresidual used in Subsection 5.4 is of this kind. It is scale-free and acts inside the direct charged-lepton Dirac block; it is not an input to Theorem 1. The lexicographic order is used only to prevent a later completion from changing an earlier structural selection. Thus a Schur correction may deform the effective low operator, but it does not reselect the carrier once the support theorem has been applied. Similarly, scale comparison may affect numerical diagnostics, but it does not change the finite determinant carrier. This is the ordering used in the remaining sections. For the primitive branch, Table 2 is evaluated in the ordered dependency

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

(16)

The first line fixes the local carrier, determinant package, $B-L$ channel, finite shadow, weak-parity factor, and boundary torsor cycle. The second line acts only after this cycle has been fixed and supplies the spectral low window, non-circulant family motion, direct-block shape diagnostics, Schur-completed matrix elements, and scale comparison.

3. Primitive Gauss-Local Reconstruction and the Boundary Torsor Cycle

This section isolates the local structural part of the reconstruction. The raw boundary datum is a codimension-three worldline defect with a projective link, a positive coefficient line, and two nonzero Gauss-local principal boundary charges of types $V_1$ and $V_2$. The faithful Gauss entry of (14) gives separated finite charge sectors. Toeplitz visibility then gives the rank-five Borel-Weil support, while the complexity entry gives the reduced degree-one positive line. The split determinant reading gives the carrier group, the even exterior package, the global ${\mathbb{Z}}_6$ form, and the finite determinant shadow. The projective-color projection of that shadow defines a finite torsor cycle, later realized as the quantized boundary shadow of the Alena-Codazzi vorticity response.

The carrier theorem is local and representation-theoretic. It is formulated for collars which supply the two principal Gauss-local charges and a relative boundary-charge algebra. The proof below uses the projective link, faithful finite-sector reconstruction, Toeplitz visibility, and reduced positive-degree closure. Section 4 supplies an Alena-Codazzi collar as a compact-leaf source model and as a realization criterion for these data. The completed locked operator, weak bridge, masses, mixing data, Pfaffian denominator, and contact coefficients are treated after the boundary cycle has been fixed.

3.1. Primitive Gauss-local defect data and boundary superselection

Let $\Gamma$ be a worldline defect in a four-dimensional Lorentzian branch. The real blow-up of the branch along $\Gamma$ has boundary fiber

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

(17)

In an optical Codazzi branch the two-eigenvalue Codazzi splitting gives two shear-free geodesic null directions in the Lorentzian principal plane. This is the standard two-eigenvalue Codazzi mechanism [38]; Lorentzian Codazzi structures and related $2+2$ optical geometries are represented by [39,40]. The corresponding projective-spinor reading identifies the link sphere with the projective line,

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

(18)

The spinor conventions are those of [7,41]. The broader twistor comparison class is represented by [8].

The local topology of a worldline complement is

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

(19)

A positive resolved transverse frame determines a positive class in the $H^2$ slot. The corresponding complex line on the projective link has degree

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

(20)

The degree-one representative is

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

(21)

Higher positive degrees define different filtered link data. The degree-one case will be recovered below by reduced positive-degree closure. The local regularity and removable-singularity background are standard [42]; the elliptic estimates used later are in the convention of [43].

Let $N_{\Gamma }$ denote the oriented rank-three transverse normal fiber along the defect. A natural scalar-sector source of transverse order at most two has associated-graded normal terms in ${Sym}^0\left(N_{\Gamma }^{*}\right)\oplus {Sym}^1\left(N_{\Gamma }^{*}\right)\oplus {Sym}^2\left(N_{\Gamma }^{*}\right)$. With the usual $Spin\left(3\right)\simeq SU\left(2\right)$ notation, these terms have types $V_0$, $V_1$, and $V_0\oplus V_2$. Thus, after the scalar trace has been separated, only two principal non-scalar types remain:

$${\left(gr{J}_{\perp }^{\le 2}\right)}_{\mathrm{ns}}=V_1\oplus V_2.$$

(22)

The spherical tensor conventions are those of [44]. The multipole notation follows the standard convention of [45].

A boundary charge on the resolved collar will mean a finite principal coefficient which is read by a surface pairing on a linking sphere and is independent of the linking radius in a source-free annulus. In the principal flat normal model this is the usual Stokes or Green identity. In a regular curved collar the same statement is transported by the frozen collar connection and the corresponding adjoint link modes.

Definition 1

(Relative Gauss-local observable algebra). Let ${\mathcal{Q}}_{\partial }$ be the finite boundary-charge algebra generated by the principal charges on the linking sphere. A relative Gauss-local observable algebra ${\mathcal{A}}_{\mathrm{loc}}$ is an annulus-supported local algebra with fixed boundary data whose admissible observables commute with ${\mathcal{Q}}_{\partial }$ in the sector representation. The annulus is source-free in the Gauss sense, namely no singular charge density is present in its interior.

Definition 2

(Gauss-local second-jet datum and primitive reduction). A worldline defect Γ will be called a Gauss-local second-jet datum if the following data are given:

(i)

the resolved projective link (18);

(ii)

a positive link line as in (20);

(iii)

a natural scalar-sector source of transverse order at most two, with the non-scalar associated-graded type (22);

(iv)

two nonzero principal boundary charges $Q_1\in V_1$ and $Q_2\in V_2$ representing the two summands of (22);

(v)

a relative Gauss-local observable algebra ${\mathcal{A}}_{\mathrm{loc}}$ on a singular-source-free punctured collar, with fixed boundary data, in the sense of Definition 1.

It is called primitive after reduced reconstruction if the positive line is reduced to (21) by the reduced visibility condition of Definition 4.

The last condition is the Gauss-local superselection condition. It is the local boundary-charge analogue of the standard sector reconstruction viewpoint for local observables [46]. In the Alena-Codazzi realization, $Q_1$ will be supplied by the first moment of the conserved translational current and $Q_2$ by the trace-free Codazzi-Gauss surface coefficient. The relative transport of this pair supplies the finite torsor transport once a projective-color boundary transport has been fixed. The central separation used below is the finite-sector form of the zero fidelity condition (12).

Lemma 1

(Second-jet reduction). Let the scalar-sector source of a worldline defect be natural and of transverse order at most two. After the scalar singlet is separated, its associated-graded non-scalar normal source has the type (22).

Proof. 

Only normal jets of order at most two can enter. The associated graded therefore lies in ${Sym}^0\left(N_{\Gamma }^{*}\right)\oplus {Sym}^1\left(N_{\Gamma }^{*}\right)\oplus {Sym}^2\left(N_{\Gamma }^{*}\right)$. For an oriented rank-three normal fiber, the three summands have $SU\left(2\right)$ types $V_0$, $V_1$, and $V_0\oplus V_2$. Removing the scalar type $V_0$ leaves exactly $V_1\oplus V_2$. □

Lemma 2

(Relative Gauss-local boundary centrality). Let $Q_a$, $a=1,2$, be the two boundary charges of Definition 2. Let ${\mathcal{A}}_{\mathrm{loc}}$ be the relative Gauss-local observable algebra on a source-free annulus with fixed boundary data. Then the charge sectors of $Q_1$ and $Q_2$ are invariant under ${\mathcal{A}}_{\mathrm{loc}}$. In any finite sector representation in which the charges are diagonal, the corresponding spectral projections lie in the center of the represented local algebra.

Proof. 

The charges are read on linking spheres. In a source-free annulus the corresponding Stokes or Green identity identifies the same boundary-charge algebra on all linking spheres in the annulus. Since ${\mathcal{A}}_{\mathrm{loc}}$ is relative to this boundary-charge algebra, its represented observables preserve the joint charge sectors. In a diagonal finite sector representation, the spectral projections of the charges are functions of the boundary-charge algebra and commute with all represented local observables. □

The superselection statement is used only at the principal boundary-charge level. The completed spectral problem may deform the effective low operator through Schur terms, but a charge-compatible completion preserves the same sector projections.

Lemma 3

(Reconstructive charge-sector separation). Assume that the two principal charges $Q_1\in V_1$ and $Q_2\in V_2$ are retained as independent components of the Gauss residual and that the fidelity defect (12) vanishes. Then, in the finite support algebra used by the reconstruction, the two charges are represented on independent central support sectors.

Proof. 

Lemma 2 gives invariant charge-sector projections for the relative Gauss-local algebra. The vanishing of (12) requires the independent Gauss residual entries to remain separately recoverable from the finite sector algebra. Hence the corresponding finite support sectors cannot be merged without lowering the rank of (11). The separated projections are central by Lemma 2. □

3.2. The universal rank-five support

The positive quantization of the primitive link is the Borel-Weil tower

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

(23)

Equivalently, $E_q\simeq {Sym}^{q-1}{\mathbb{C}}^2$ is the irreducible $SU\left(2\right)$ module of spin $(q-1)/2$ and dimension q. The Borel-Weil theorem is used in its standard homogeneous-bundle form [10]. The same zero-mode count is obtained from monopole harmonics on $S^2$ [11], and the Taub-NUT Dirac problem gives a useful comparison [47].

The finite visibility rule on the rank-one projective link is

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

(24)

This is the Clebsch-Gordan decomposition for $E_q\otimes E_q^{*}$, after the scalar summand has been removed. In the quantization language, (24) is the ${\mathbb{CP}}^1$ Berezin-Toeplitz cutoff [14]. The same finite-mode content appears in the fuzzy-sphere comparison [15] and in finite-matrix brane models [48].

The visibility condition (24) says that the irreducible type $V_{\ell }$, $\ell \ge 1$, is visible on $E_q$ exactly for $q\ge \ell +1$. Hence the first visible block for $V_{\ell }$ is $E_{\ell +1}$.

Definition 3

(Boundary-central Toeplitz support). A finite positive support is a finite sum $F={⨁}_q{m}_q{E}_q$ of Borel-Weil blocks. A type $V_{\ell }$ is visible in an isotypic component $E_q^{\oplus m_q}$ if $V_{\ell }\subset {End}_0\left(E_q\right)$. The support is boundary-central for the two-channel source (22) if the two nonzero principal charges $Q_1\in V_1$ and $Q_2\in V_2$ are represented in two independent central sectors of the $SU\left(2\right)$-equivariant local support algebra. The size of F is $rankF={\sum }_q{m}_{q}q$.

Lemma 4

(Central sectors of a finite $SU\left(2\right)$ support). Let $F={⨁}_q{m}_q{E}_q$ be a finite positive support. Then

$${End}_{SU\left(2\right)}\left(F\right)\simeq \underset{q}{⨁}{M}_{m_q}\left(\mathbb{C}\right),Z\left({End}_{SU\left(2\right)}\left(F\right)\right)\simeq \underset{q}{⨁}\mathbb{C}{id}_{E_q^{\oplus m_q}}.$$

(25)

Thus central support sectors are isotypic sectors of the selected $SU\left(2\right)$ support.

Proof. 

Schur’s lemma gives ${Hom}_{SU\left(2\right)}(E_q,E_{q^{\prime }})=0$ for $q\ne q^{\prime }$ and ${End}_{SU\left(2\right)}\left(E_q\right)=\mathbb{C}$. The stated decomposition and its center follow immediately. □

Theorem 1

(Primitive Gauss-local rank-five support). Let Γ be a Gauss-local second-jet datum in the sense of Definition 2, with positive line reduced to (21). Among boundary-central Toeplitz-visible supports for the two nonzero principal charges $Q_1\in V_1$ and $Q_2\in V_2$, the minimal support is

$$V_{\Gamma }=C_{\Gamma }\oplus W_{\Gamma },C_{\Gamma }=E_3,W_{\Gamma }=E_2.$$

(26)

In particular, $rank{V}_{\Gamma }=5$.

Proof. 

By (24), the first visible block for $V_1$ is $E_2$, and the first visible block for $V_2$ is $E_3$. Boundary centrality gives two independent finite charge sectors. By Lemma 4, such sectors are isotypic support sectors. A sector which sees $V_1$ has rank at least 2, and a sector which sees $V_2$ has rank at least 3. The two sectors are independent, so the total rank is at least 5. The support (26) realizes this bound. □

Proposition 2

(Separated support ladder). Let the independent principal Gauss-local source types be $V_1,\dots ,V_r$, and assume boundary-central faithful reconstruction for these r channels. Among Toeplitz-visible positive supports, the minimal separated support is

$$F_r^{\mathrm{sep}}={\displaystyle \underset{\ell =1}{\overset{r}{⨁}}}{E}_{\ell +1},rank{F}_r^{\mathrm{sep}}={\displaystyle \sum _{\ell =1}^r}(\ell +1)={\displaystyle \frac{r(r+3)}{2}}.$$

(27)

Proof. 

By (24), the first block on which $V_{\ell }$ is visible is $E_{\ell +1}$. Boundary-central faithful reconstruction requires the independent principal charge classes to be represented on independent central support sectors. By Lemma 4, these sectors are isotypic sectors. Hence each $V_{\ell }$ requires a separated sector of rank at least $\ell +1$, and (27) realizes the lower bound. □

If only raw Toeplitz visibility of the combined type $V_1\oplus V_2$ is required, (24) gives the smallest single Borel-Weil block as $E_3$. The separated finite-support problem below is therefore stronger than raw visibility.

The raw single-block visibility recorded here clarifies the role of boundary centrality in Theorem 1. The selected rank-five carrier is the first separated realization of two Gauss-local charge sectors.

The primitive degree is also detected by the first-threshold structure. If the link line is replaced by a positive degree-n line $𝒪\left(n\right)$, the q-th block is $H^0({\mathbb{CP}}^1,𝒪\left(n(q-1)\right))$. The first threshold for $V_{\ell }$ is then

$$q_{\ell }\left(n\right)=1+⌈{\displaystyle \frac{\ell }{n}}⌉.$$

(28)

Among positive degrees, (28) separates the first thresholds of $V_1$ and $V_2$ only for $n=1$. For every $n\ge 2$, the two thresholds coincide.

The degree-one sector is therefore the unique positive sector which separates the two principal charges at their first visible blocks. Non-primitive positive degrees define different filtered link data.

The primitive sector can also be recovered by allowing the positive degree to vary and then imposing reduced closure. For the closure used below, the first-threshold statement is combined with boundary-central minimality, absence of unsourced low-visible integer types, and the finite determinant shadow. For $L=𝒪\left(n\right)$, the block $H^0({\mathbb{CP}}^1,𝒪\left(n(q-1)\right))$ has dimension $n(q-1)+1$. Since two independent central charge sectors must lie in distinct isotypic support sectors, the minimal boundary-central support for the two-channel source is realized by the two blocks $q=2$ and $q=3$. Its rank is

$$rank{F}_{min}\left(n\right)=(n+1)+(2n+1)=3n+2.$$

(29)

Definition 4

(Reduced primitive visibility). A boundary-central two-channel support is reduced if its diagonal integer Toeplitz-visible content contains no principal source type beyond the prescribed Gauss-local types $V_1$ and $V_2$.

Proposition 3

(Reduced tri-quantized primitive degree). Among positive link degrees, reduced primitive visibility forces the degree-one line. For this degree the minimal boundary-central support is the rank-five support (26), and the finite-coefficient shadow in (46) is the full ${\mathbb{Z}}_6$ generator.

Proof. 

For the minimal boundary-central support $F_{min}\left(n\right)$, the diagonal integer Toeplitz-visible content is $V_1,\dots ,V_{2n}$. In the positive-degree comparison, the entries $V_{\ell }$ with $\ell \ge 3$ are removable unsourced principal visibility in the sense of Proposition 1. Thus Definition 4 gives $2n\le 2$. Since the link degree is positive, $n=1$. The resulting support has rank 5, and Theorem 1 identifies it with (26). For $n=1$, the class (46) is the generator of the finite coefficient group. □

Theorem 2

(Minimal self-reconstructing boundary carrier). Let Γ be a Gauss-local second-jet datum with projective link (18), positive line (20), and nonzero principal charges $Q_1\in V_1$ and $Q_2\in V_2$. Assume that the Gauss, fidelity, visibility, and support-complexity entries of (14) are closed on the selected finite support. Then the reduced positive degree is the primitive degree (21), and the minimal separated support is (26).

Proof. 

The vanishing fidelity defect gives the independent central support sectors of Lemma 3. The complexity entry of (14) removes the unsourced principal visibility as in Proposition 3, so the positive degree is $n=1$. At this degree, Proposition 2 with $r=2$ gives the minimal separated support, which is (26) in the notation of Theorem 1. □

The same representation count gives the block-parity rule used later in the Schur completion.

Proposition 4

(Integer-source block parity and low-block rigidity). Let $P_{23}$ be the orthogonal projection onto $E_2\oplus E_3$ inside the positive Borel-Weil tower. The mixed blocks between $E_2$ and $E_3$ carry only half-integer $SU\left(2\right)$ types. Hence the integer principal channels $V_1$ and $V_2$ act on the diagonal blocks of the selected support. Moreover, no integer type $V_{\ell }$ with $\ell \ge 3$ occurs in ${End}_0(E_2\oplus E_3)$. Consequently, a principal integer multipole of type $V_{\ell }$, $\ell \ge 3$, has zero compression to the selected low support.

Proof. 

One has ${End}_0\left(E_2\right)=V_1$ and ${End}_0\left(E_3\right)=V_1\oplus V_2$. The mixed blocks $Hom(E_2,E_3)$ and $Hom(E_3,E_2)$ have half-integer types $V_{1/2}\oplus V_{3/2}$. The principal source types are integer types, so their principal action is diagonal with respect to the selected split. The remaining low-block statement follows by equivariance. □

Higher normal moments may still enter the completed normal operator through low-high Schur terms. Proposition 4 states that such moments do not reselect the primitive low support at the principal Toeplitz level and that the selected split is compatible with the integer source channels. Under the Callias-Schur gap used later, higher moments deform the effective low operator rather than the carrier (26). The support theorem therefore fixes the finite carrier on which the determinant package, the torsor cycle, and the completed spectral diagnostics are evaluated.

3.3. The determinant carrier package

The minimal self-reconstructing carrier theorem has fixed the finite carrier (26). The structures recorded in this subsection are read after this selection has been made. The two summands are the independent finite charge sectors associated with the two principal Gauss-local charges, so the natural compact basis changes are split unitary changes. Put

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

(30)

Thus $dimC=3$ and $dimW=2$. Each selected Borel-Weil block carries its standard invariant Hermitian structure. The split compact basis group is initially $U\left(C\right)\times U\left(W\right)$. Fixing the total split top exterior form imposes

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

(31)

Hence the compact determinant-preserving carrier group is

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

(32)

Equivalently, the corresponding global form is

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

(33)

The line-operator sensitivity to the global form (33) is the standard one in gauge theory [49].

The quotient in (33) fixes the usual congruence between color triality, weak parity, and hypercharge. Let $t\in {\mathbb{Z}}_3$ denote color triality, and let $p\in {\mathbb{Z}}_2$ denote the weak parity of the $SU\left(2\right)$ representation. With the hypercharge convention (36), a representation descending to the global group (33) satisfies

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

(34)

Indeed, the standard covering map may be taken in the form $(A,B,z)\mapsto (z^{2}A,z^{-3}B)$. The generator of the kernel is represented by $z=e^{2\pi i/6}$. On a representation with data $(t,p,Y)$ it acts with phase $z^{-2t+3p-6Y}$, so descent to (33) gives (34). The entries in Table 4 obey this congruence.

After the separated carrier has been fixed, the determinant reduction is applied to the split compact basis group $U\left(C\right)\times U\left(W\right)$. The split top-form condition (31) gives (32) and hence the global form (33).

The determinant-compatible finite module is the even exterior package

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

(35)

Its dimension is $1+\left({\displaystyle \genfrac{}{}{0pt}{}{5}{2}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{5}{4}}\right)=16$. The exterior organization is the usual finite package appearing in the $SU\left(5\right)$ and $Spin\left(10\right)$ comparison class [16]. Related Clifford-ideal descriptions give a separate algebraic comparison [17]; further Clifford and ideal-based variants are represented by [50]. In the present construction the exterior package is applied to the carrier selected by Theorem 2.

Let $N_C$ and $N_W$ be the exterior degree operators associated with C and W. The hypercharge convention compatible with (31) is

$$Y=-{\displaystyle \frac{1}{3}}{N}_C+{\displaystyle \frac{1}{2}}{N}_W.$$

(36)

The electric charge is $Q=T_3+Y$, with $T_3$ acting on the weak factor. The weak-hypercharge superselection comparison is represented by [51]. The even package is recorded in Table 4. The identifications using dual representations are made with the top-form convention (31).

The same finite package gives the hypercharge-square trace

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

(37)

The exterior bidegree also carries the independent structural charge

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

(38)

It depends only on the color degree and is independent of (36). The corresponding $Spin\left(10\right)$-type grading is

$$X_{\mathrm{Spin}}=5(B-L)-4Y.$$

(39)

The resulting degree readings are listed in Table 5. Octonionic ladder and internal-space comparisons give related finite-structure examples [52], while recent algebraic unification variants are represented by [53,54].

Since the finite package (35) is read after the carrier selection, its local anomaly and normalization counts are consequences of the selected determinant-compatible package. Let $I_3$ and $I_2$ denote the total nonabelian Dynkin traces of the color and weak generators on (35), with the fundamental representation normalized by $T\left(\mathbf{fund}\right)={\displaystyle \frac{1}{2}}$. Then

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

(40)

Together with (37), this gives the canonical hypercharge normalization

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

(41)

Table 6 records the local anomaly consequences of the determinant-compatible even package (35) with hypercharge (36). The same package satisfies the Witten $SU\left(2\right)$ parity count and has the normalization (41), obtained from (37).

Table 7, together with Table 6, records the corresponding local checks for (38) and for the mixed Abelian coefficients with (36). Thus the Abelian plane $span\{Y,B-L\}$ is used below as the anomaly-free Schur-channel plane on the same determinant-compatible package.

The determinant role remains assigned to the top-form kernel of Proposition 6. The $B-L$ degree is used below as an anomaly-free Schur-channel filtration on the same finite package. The finite-trace direction in this plane which is orthogonal to the determinant degree is the $Spin\left(10\right)$-type degree (39).

Proposition 5

(Finite-trace orthogonal Abelian channel). On the determinant-compatible even package (35), the degree $X_{\mathrm{Spin}}$ is finite-trace orthogonal to Y. More precisely,

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

(42)

Proof. 

The degree counts are obtained from Table 4 and Table 5, with multiplicities given by the dimensions of the displayed summands. The identity ${Tr}_{{\Lambda }^{\mathrm{even}}V}\left(Y{X}_{\mathrm{Spin}}\right)=0$ follows from (39) and (37). □

The local determinant count is a finite degree count on (35). For a general split degree operator

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

(43)

the infinitesimal form of the top-form condition (31) is

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

(44)

The local determinant-obstruction factors on the even exterior package are

The last quadratic factor is positive definite. Hence the common kernel of the local finite factors in (45) is exactly (44). This is the same infinitesimal kernel as the stabilizer of the split top exterior form.

Proposition 6

(Top-form determinant kernel). For the selected carrier (30), the infinitesimal stabilizer of the split top exterior line is the common local kernel of the finite determinant-obstruction factors in (45). The normalized degree-counting representative of this kernel is (36).

Proof. 

This is the common linear factor $3a+2b$ in (44) and (45); the remaining quadratic factor in the cubic row is positive definite, and the usual integral normalization gives (36). □

Remark 1

(Structural carrier package). The reduced primitive-degree closure gives the primitive rank-five support (26) as the minimal positive closure. Together with the split top-form constraint (31), this gives the compact carrier group (32). The determinant-compatible even exterior package is (35), with degree readings (36), (38), and (39). The local checks in Table 6 and Table 7, together with the finite-trace orthogonality (42), are used as consistency data for this carrier.

In a determinant-sensitive family reading, the local finite factors in (45) are the finite part seen by the Bismut-Freed determinant form for a regular boundary-admissible family [27]. The local characteristic-class language is standard [55]; secondary flat or torsion data belong to the corresponding global determinant problem [56]. Thus the local determinant kernel fixes the split carrier algebra, while residual flat and torsion determinant holonomies are completed-branch data.

The primitive class has a finite-coefficient shadow after the determinant global form (33) has been fixed. By full finite-shadow conservation, the coefficient reduction is first retained at the ${\mathbb{Z}}_6$ level:

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

(46)

Since $L_{\Gamma }\simeq 𝒪\left(1\right)$, (46) is the generator of the finite coefficient group. Its projective-color projection is

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

(47)

The corresponding weak projection gives the compatible parity class in $H^2({\mathbb{CP}}^1,{\mathbb{Z}}_2)$. Thus the central ${\mathbb{Z}}_3$ factor is the projective-color shadow of the full ${\mathbb{Z}}_6$ determinant quotient, while the ${\mathbb{Z}}_2$ factor is read through the chirality/exterior-parity lock in (102).

It is useful to keep the full finite shadow before the projective-color quotient is chosen. Let ${\mathcal{T}}_{\Gamma }^{\left(6\right)}$ denote the finite torsor with coefficient class (46). Its regular carrier is

$${\mathcal{H}}_{\Gamma }^{\left(6\right)}=C^0\left({\mathcal{T}}_{\Gamma }^{\left(6\right)}\right)\simeq \mathbb{C}\left[{\mathbb{Z}}_6\right].$$

(48)

After a generator has been chosen, let $S_6$ denote the regular shift on this carrier. The two coprime factor shifts are represented by $S_3=S_6^2$ and $S_2=S_6^3$. Equivalently,

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

(49)

Thus the projective-color carrier used below is the ${\mathbb{Z}}_3$ factor of (49), with the ${\mathbb{Z}}_2$ factor reserved for the weak-parity part of the lock (102). If a completed phase lift is retained for the two finite factors, the corresponding phases satisfy

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

(50)

The last relation is a finite-shadow coherence condition. It does not change the selected carrier or the determinant group.

The weak-parity factor may be recorded by the finite ${\mathbb{Z}}_2$ response on the second factor of (49). Let $S_2$ be the regular ${\mathbb{Z}}_2$ shift. For a completed parity phase ${\theta }_2$, put $d_2=S_2-e^{i{\theta }_2}\mathbf{1}$.

Lemma 5

(Weak-parity finite-shadow factor). The vertex Laplacian of the ${\mathbb{Z}}_2$ factor is

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

(51)

In the flat parity normalization ${\theta }_2=0$, the two eigenvalues are 0 and $4{\nu }_2$.

Proof. 

The regular ${\mathbb{Z}}_2$ shift satisfies $S_2=S_2^{†}$ and $S_2^2=\mathbf{1}$. Substitution into $d_2^{†}{d}_2$ gives (51). The spectrum follows from the two eigenvalues $+1$ and $-1$ of $S_2$. □

Thus the projective-color torsor is the active central family factor, while the ${\mathbb{Z}}_2$ factor is a parity component of the lock. If the complementary parity mode is kept in the isolated low window, the central multiplicity is doubled. This rank effect is used in Corollary 2.

The odd companion of the exterior package is supplied by Clifford multiplication by $V\oplus V^{*}$ on ${\Lambda }^{•}V$. Its $W\oplus W^{*}$ part gives the one-Higgs weak bridge after the locked low operator has been fixed. Quillen superconnections give the structural comparison for such odd insertions [35]. Defect-localized zero modes and domain-wall fermions give related analytic comparison classes [57,58]. The full $B-L$ filtered use of this bridge belongs to the completed locked cycle.

3.4. The projective-color torsor complex

The projective-color coefficient class in (47) is the ${\mathbb{Z}}_3$ projection of the full finite shadow (46). Equivalently, it is the ${\mathbb{Z}}_3$ factor of the master carrier (49), with the ${\mathbb{Z}}_2$ factor used later as the weak-parity part of the lock (102). It fixes the regular finite central-response torsor, denoted by ${\mathcal{T}}_{\Gamma }$, with coefficient class

$$\left[{\mathcal{T}}_{\Gamma }\right]={\left[c_1\left(L_{\Gamma }\right)\right]}_3.$$

(52)

Its vertex space is the regular central carrier

$${\mathcal{H}}_{\mathrm{cen}}=C^0\left({\mathcal{T}}_{\Gamma }\right)\simeq \mathbb{C}\left[{\mathbb{Z}}_3\right].$$

(53)

This is the finite central response space used in the completed family sector. The torsor fixes the central factor; the locked-kernel rank, the sectoral splitting, and the physical mixing data remain completed spectral quantities. Among cyclic carriers, $C_3$ is the smallest faithful realization of the projective-color shadow whose real regular representation has a singlet and one two-dimensional torsor complement. This makes $C_3$ the minimal active finite vortex-shadow carrier used below. In the charged-lepton direct block, the same vertex carrier later gives this real decomposition.

Let ${\mathcal{E}}_{\Gamma }$ be a finite Hermitian coefficient system over ${\mathcal{T}}_{\Gamma }$. For the structural central carrier one may take ${\mathcal{E}}_{\Gamma }=\mathbb{C}$; in the locked completion it is tensored with the selected exterior and low-mode data. The finite edge resolution is

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

(54)

After choosing an ordering $a\in {\mathbb{Z}}_3$ of the torsor, the covariant edge differential is written

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

(55)

Here $U_a$ is the unitary edge transport of the coefficient system and $e^{i{\theta }_{\Gamma }}$ is the flat determinant-torsor phase. The edge space in (54) is an auxiliary finite resolution of the torsor response, with the family carrier remaining the vertex space (53).

The projective-color finite Hodge-spectral module is the finite package

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

(56)

Here $C^{•}({\mathcal{T}}_{\Gamma },{\mathcal{E}}_{\Gamma })$ is the complex (54), and

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

(57)

The algebra $C^0\left({\mathcal{T}}_{\Gamma }\right)$ is the vertex algebra of the torsor, while ${\mathcal{A}}_{\mathrm{fam}}$ is the algebra of central family blocks tested by the Schur completion. The graph-Hodge comparison is represented by [59]. The vector-bundle Laplacian reading of unitary edge transports is the comparison of [60]. Finite-dimensional spectral-module comparisons are represented by [61].

The finite Wilson defect of the torsor transport is

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

(58)

The torsor residual is

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

(59)

The flat minimal branch has $R_{\mathrm{tor}}^{\Gamma }=0$. More generally, the non-central part of (58) records the finite torsor curvature seen by the completed branch.

Thus the minimal flat torsor branch is the subcase $R_{\mathrm{tor}}^{\Gamma }=0$ of (59).

In a flat coefficient trivialization $U_a=\mathbf{1}$, put

$$d_{\theta }=S-e^{i{\theta }_{\Gamma }}\mathbf{1},S{e}_a=e_{a+1}.$$

(60)

The associated vertex Laplacian is

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

(61)

Thus the adjacent central response is the vertex Laplacian of the torsor complex. The quadratic boundary action whose Hessian is (61) is obtained from the norm of $d_{\theta }\xi$:

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

(62)

The corresponding Hermitian response is ${\nu }_{\Gamma }{\Delta }_{\theta }$. Its eigenvalues are

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

(63)

In particular, $\parallel {\nu }_{\Gamma }{\Delta }_{\theta }\parallel \le 4{\nu }_{\Gamma }$.

Proposition 7

(Wilson-Laplacian determinant). In the flat scalar trivialization (60), the torsor Wilson defect and the vertex Laplacian satisfy

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

(64)

More generally, if the coefficient system has rank m and unitary holonomy ${\mathcal{W}}_{\Gamma }$, then

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

(65)

Thus the torsor Wilson residual is the determinant obstruction to the harmonic part of the finite Hodge complex.

Proof. 

In the trivialization used in (60), one has ${\mathcal{W}}_{\Gamma }=e^{-3i{\theta }_{\Gamma }}$. Since $S^3=\mathbf{1}$, the characteristic polynomial of S gives $det{d}_{\theta }=1-e^{3i{\theta }_{\Gamma }}$. Hence (64) follows. For a rank-m coefficient system, unitary gauge changes at the vertices move the edge transports to a single cyclic holonomy. The resulting block determinant is $det(\mathbf{1}-{\mathcal{W}}_{\Gamma })$ up to a unit-modulus factor. Taking $d_{\Gamma }^{†}{d}_{\Gamma }$ gives (65). □

Corollary 1

(Torsor cohomology). For the finite complex (54),

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

(66)

In particular, the scalar flat branch has $(b_0,b_1)=(1,1)$, while a scalar branch with ${\mathcal{W}}_{\Gamma }\ne 1$ has $(b_0,b_1)=(0,0)$.

Proof. 

A section in $ker{d}_{\Gamma }$ is determined by one vertex value. After one turn around the cycle this value is acted on by ${\mathcal{W}}_{\Gamma }$, so $H^0\simeq ker(\mathbf{1}-{\mathcal{W}}_{\Gamma })$. Since the complex has only degrees zero and one, $H^1=C^1({\mathcal{T}}_{\Gamma },{\mathcal{E}}_{\Gamma })/im{d}_{\Gamma }$. The spaces $C^0$ and $C^1$ have the same finite rank, and therefore $dim{H}^1=dim{C}^1-rank{d}_{\Gamma }=dim{C}^0-rank{d}_{\Gamma }=dimker{d}_{\Gamma }$. □

Proposition 8

(Scalar torsor spectral invariants). For the unscaled scalar Laplacian, put ${\mu }_k={\lambda }_k/{\nu }_{\Gamma }$. The elementary symmetric invariants of (63) are

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

(67)

Equivalently, the characteristic polynomial is

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

(68)

Proof. 

The trace and the second elementary coefficient are computed from (63), using the three cubic roots of unity. The product is (64). The characteristic polynomial is the monic polynomial with the elementary coefficients in (67). □

Thus the normalized trace used below is independent of the torsor flux, while the determinant is the Wilson obstruction. The scalar operator (61) is therefore the magnetic graph Laplacian on the projective-color cycle. Local edge phases can be gauged into the cycle holonomy (58). Thus ${\mathcal{W}}_{\Gamma }$ is the gauge-invariant flux datum, and ${\theta }_{\Gamma }$ is its flat-trivialization coordinate; (63) records the corresponding central-response spectrum.

In the vortex-shadow reading, the cycle holonomy is the finite circulation datum, the torsor cohomology (66) records the flat-branch obstruction, and the trace (74) gives the flux-independent central seed. The determinant in (64) remains the Wilson obstruction of this finite shadow.

The same torsor complex gives the finite test for non-circulant central response. Let $\omega =e^{2\pi i/3}$ and let $Z{e}_a={\omega }^a{e}_a$. The clock and shift obey $ZS=\omega SZ$. Any central $3\times 3$ block T has the Weyl expansion

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

(69)

The circulant part is the $a=0$ component. The non-circulant projection is

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

(70)

For a Weyl monomial $T_{ab}=Z^a{S}^b$, the mixed torsor commutator is

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

(71)

Consequently, the mixed torsor curvature vanishes exactly on the circulant part. Equivalently, for a central block T,

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

(72)

Proposition 9

(Torsor spectral seminorm). For the flat torsor differential (60), the quadratic form

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

(73)

is a spectral seminorm on ${\mathcal{A}}_{\mathrm{fam}}=End\left({\mathcal{H}}_{\mathrm{cen}}\right)$. Its kernel is the circulant family algebra $\mathbb{C}\left[S\right]$. Equivalently, (72) identifies the non-circulant detector with this seminorm.

Proof. 

It is enough to check the Weyl basis. Formula (71) vanishes precisely for $a=0$. Orthogonality of the Weyl basis under the central trace gives (72). Hence ${\mathcal{L}}_{\Gamma }\left(T\right)=0$ exactly for the $a=0$ component of (69), namely for $T\in \mathbb{C}\left[S\right]$. □

Thus $\mathbb{C}\left[S\right]$ is the flat circulant family algebra of the projective-color finite Hodge-spectral module. In the vortex-shadow reading it is the flat circulant part of the boundary transport. A completed family block which remains in this kernel has a common Fourier basis and is invisible to the mixed torsor curvature. Physical CKM motion therefore requires a Schur-visible component in the nonzero part of (73).

The relative transport of the two principal Gauss-local channels supplies a natural source for the clock degree. In the central-character reading, a spherical component of weight m contributes the character $Z^m$ modulo three on the projective-color torsor. A Schur-visible transition between a component $m_1$ of the $V_1$ charge and a component $m_2$ of the $V_2$ charge therefore carries the central degree $Z^{m_2-m_1}$, possibly multiplied by a shift degree. Hence the transition is non-circulant exactly when $m_2-m_1\not\equiv 0(mod3)$.

In the central-character model of the projective-color torsor, the criterion is read directly from (69): a Schur-visible relative transport component between $V_1$ and $V_2$ contributes to (70) unless the two spherical weights are congruent modulo three.

Finally, the finite trace of the torsor Laplacian fixes the central part of the determinant-shadow count. By (67), the normalized central trace is

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

(74)

Together with (37), this gives the finite coefficient

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

(75)

Thus the determinant-shadow trace uses the flux-independent part of the projective-color finite Hodge-spectral module. The Wilson determinant (64) records the flux obstruction, while (75) records the central trace normalization. The universal analytic normalization of the corresponding determinant-curvature term belongs to the completed boundary-admissible family.

Remark 2

(Structural boundary-cycle package). The primitive Gauss-local support (26), the determinant package of Remark 1, and the projective-color finite Hodge-spectral module (56) define the finite boundary-cycle package. Its structural data are the split carrier group (32), the exterior module (35), the degree readings (36) and (38), the local checks in Table 6 and Table 7, the full finite shadow (46), the torsor carrier (53), the torsor Laplacian (61), the Wilson-Laplacian determinant identities (64) and (65), the torsor cohomology (66), the spectral invariants (67) and characteristic polynomial (68), the family spectral seminorm (73), and the finite determinant trace (75).

The finite structures in this section are therefore structural consequences of the primitive boundary cycle. The carrier, determinant kernel, exterior module, finite shadow, projective-color finite Hodge-spectral module, and torsor trace are fixed before the closed spectral branch is solved. The Alena-Codazzi collar supplies a realization mechanism for the primitive source data and for the continuum vorticity response in Section 4. The finite module records only the corresponding quantized projective-color boundary shadow. The locked low representative, weak bridge, Pfaffian denominator, family splittings, masses, mixing data, and contact coefficients are evaluated later on this boundary cycle through the Schur-Kuranishi completion.

4. Alena-Codazzi Realization of the Primitive Boundary Cycle

The preceding section reconstructed the primitive boundary cycle from the projective link, the two Gauss-local boundary charges, the reduced positive-degree closure, the determinant package, and the projective-color finite Hodge-spectral module. The present section fixes a source-realization layer for these data. Its role is to exhibit a non-empty current-residual Alena-Codazzi class whose singular associated-graded boundary data supply the inputs of Definition 2. The Alena-Codazzi collar supplies a residual scalar, a conserved translational current, a scalar Codazzi multiplier, a compact-leaf source mechanism, the continuum vorticity response, and the two principal boundary charges required by that definition.

The section has two levels. First, under the regular thin-core, two-channel, exact-calibration, and relative Gauss-local hypotheses, the collar realizes the Gauss-local source data to which Theorem 2 applies. Second, under torsor-admissible boundary transport, the transported charge pair realizes the projective-color finite Hodge-spectral module (56); flat torsor closure is the residual condition (59). The completed locked low operator, weak bridge, masses, mixing data, Pfaffian denominator, and contact coefficients enter only after this boundary cycle has been fixed.

4.1. The Alena-Codazzi primitive realization class

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

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

(76)

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 density (76) is the collar residual of the Alena Tensor variational family. The Alena Tensor identification supplies the branch-stress interpretation [5]. The current and vortex terms used here are the residual-collar data of [6]. The continuum, variational, and branch-potential inputs are those of [19,20,21]. Only the current, vorticity-response, and branch-stress slots of this family enter the realization criterion below. At this level only the finite boundary shadow of this vorticity slot is retained by the projective-color finite Hodge-spectral module.

The associated translational current is

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

(77)

The frozen amplitude block is assumed non-degenerate:

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

(78)

This condition is used as a local persistence condition for the scalar collar. The support theorem uses only the primitive link and the boundary-central two-channel source.

The product-rule Hilbert response of (76) is written in the frozen split form

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

(79)

The first term records the response of the residual scalar density. The second term records the branch-response tensor carried by $p_{\Lambda }$. The collar is called split-conserved when

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

(80)

on the source-free punctured collar. This split is a coefficient split. It supplies the current coefficient and the trace-adjusted Codazzi coefficient before the link two-jet is read.

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

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

(81)

The tensor (81) is the coefficient used in the multiplier equation. The factor $1/3$ is the rank-three transverse trace adjustment. The normal trace-free $V_2$ coefficient used later is the associated-graded transverse component of this Codazzi coefficient. A nonzero scalar $\phi$ is a Codazzi multiplier when

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

(82)

satisfies the trace-adjusted Codazzi condition on the punctured collar. With

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

(83)

the multiplier equation is

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

(84)

On a connected non-degenerate open set where $B_{\mu \nu }$ is invertible, with inverse ${\beta }^{\mu \nu }$, the multiplier one-form is fixed by B:

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

(85)

Proposition 10

(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 for (81) if and only if (84) holds with $\theta ={\theta }^B$ and ${\theta }^B$ is exact on Ω. In that case, if ${\theta }^B=d{f}_B$, the scalar reduction is

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

(86)

with one nonzero constant $C_{\phi }$. Together with (77), the remaining compatibility equation is

$$U\left({\mu }_{\zeta }{R}_{\omega }\right)+C_{\phi }{e}^{f_B}{\theta }^B\left(U\right)=\left(1-{\mu }_{\zeta }{R}_{\omega }-C_{\phi }{e}^{f_B}\right)\left({\nabla }_{\mu }^{\left(k\right)}{U}^{\mu }+U(log{p}_{\Lambda })\right).$$

(87)

Proof. 

Substitution of (82) into the trace-adjusted Codazzi condition gives (84). Contracting with ${\beta }^{\mu \nu }$ gives (85). Hence a nonzero local multiplier exists precisely when the full multiplier equation holds with this one-form and the one-form is exact. The scalar reduction follows from (76). Applying (77) to the reduced scalar gives (87). □

Definition 5

(Alena-Codazzi primitive realization class). A current-residual Alena-Codazzi collar belongs to the primitive realization class if the following conditions hold on the source-free punctured collar:

(i)

the residual scalar has the form (76);

(ii)

the translational current satisfies (77);

(iii)

the frozen amplitude block satisfies (78);

(iv)

the split-conserved condition (80) holds;

(v)

the residual scalar is a Codazzi multiplier in the sense of Proposition 10;

(vi)

the thin-core limit contains a primitive positive multiplicity-one regular component;

(vii)

the frozen collar has nonzero Codazzi gap on that component;

(viii)

the associated normal two-jet has nonzero $V_1$ and $V_2$ coefficients.

If, in addition, a boundary-admissible normal representative has an open Callias-Schur gap and subcritical low-high coupling, the collar will be called Schur-admissible.

Conditions (i)-(viii) realize the Gauss-local data of Definition 2. Schur-admissibility is an analytic persistence condition for the later low-sector representative. The realization of (54) requires the torsor-admissible boundary transport of Definition 7.

4.2. Non-empty compact-leaf collars and compatibility with the current-residual multiplier

A local model for the realization class is obtained from a warped compact-leaf collar. Let ${\Sigma }_g$ be a compact hyperbolic surface of genus $g\ge 2$, with metric $\gamma$. On a frozen collar interval, the branch is taken in a warped form with a constant anisotropy parameter $s=tanh\chi \in (0,1)$ on each leaf and with a positive warp factor $a\left(s\right)$. On the source-free part of the collar, the two-eigenvalue trace-adjusted Codazzi equations are imposed through the corresponding warped first integrals; the integration constant entering the gap is denoted by $F_0$. The curvature convention is the standard one used in the warped-product calculation [36].

The normalized vorticity closure in this sector has the scalar leaf form

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

(88)

where $D_o\ne 0$ is the frozen vorticity-flux scale. Since the right-hand side has nonzero mean on a compact leaf, a smooth global solution of (88) is obstructed. A regularized one-core closure is obtained by choosing a non-negative mollifier ${\rho }_{\epsilon }$ on ${\Sigma }_g$ with unit integral and setting

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

(89)

where the coefficient is fixed by the zero-mean condition,

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

(90)

The elliptic solvability of (89) is the standard compact zero-mean Poisson statement [43]. If ${\rho }_{\epsilon }\rightharpoonup {\delta }_p$, the limiting closure is distributional with one leaf source at p. Following this source along the Lorentzian principal plane gives the local worldline core. This is the continuum source mechanism whose finite projective-color boundary shadow is recorded by (56).

In what follows, source-free means singular-source-free in the Gauss-local sense. The smooth zero-mode-compensating background in (89) is treated as part of the frozen compact-leaf geometry. The boundary charges are read from the singular associated-graded part, equivalently after subtracting a smooth local particular solution of the background equation.

The same warped sector has a nonzero two-eigenvalue Codazzi gap when the integration constant $F_0$ is nonzero:

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

(91)

Thus, for $F_0\ne 0$ and $0<s<1$, the frozen compact-leaf model lies in the non-degenerate optical sector used by the primitive link construction. The compact leaf supplies the source mechanism, while the punctured collar supplies the singular-source-free region on which the Codazzi multiplier and the Gauss-local charge readings are applied.

Proposition 11

(Resolved compact-leaf source collar). For every compact hyperbolic leaf ${\Sigma }_g$, $g\ge 2$, and every frozen parameter range with $0<s<1$, $D_o\ne 0$, and $F_0\ne 0$, the regularized equation (89) has smooth solutions ${\alpha }_{\epsilon }$ after the additive constant has been fixed. As ${\rho }_{\epsilon }\rightharpoonup {\delta }_p$, the source term converges to a one-core distributional closure on the compact leaf. Away from the core, after subtraction of the smooth compact-leaf background, the collar is singular-source-free and has nonzero Codazzi gap (91).

Proof. 

The right-hand side of (89) has zero integral by (90). The compact Poisson equation therefore has a smooth solution, unique up to an additive constant. The weak convergence of the mollifier gives the distributional one-core limit. The gap statement is (91). The source-free statement uses the singular-source convention fixed above. □

Within the compact-leaf realization class, smooth frozen-background terms do not change the carrier selected by Theorem 1, provided the singular associated-graded charges $Q_1\in V_1$ and $Q_2\in V_2$ and the nonzero gap condition are unchanged. The support theorem uses the primitive link and the singular boundary-charge sectors; smooth compact-leaf background terms are removed by the subtraction fixed above.

The compact-leaf model gives a non-empty punctured optical collar. It remains to relate an exact Codazzi-calibrated collar of this type to the current-residual scalar form (76). The following proposition gives the local compatibility statement needed for the realization criterion.

Proposition 12

(Compatibility with the Alena current-residual form). Let Ω be a connected source-free punctured collar in the non-degenerate compact-leaf sector. Let $B_{\mu \nu }$ be invertible on Ω. Assume that the collar is Codazzi-calibrated, i.e. there exists a smooth nonzero scalar ${\varphi }_C$ such that $A_{\mu \nu }={\varphi }_C{B}_{\mu \nu }$ satisfies the trace-adjusted Codazzi equation. Assume also that $p_{\Lambda }>0$, that the branch flow field U is smooth on Ω, and that the residual vorticity slot can be represented on the chosen collar by a smooth product $h={\mu }_{\zeta }{R}_{\omega }$ with the required sign. Then, after possibly restricting Ω to a smaller collar along the U-flow, the same calibrated tensor admits a local Alena current-residual representative:

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

(92)

Moreover ${\varphi }_C=C_{\varphi }{e}^{f_B}$ with ${\theta }^B=d{f}_B$, where ${\theta }^B$ is defined by (85).

Proof. 

Since $A_{\mu \nu }={\varphi }_C{B}_{\mu \nu }$ is Codazzi and $B_{\mu \nu }$ is invertible, Proposition 10 gives

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

(93)

Thus ${\theta }^B$ is exact on the connected collar, and ${\varphi }_C=C_{\varphi }{e}^{f_B}$ after the multiplicative constant has been absorbed into $C_{\varphi }$.

Let $y={\zeta }^2$. The current equation in (92) is the transport equation

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

(94)

For positive initial data on a local transverse section, the solution of (94) remains positive along the U-flow. On a sufficiently small collar, $e^{f_B}$ is bounded. The magnitude of $C_{\varphi }$ and the positive initial value of y may therefore be chosen so that $h:=1-C_{\varphi }{e}^{f_B}-y$ has the required sign on the chosen collar. Define

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

(95)

If $R_{\omega }$ is already fixed and has nonzero sign on the collar, the positive factor ${\mu }_{\zeta }$ is obtained by division by $R_{\omega }$ after the sign convention has been fixed. Otherwise (95) defines the residual product $h={\mu }_{\zeta }{R}_{\omega }$ used in the collar reduction. Then (92) holds. Substitution of (95) into (94) gives (87). The calibrated tensor $A_{\mu \nu }$ is unchanged, so the nonzero Codazzi gap of the punctured collar is unchanged. □

Proposition 12 is a local compatibility result inside the exact Codazzi-calibrated class. Stability of the multiplier equation in a full field neighborhood would require an implicit-function theorem for the nonlinear map $(\phi ,B)\mapsto C\left(\phi B\right)$, with the period obstruction for ${\theta }^B$ included. The stability used below is the stability of the primitive, two-channel, and gapped realization class, and is formulated after the thin-core and moment data have been introduced.

4.3. Thin-core realization, moment genericity, and Gauss-local charges

The compact-leaf construction of Proposition 11 gives a singular-source-free punctured collar away from the regularized core. Proposition 12 supplies the local current-residual representative inside the exact calibrated class. A thin-core limit is now used only to produce the worldline component and the primitive link class. The geometric-measure compactness input is the usual compactness of integral currents [62], in the form used in standard geometric-measure theory [63]. The vortex-concentration comparison is the one used in Ginzburg-Landau compactness [64], with the Jacobian-current formulation of [65].

Theorem 3

(Thin-core Alena-Codazzi current-residual origin). Let $\left\{{\mathcal{C}}_{\epsilon }\right\}$ be a family of current-Codazzi closed Alena collars on punctured tubular neighborhoods of regularized cores. Assume:

(i)

the core energy and the concentrated current mass are uniformly bounded;

(ii)

the boundary charge on a linking sphere is controlled and has a nonzero limit;

(iii)

the regularized cores have no interior boundary in the source-free collar;

(iv)

the Codazzi residual is penalty-dominant on the punctured collar;

(v)

the frozen Codazzi gap is nonzero away from the core;

(vi)

the limiting regular component under consideration is primitive and multiplicity one.

Then, after passing to a subsequence, the regularized cores converge as integral 1-currents to an integral 1-current with no interior boundary in the source-free collar. On the primitive multiplicity-one regular component, the limit is represented by a worldline Γ. The real blow-up along Γ gives the link (18), and the controlled primitive boundary charge gives the degree-one line (21).

Proof. 

The uniform mass bound and the boundary control give compactness of the associated integral 1-currents. The absence of interior boundary is preserved in the limit. The penalty-dominant Codazzi residual gives the trace-adjusted Codazzi closure on the complement of the limiting core, while Proposition 10 supplies the scalar multiplier form on the punctured collar. On a multiplicity-one regular component, the limiting current is represented by a smooth worldline. The primitive nonzero boundary charge fixes the degree-one transverse class, and the nonzero frozen gap gives the optical two-eigenvalue splitting used by the link construction. □

The two non-scalar source coefficients are most transparent in a moment-resolved normal model. Let $N_{\Gamma }$ be the oriented transverse normal fiber and let $\omega \in S\left(N_{\Gamma }\right)$. A second-order moment-resolved core has a first moment $m\in N_{\Gamma }$ and a trace-free second moment $M_0\in {Sym}_0^2\left(N_{\Gamma }\right)$. The scalar trace of the second moment belongs to the $V_0$ channel and is not part of the non-scalar support problem.

Definition 6

(Second-order moment-resolved source). A defect source will be called second-order moment-resolved if, in defect-adapted normal coordinates, the associated-graded scalar-sector source up to transverse order two is represented by a normalized core profile with first moment $m\in N_{\Gamma }$ and trace-free second moment $M_0\in {Sym}_0^2\left(N_{\Gamma }\right)$.

The induced non-scalar link map is

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

(96)

The map (96) is the standard $SU\left(2\right)$-equivariant identification of oriented normal vectors with degree-one spherical harmonics and of trace-free quadratic forms with degree-two spherical harmonics. Hence it is an isomorphism, and the two-channel condition $m\ne 0$, $M_0\ne 0$ is open and dense in the finite moment space.

The positivity of a core profile does not obstruct a prescribed trace-free second moment. A sufficiently large scalar trace may be added to the raw second moment to make the covariance positive, and this added trace contributes only to the scalar channel.

The Alena-Codazzi scalar two-jet has the same representation content. On a primitive regular component let ${\nu }_{\Gamma }^{\perp }$ be the degree-one transverse trace on the resolved link.

For a current-Codazzi closed Alena collar in the primitive realization class, Lemma 1 gives the non-scalar associated-graded normal two-jet in the form

$${\left(gr{J}_{\perp }^{\le 2}\right)}_{\mathrm{ns}}=c_1\left(\phi \right){\nu }_{\Gamma }^{\perp }+c_2\left(\phi \right)\left({\nu }_{\Gamma }^{\perp }\otimes {\nu }_{\Gamma }^{\perp }-{\displaystyle \frac{1}{3}}{id}_{N_{\Gamma }}\right).$$

(97)

The first term has type $V_1$, the trace-free quadratic term has type $V_2$, and the locus $c_1\left(\phi \right)c_2\left(\phi \right)\ne 0$ is the separated two-channel locus.

The two components in (97) have a boundary-charge reading. For the $V_2$ channel the principal normal model is the punctured flat fiber $N_{\Gamma }\setminus \left\{0\right\}\simeq {\mathbb{R}}^3\setminus \left\{0\right\}$. Let $A_{ij}$ be the trace-free principal Codazzi representative.

Lemma 6

(Principal Codazzi-Gauss charges). On the principal punctured normal model, the trace-free Codazzi representative satisfies

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

(98)

The singular $\ell =2$ coefficient of u is the surface charge

$$Q_2(M;r)={\int }_{S_r^2}\left(u{\partial }_n{H}_M-H_M{\partial }_{n}u\right)dS,H_M\left(x\right)=M_{ij}{x}^i{x}^j,$$

(99)

with $M\in {Sym}_0^2\left(N_{\Gamma }\right)$. It is independent of r in a source-free annulus. The $V_1$ coefficient is the corresponding first normal moment of the conserved current (77).

Proof. 

For fixed j, put ${\alpha }_j=A_{ij}d{x}^i$. The principal Codazzi condition gives $d{\alpha }_j=0$. Since $H^1({\mathbb{R}}^3\setminus \left\{0\right\})=0$, one has $A_{ij}={\partial }_i{v}_j$. Symmetry gives $v_j={\partial }_{j}u$, and the trace-free condition gives $\Delta u=0$. Green’s identity applied to the annulus between two linking spheres gives the equality of (99) at the two radii. The current statement follows from Stokes’ theorem applied to (77). □

The two charges supplied by Lemma 6 are invariant in a source-free annulus: the $V_2$ charge is a Green pairing on a linking sphere, and the $V_1$ charge is a flux or moment of the conserved current. Since the local algebra is relative to the fixed boundary-charge algebra, Lemma 2 puts their finite-sector spectral projections in the center of the represented local algebra.

The principal flat charge is transported on a regular curved collar by the corresponding Green identity. Let $\mathfrak{C}$ denote the first-order Codazzi operator acting on the trace-free $V_2$ representative, and let ${\mathfrak{C}}^{*}$ be its formal adjoint.

The principal flat charge is transported on a regular curved collar by the Green identity

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

(100)

If $\mathfrak{C}A=0$ and ${\mathfrak{C}}^{*}\Psi =0$ on the source-free collar, the boundary pairing is independent of r. The $V_1$ current charge is transported by (77).

Thus the Alena-Codazzi realization supplies the two charge sectors required in Definition 2: $Q_1$ is the phase-current moment and $Q_2$ is the trace-free Codazzi-Gauss surface charge. The realization of the torsor differential additionally requires a unitary finite transport of this charge pair.

4.4. Torsor-compatible boundary transport

The projective-color torsor complex of Subsection 3.4 requires more than the existence of the two boundary charges. The charge pair must also determine unitary edge transports in (55). This is a finite transport condition on the boundary data, while the source type remains the two-channel type fixed in Definition 2.

Definition 7

(Torsor-compatible boundary transport). A Schur-admissible Alena-Codazzi collar in the primitive realization class will be called torsor-admissible if the following data are fixed on the singular-source-free collar:

(i)

the $V_1$ current moment has a compact unitary phase calibration on the linking spheres;

(ii)

the $V_2$ Codazzi-Gauss charge is read in a transported unitary Green-adjoint frame;

(iii)

the relative transport of the pair $(Q_1,Q_2)$ descends to the projective-color torsor (52);

(iv)

the descended relative transport gives unitary edge transports $U_a$ in (55);

(v)

the edge transport is subcritical with respect to the separating Callias-Schur gap of the chosen normal representative.

It is called torsor-compatible if the Wilson defect (58) is central in the represented boundary-charge algebra. The minimal flat torsor branch is the subcase $R_{\mathrm{tor}}^{\Gamma }=0$.

The first two conditions fix unitary normalizations of the two boundary readings. The third and fourth conditions say that the relative charge transport is a transport on the finite torsor (52). Centrality of (58) is the finite compatibility condition for the structural torsor cycle. Flat closure is the residual condition (59). A non-central Wilson defect is a completed-branch curvature datum.

For a torsor-admissible Alena-Codazzi collar, the transported boundary charge pair induces the projective-color torsor complex (54) on the selected boundary data. The $V_2$ charge is transported by (100), the $V_1$ charge by (77), and the relative transported pair gives the unitary edge transports $U_a$ in (55). Torsor-compatibility makes (58) central; in the minimal flat branch the adjacent response, mixed-curvature detector, and finite torsor trace are (61), (72), and (74).

Thus the compact-leaf Alena-Codazzi mechanism supplies the source side of the boundary cycle in two steps. The primitive realization class supplies the carrier data. Torsor-admissible transport supplies the finite edge transports needed for the projective-color torsor complex, while flat torsor closure is imposed by (59).

4.5. Stability and realization criteria

The stability used in the realization criteria is stability inside the constrained current-Codazzi class. The multiplier equation is a differential constraint. The stable features used here are the primitive transverse degree, the nonzero two-channel moment condition, the nonzero Codazzi gap, the calibrated finite torsor transport when imposed, and the open Callias-Schur spectral gap.

The primitive degree is topological and persists under deformations which do not cross a degeneracy of the resolved transverse frame. The two-channel condition is open by (96). The nonzero gap condition is open on a fixed compact subcollar. The spectral persistence statement used later is the standard isolated-cluster statement of analytic perturbation theory [24], applied to the Callias-Schur low sector. The Callias comparison is the one of [22,23].

The stability used below is the standard one. Under small perturbations inside the constrained current-Codazzi class, the primitive degree remains locally constant, the two-channel condition remains open in the moment space described by (96), and a nonzero frozen gap remains nonzero on a fixed compact subcollar. For a Schur-admissible representative, the Riesz projection around an isolated cluster persists under perturbations smaller than one half of the spectral separation, so its rank is unchanged.

A stronger open-neighborhood statement for the Codazzi multiplier would require solving the nonlinear equation $(\phi ,B)\mapsto C\left(\phi B\right)$ after perturbation, with the exactness or period obstruction for ${\theta }^B$ included. The persistence of torsor-admissible transport is a finite boundary condition on the transported charge pair. Both are separate from the support theorem.

The first local realization criterion combines the compact-leaf model, the current-residual multiplier, the thin-core limit, the moment-generic two-jet, and the relative Gauss-local charge centrality. The inputs used in the criterion have the following status.

Table 8. Status of the inputs in the Alena-Codazzi realization criteria.

Input	Status
Compact-leaf Poisson source	Constructed by Proposition 11
Nonzero frozen Codazzi gap	Constructed in the compact-leaf sector by (91)
Exact Codazzi multiplier	Criterion inside the current-Codazzi class
Thin-core regular component	Regularity hypothesis in Theorem 3
Two-channel moment data	Open dense finite moment condition
Relative Gauss-local algebra	Boundary-charge algebra input; sector separation is supplied by the zero fidelity entry (12)
Torsor-admissible transport	Finite boundary realization condition
Flat torsor closure	Residual condition (59)
Schur-admissible low sector	Optional analytic persistence condition

Table 8. Status of the inputs in the Alena-Codazzi realization criteria.

Input	Status
Compact-leaf Poisson source	Constructed by Proposition 11
Nonzero frozen Codazzi gap	Constructed in the compact-leaf sector by (91)
Exact Codazzi multiplier	Criterion inside the current-Codazzi class
Thin-core regular component	Regularity hypothesis in Theorem 3
Two-channel moment data	Open dense finite moment condition
Relative Gauss-local algebra	Boundary-charge algebra input; sector separation is supplied by the zero fidelity entry (12)
Torsor-admissible transport	Finite boundary realization condition
Flat torsor closure	Residual condition (59)
Schur-admissible low sector	Optional analytic persistence condition

Theorem 4

(Alena-Codazzi realization criterion for primitive Gauss-local data). Let $\left\{{\mathcal{C}}_{\epsilon }\right\}$ be a family of Alena-Codazzi collars satisfying the following conditions:

(i)

the punctured collars are current-Codazzi closed in the sense of Proposition 10;

(ii)

the compact-leaf source mechanism of Proposition 11 supplies a non-degenerate singular-source-free punctured collar;

(iii)

the exact Codazzi-calibrated collar is compatible with the current-residual form as in Proposition 12;

(iv)

the thin-core assumptions of Theorem 3 hold on a primitive multiplicity-one regular component;

(v)

the associated normal two-jet is two-channel generic in the sense of (97);

(vi)

the local observable algebra is relative Gauss-local with fixed boundary data in the sense of Definition 1.

Then the limit supplies a Gauss-local second-jet datum in the sense of Definition 2. Consequently, Theorem 2 applies once the Gauss, fidelity, visibility, and support-complexity entries of (14) are closed on the finite support, and the minimal separated support is the carrier (26). If the collar is Schur-admissible, this support is represented by an isolated low-sector bundle which persists under the stability conditions stated in Subsection 4.5.

Proof. 

Proposition 10 gives the current-Codazzi closure on the punctured collar. Proposition 11 gives a compact-leaf source model with nonzero frozen gap, and Proposition 12 gives its local current-residual representative in the exact calibrated class. Theorem 3 supplies the limiting worldline and the positive transverse class on the regular component. The two-jet form (97) gives the two nonzero associated-graded source types $V_1$ and $V_2$. Lemma 6 and the following Gauss-local centrality statement give their boundary-charge reading and Gauss-local sector centrality. These are the data of Definition 2. The carrier statement follows from Theorem 2 applied to the corresponding closed support entries of (14). The Schur-admissible persistence statement follows from the stability conditions stated in Subsection 4.5. □

The torsor-cycle realization is the calibrated refinement of Theorem 4.

The torsor-cycle realization is the calibrated refinement of Theorem 4. If its hypotheses hold and the limiting primitive collar is Schur-admissible and torsor-admissible in the sense of Definition 7, then the transported charge pair realizes the boundary torsor complex of Remark 2. The minimal flat branch is selected by the residual condition (59).

Thus the compact-leaf construction gives a non-empty source model for the primitive boundary cycle. Under the exact-calibration, thin-core regularity, two-channel, and relative Gauss-local hypotheses, it gives a current-residual realization of the source data used by the support mechanism. Under torsor-admissible boundary transport, it also realizes the projective-color torsor complex. The flat torsor branch is selected by the residual closure (59). The completed locked representative and the quantitative Schur-Kuranishi data are evaluated on this realized boundary cycle in the following sections.

5. Locked Low Cycle and $B-L$-Filtered Schur Completion

The previous sections fixed the primitive carrier, the determinant-compatible exterior package, the projective-color torsor complex, and a compact-leaf Alena-Codazzi realization criterion. This section treats the completed locked low representative on that boundary cycle. The finite carrier and the torsor complex determine the support, the determinant direction, the central response carrier, and the allowed exterior channels. A reduced primitive normal representative removes charge-invisible low multiplicities before the projective-color factor is attached. The completed Schur operator supplies the actual low matrix elements.

The guiding mechanism is a $B-L$-filtered two-stage Schur reconstruction. Hypercharge Y fixes the determinant direction in the selected package, while $B-L$ filters the finite Schur-Kuranishi channels. The weak bridge is $B-L$ preserving and gives the direct Dirac/Yukawa blocks. The singlet Majorana channel has $|\Delta (B-L\left)\right|=2$ and supplies the denominator for the neutral Schur complement. CKM therefore tests Schur-visible non-circulant motion in the direct Dirac sector, whereas PMNS tests the inverse neutral Majorana denominator.

5.1. Completed low operator and locked central factorization

Let $P_q$ denote the Riesz projection onto the isolated positive link block $E_q$ in a boundary-admissible normal representative. Along a smooth branch of resolved links, the projected connection is

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

(101)

This is the finite Berry-Wilczek-Zee connection on the moving zero-mode bundle [25,26]. In the rank-five support (26), the projected connection takes values in $u\left(C\right)\oplus u\left(W\right)$. After the split top-form condition (31), its determinant-preserving part takes values in $s\left(u\right(3)\oplus u(2\left)\right)$. The standard perturbation control of the isolated projection is the one of [24], and the Callias-type comparison for the normal representative is the one of [22,23].

The fermionic low operator is obtained after locking branch chirality with exterior parity. If ${\gamma }_{\mathrm{br}}^5$ denotes branch chirality and ${\Gamma }_V={(-1)}^{N_C+N_W}$ denotes exterior parity on ${\Lambda }^{•}V$, the locked involution is

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

(102)

Let $P_{\pm }$ denote the corresponding locked projections. A completed locked low operator has the finite odd form

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

(103)

Here $D_E$ is the projected even-package Dirac part, $c(\Phi +{\Phi }^{†})$ is the exterior-odd weak bridge, $Q_{\mathrm{hol}}$ records holomorphic link transport, and $Q_C$ records the completed Codazzi Schur response. The entries of (103) are completed-branch data on the support fixed in Section 3.

The even exterior package (35) is the low locked block. The ${\mathbb{Z}}_2$ projection of the finite shadow (46) is used at this level as the weak parity component of the lock (102). After this parity lock, the residual projective-color factor is the central carrier (53). A complementary parity block may occur in the completed normal representative only if it remains outside the isolated Riesz window. Thus mirror separation is an analytic gap condition on the completed representative.

The high sector enters through the usual Feshbach-Schur compression. For a local splitting $\mathcal{H}={\mathcal{H}}_L\oplus {\mathcal{H}}_H$ of the gauge-fixed normal family, with $K_{HH}$ invertible on the high window, the first effective low operator is

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

(104)

The remainder $R_{\ge 3}$ records higher Schur terms of the completed branch. Charge-compatible completions are those for which (104) preserves the boundary-sector projections supplied by Lemma 2.

The higher integer source moments have a controlled status in this compression. Let P denote the selected low projection and let $Q=1-P$. If $T_{\ell }$ is a principal integer moment of type $V_{\ell }$ with $\ell \ge 3$, then Proposition 4 gives $P{T}_{\ell }P=0$ on the primitive support. Hence such a moment cannot reselect the carrier and its first low-sector contribution is Schur-mediated.

Proposition 13

(Schur suppression of higher integer moments). Let $T_{\ell }$ be a charge-compatible principal integer moment with $\ell \ge 3$, acting between the selected low window and its complement. Let $g_{\mathrm{CSch}}={\parallel K_{HH}^{-1}\parallel }^{-1}$ be the high-window Schur gap. If $\parallel T_{\ell }\parallel \ll g_{\mathrm{CSch}}$, then the leading induced low operator satisfies

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

(105)

Proof. 

The direct low-low compression vanishes by Proposition 4. Expanding the Feshbach-Schur term in (104) gives the displayed second-order term. The norm estimate follows from $\parallel K_{HH}^{-1}\parallel =g_{\mathrm{CSch}}^{-1}$. □

Under the hypotheses of Proposition 13, principal integer moments with $\ell \ge 3$ do not reselect the primitive support, the determinant carrier, the exterior package, or the finite shadow. Their first contribution to the completed low dynamics is the Schur-mediated term estimated in (105).

The projective-color carrier becomes a physical low-sector factor only after the locked normal representative has been isolated by the Callias-Schur gap. This is the low spectral isolation condition. Let $P_{\mathrm{prim}}$ be the primitive locked low projection before the projective-color central response is turned on, and let ${\Delta }_{\mathrm{CSch}}$ be its separation from the complementary spectrum. Assume that, after the ${\mathbb{Z}}_2$ part of (46) has been locked by (102), the unperturbed central low window is $P_{\mathrm{prim}}\otimes {\mathcal{H}}_{\mathrm{cen}}$.

Proposition 14

(Locked central low-sector factorization). Let the unperturbed locked normal representative have the isolated low window $P_{\mathrm{prim}}\otimes {\mathcal{H}}_{\mathrm{cen}}$ with gap ${\Delta }_{\mathrm{CSch}}$. Let $R_{\mathrm{cen}}$ be a charge-compatible central and low-high Schur perturbation. If $\parallel R_{\mathrm{cen}}\parallel <{\Delta }_{\mathrm{CSch}}/2$, then the Riesz projection of the completed representative has rank

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

(106)

The corresponding low bundle is identified with the unperturbed product by spectral transport. The effective operator on this bundle may contain central Schur terms.

Proof. 

Let $\mathcal{C}$ be a contour enclosing the unperturbed product low spectrum and separated from the complementary spectrum by ${\Delta }_{\mathrm{CSch}}$. The norm bound on $R_{\mathrm{cen}}$ keeps the perturbed spectrum inside the same contour. The Riesz projection defined by $\mathcal{C}$ therefore persists and has the same rank. The identification of the bundles follows by spectral transport along the perturbation parameter. □

Definition 8

(Reduced primitive normal representative). A primitive locked normal representative is reduced if, after the boundary-charge sector and exterior channel have been fixed, the pre-torsor Riesz window contains no multiplicity factor invisible to the fixed boundary-charge and exterior data. Equivalently, the finite endomorphism algebra of the remaining primitive multiplicity factor is scalar.

For a reduced primitive normal representative, the finite endomorphism algebra of the remaining primitive multiplicity factor is scalar. Hence the primitive locked low projection before the projective-color central response is turned on has

$$rank{P}_{\mathrm{prim}}=1.$$

(107)

Corollary 2

(Weak-parity rank obstruction). Assume that the projective-color factor (53) and both weak-parity modes of (49) remain in the same isolated Riesz window. Then the central multiplicity is six. Hence the rank-three low cluster requires a fixed weak-parity component or a parity gap separating the complementary mode. In the flat parity normalization, this complementary gap is $4{\nu }_2$ by Lemma 5.

Proof. 

The full finite carrier factorizes as in (49). The projective-color factor has dimension three and the parity factor has dimension two. If both parity modes are retained in the same low window, the central factor has dimension six. The flat-gap value is the nonzero eigenvalue in (51). □

Corollary 3

(Rank-three torsor low cluster). For a reduced primitive normal representative satisfying the hypotheses of Proposition 14 and the weak-parity isolation of Corollary 2, the completed projective-color low window has rank three:

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

(108)

Proof. 

By (53), $dim{\mathcal{H}}_{\mathrm{cen}}=3$. The claim follows from (106) and (107). □

The statement is a rank and bundle statement. The completed low operator need not split as a tensor sum on the transported product window. Determinant-line and regular-family readings of the moving low bundle belong to the Bismut-Freed comparison class [27].

5.2. Weak bridge and lock tangents

The weak bridge is read from the Clifford-odd tangents to the chirality/exterior lock on the exterior package (35). A first-order deformation $\delta \Gamma$ of (102) is tangent to the space of involutions when $\{{\Gamma }_{\mathrm{lock}},\delta \Gamma \}=0$. Clifford multiplication by $V\oplus V^{*}$ is exterior-odd, hence gives such tangents. The direct weak bridge is the part of this odd tangent which is color-singlet and $B-L$ preserving.

In sector notation, the finite Yukawa blocks are compressed matrix elements of the zero-order part of (103):

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

(109)

The exterior package fixes the possible pairs $(A,B)$; the entries in (109) are completed-branch data. In a local orthonormal low-mode frame, (101) 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 one induced by (31).

After the central factor (53) has been isolated, each direct Hermitian family block $H_f=Y_f{Y}_f^{†}$ has a Schur-visible first family-motion part $H_f^{\left(1\right)}$. Its finite Hodge-Schur bookkeeping is

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

(110)

Here $H_f^{\mathrm{circ}}$ is the circulant 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 (70). Thus only ${\Gamma }_f$ is tested by the mixed torsor curvature (72) at first order.

Thus the carrier (26), the exterior package (35), and the locked central factorization of Proposition 14 determine the gauge representation, the finite central response carrier, and the allowed weak-bridge Dirac channels. The numerical singular values of the Yukawa blocks (109) are completed-branch matrix elements.

Table 9, with the degree readings (36) and (38), shows that the unique color-singlet $B-L$ preserving linear Clifford-odd lock tangent on the selected carrier is $W\oplus W^{*}$. Its two hypercharge readings are $\pm {\displaystyle \frac{1}{2}}$.

5.3. $B-L$ filtration and determinant-tangent status

The degree readings (36) and (38) play different roles. The hypercharge degree is the normalized generator of the determinant kernel in Proposition 6. The $B-L$ degree is used here as a channel filtration on the same anomaly-free Abelian plane. Since the weak bridge changes only the weak exterior degree, it preserves $B-L$. The one-Higgs Dirac channels, including the quark, charged-lepton, and neutral Dirac neutrino channels, are therefore $\Delta (B-L)=0$.

The singlet Majorana channel has $|\Delta (B-L\left)\right|=2$. The active Majorana channel has the same $B-L$ status and enters as a two-weak-factor effective class. A baryon contact class, when it is present, belongs to the doubled top-form sector and is $B-L$ preserving. These channels occupy different finite Schur-Kuranishi blocks on the same carrier. This separation is the finite analogue of the usual distinction between Dirac, Majorana, and baryon-contact operators in unified exterior descriptions [16]. Algebraic comparison models with Clifford ideals give related finite-channel decompositions [17].

The channel separation used below is the one recorded in Table 10. On the determinant-compatible package (35), weak-bridge Dirac blocks preserve $B-L$, singlet and active Majorana blocks have $|\Delta (B-L\left)\right|=2$, and the doubled top-form baryon contact class is separated from the weak bridge by determinant bidegree and from the Majorana channels by its $B-L$ status.

The scalar trace separated before (22) reappears only in the scalar part of the completed branch. By Schur’s lemma, its compression to the selected support is block-scalar, of the form $\alpha P_C+\beta P_W$. It has no clock degree and does not change the Toeplitz support. The determinant-tangent scalar part is therefore one-dimensional at first order: tangency to the infinitesimal form of (31) imposes $3\alpha +2\beta =0$. Thus the first determinant-tangent scalar correction may be written

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

(111)

The primitive scalar unit used in the minimal branch is

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

(112)

The value $S_{\mathrm{prim}}=4{\pi }^2$ is the Bohr-Sommerfeld closed-cell normalization used for the primitive link phase cell. The resolved ${\mathbb{CP}}_{\Gamma }^1$ 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 (35). The finite central trace part of the corresponding projective-color reading is the torsor coefficient (75).

In the minimal determinant-tangent branch, (111) leaves one scalar u. With the primitive normalization (112), 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 (75).

Remark 3

(Normalization status of the primitive unit). The top-form constraint and the selected carrier fix the one-dimensional direction (111). The absolute normalization in (112) depends on the high-block normalization in the Schur compression (104). In particular, replacing $K_{HH}$ by $c{K}_{HH}$ rescales the second term in (104) by $c^{-1}$. Thus (112) is the minimal-branch normalization of the determinant-tangent unit.

5.4. Charged-lepton torsor shape and balance

The charged-lepton block is the color-singlet direct Dirac block $L_L\leftrightarrow e_L^c$ in Table 10. It is separated from the quark CKM sector, which is a relative-basis reading of two direct Dirac blocks, and from the neutral lepton sector, where the light mass matrix is read through the Majorana Schur complement. Hence the scale-free shape diagnostic below is used only in the charged-lepton direct block.

Let $x_{\ell }$ denote the positive charged-lepton amplitude vector on the three-sector torsor carrier,

$$x_{\ell ,r}=\sqrt{m_{\ell ,r}},r\in {\mathbb{Z}}_3.$$

(113)

For the flat projective-color torsor, the family singlet is the harmonic vertex mode of (61). The real regular torsor representation therefore decomposes as this harmonic line and its real two-dimensional Hodge complement. Thus $P_{\mathbf{1}}$ is the harmonic projection and $P_{\mathrm{T}}=1-P_{\mathbf{1}}$ is the positive Hodge-complement projection. 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

$$x_{\ell ,r}=C_{\ell }\left(a_{\ell }+cos\left({\varphi }_{\ell }+{\displaystyle \frac{2\pi r}{3}}\right)\right),r\in {\mathbb{Z}}_3.$$

(114)

This is the real coordinate form of the decomposition of the vertex carrier, not a mass prediction.

The scale-free charged-lepton balance residual is defined by

$$R_{\mathrm{bal}}^{\ell }=log{\displaystyle \frac{\parallel P_{\mathbf{1}}{x}_{\ell }{\parallel }^2}{\parallel P_{\mathrm{T}}{x}_{\ell }{\parallel }^2}}-{\Sigma }_{\ell }.$$

(115)

The scalar ${\Sigma }_{\ell }$ records the finite Schur correction to the Hodge-norm balance between the harmonic line and the torsor complement. The zero-correction case ${\Sigma }_{\ell }=0$ is the leading balance reading. In a completed charged-lepton branch, ${\Sigma }_{\ell }$ would be supplied by the finite Schur tensor of the direct Dirac block.

Lemma 7

(Charged-lepton balance form). For the torsor coordinate form (114), the residual (115) closes if and only if

$$2a_{\ell }^2=e^{{\Sigma }_{\ell }}.$$

(116)

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 (115) gives (116). □

The corresponding charged-lepton Koide quotient is

$$Q_{\ell }={\displaystyle \frac{{\sum }_{r\in {\mathbb{Z}}_3}{m}_{\ell ,r}}{{\left({\sum }_{r\in {\mathbb{Z}}_3}\sqrt{m_{\ell ,r}}\right)}^2}}={\displaystyle \frac{1+e^{-{\Sigma }_{\ell }}}{3}}.$$

(117)

Thus the Koide value $Q_{\ell }=2/3$ [33] is the zero-correction Hodge-balance limit of (115). A nonzero observed balance defect is interpreted here as a target value for the charged-lepton Schur correction, not as a new carrier-selection condition. Near the zero-correction branch, writing $a_{\ell }=1/\sqrt{2}+\delta a_{\ell }$ gives ${\Sigma }_{\ell }=2\sqrt{2}\delta a_{\ell }+O\left(\delta a_{\ell }^2\right)$ from (116). Hence the logarithmic balance defect and the displacement of the torsor-shape parameter are two coordinates on the same residual.

A completed charged-lepton Schur tensor would have to supply

$${\Sigma }_{\ell }={Tr}_{\mathrm{bal}}\left({\mathcal{S}}_{\ell }\right),{\varphi }_{\ell }={Hol}_{\mathrm{clock}}\left({\mathcal{S}}_{\ell }\right),$$

(118)

after the direct charged-lepton block has been isolated. The first component in (118) fixes the balance defect in (116). The second component would be a clock residual for the torsor angle. The balance residual alone does not determine ${\varphi }_{\ell }$ or the overall scale $C_{\ell }$.

5.5. Direct Dirac family sectors and CKM

The projective-color torsor supplies the central response carrier (53). Its finite differential and mixed-curvature detector were constructed in Subsection 3.4. The role of the present subsection is to apply that detector to the $B-L$ preserving direct Dirac blocks.

A leading family response contained in the circulant algebra $\mathbb{C}\left[S\right]$ is diagonalized by the finite Fourier basis. If $Y_u{Y}_u^{†},Y_d{Y}_d^{†}\in \mathbb{C}\left[S\right]$ and both have simple spectra, then the discrete Fourier basis diagonalizes both blocks, up to phases and ordering. Hence a common circulant family baseline gives central sectors but no physical quark mixing.

The nontrivial CKM matrix must therefore come from a non-circulant component of the completed low operator. In the Schur formulation, this component is visible when a central-degree detector appears in the low-high block which moves the isolated projection. For a branch parameter X, the detector has the form

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

(119)

The inverse Sylvester map transfers such a component to the motion of the low projection and hence to the finite Berry-Wilczek-Zee connection [26]. For two branch directions $X,Y$, the curvature of the projected low bundle is read in the standard projected form

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

(120)

Thus a Schur-visible detector contributes to family mixing through the induced motion of the isolated low projection. By Proposition 9, the finite test for this central part is the non-circulant projection (70). Relative $V_1/V_2$ charge transport gives the corresponding source when the transported weights are noncongruent modulo three.

A CP-sensitive central invariant is obtained after the two direct Dirac sectors have been compared. Let ${\Gamma }_u$ and ${\Gamma }_d$ 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

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

(121)

It vanishes when the two corrections are contained in a common commuting family algebra. The normalization relating (121) to the physical quark invariant is a completed Schur-Berry datum, with the standard comparison convention given by [31].

A minimal Hermitian non-circulant detector pair is represented by the clock-sine and shifted edge operators. The completed Schur-Berry edge phase is denoted by ${\vartheta }_{\Gamma }$; in a one-phase lift it is identified with the torsor edge phase of (55), subject to the finite-shadow coherence (50). Put

$$K_Z={\displaystyle \frac{Z-Z^{†}}{2i}},H_S\left({\vartheta }_{\Gamma }\right)=e^{-i{\vartheta }_{\Gamma }}S+e^{i{\vartheta }_{\Gamma }}{S}^{†}.$$

(122)

Their finite central loop has the invariant

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

(123)

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

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

(124)

Here $3{\vartheta }_{\Gamma }$ is the rephasing-invariant CKM phase in the one-phase reduction. The condition fixes the completed loop phase after the three CKM magnitudes and the quark Jarlskog invariant have been specified. It does not modify the carrier, the torsor complex, or the non-circulant detector.

A Schur-visible central detector contained in $\mathbb{C}\left[S\right]$ is simultaneously diagonalized with the circulant family baseline and does not generate a nonabelian family curvature; equivalently, (70) vanishes in that subcase. A detector pair containing the two degrees in (122) has nonzero commutator for generic ${\vartheta }_{\Gamma }$ and gives the central loop invariant (123). Relative $V_1/V_2$ charge transport supplies such a clock degree when the relevant weights are not congruent modulo three.

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

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

(125)

Thus

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

(126)

The standard CKM conventions are those of [28,29,30]. The CP-sensitive normalization is the one associated with (121).

Proposition 15

(Direct Dirac CKM mechanism). In the $B-L$ preserving direct Dirac sector, CKM mixing is generated only by the relative non-circulant Schur-visible corrections to the up and down blocks. To first order it is given by (126), and the finite test for the relevant correction is (72).

Proof. 

The leading circulant blocks have a common Fourier basis by simultaneous diagonalization in $\mathbb{C}\left[S\right]$. A circulant correction remains diagonal in that basis and gives no off-diagonal term in (125). Hence only the non-circulant part of the Schur-visible correction contributes to the first basis motion. Proposition 9 identifies this part with the mixed torsor curvature norm (72). Taking the relative basis change between the down and up sectors gives (126). □

Corollary 4

(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

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

(127)

Proof. 

Equation (125) gives the first-order basis motion with denominators bounded below by ${\delta }_f$. Taking the Frobenius norm in (126) gives the corresponding sum of up and down contributions. The identity (72) gives $\parallel {\Pi }_{\mathrm{noncirc}}{\Gamma }_f{\parallel }_{\mathrm{F}}=3^{-1/2}{\parallel [d_{\theta },{\Gamma }_f]\parallel }_{\mathrm{F}}$ for the flat torsor differential (60). The second-order remainder is controlled by the same simple-spectrum gaps. Substitution gives (127). □

Definition 9

(Minimal non-circulant Hodge-Schur direct branch). After the carrier, finite shadow, projective-color torsor, and $B-L$ filtration have been fixed, a direct quark branch is called minimal non-circulant Hodge-Schur if the common circulant baseline is removed as in (110), the branch is non-flat in the seminorm (73), and the relative up-down correction minimizes the residual

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

(128)

among nonzero Schur-visible direct corrections with nontrivial relative $V_1/V_2$ clock degree modulo three. Here ${\delta }_f$ is the direct-sector gap used in Corollary 4. 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 3 have been fixed. It selects the first non-flat direct Dirac representative on the projective-color boundary cycle. In this representative, the CKM displacement is measured by (128), while common circulant response remains in the kernel of (73).

The determinant-shadow scale used in Subsection 6.1 may be read as the first path unit of this minimal direct-branch diagnostic. In that diagnostic, the Cabibbo entry is the primitive path, while higher quark mixings and the loop CP quantity are assigned to longer central paths. This assignment belongs to the completed minimal branch. A change of the torsor-path normalization changes these readings without changing the carrier, the torsor complex, or the detector (72).

5.6. Neutral Schur complement and Pfaffian orientation

The neutral sector is a second Schur problem. Let $D_{\nu }$ be the neutral Dirac bridge, let $R_N$ be the heavy singlet family denominator, let $M_R$ be its scale, and let $A_L$ denote the active Majorana correction. In the neutral basis $({\nu }_L,{\nu }_R^c)$, the finite neutral block is

$$K_N=\left(\begin{array}{cc}{A}_L& D_{\nu }^T\\ D_{\nu }& M_R{R}_N\end{array}\right).$$

(129)

The Majorana scale in the minimal Pfaffian branch is assigned to the $B-L$ breaking determinant line. Its half-flux action is

$$S_M={\displaystyle \frac{1}{4}}{S}_{\mathrm{prim}},M_R=M_{\mathrm{Pl}}{e}^{-S_M}.$$

(130)

This is a Pfaffian square-root reading of the neutral determinant class and is a conditional branch assumption beyond the carrier theorem. The Pfaffian branch is a Pfaffian structure of the neutral family operator, when such a structure exists. A holomorphic square-root reading of $L_{\Gamma }$ is excluded: a line M with $M^{\otimes 2}\simeq L_{\Gamma }$ would imply $2c_1\left(M\right)=1$ in $H^2({\mathbb{CP}}^1,\mathbb{Z})$, which is impossible.

The neutral Pfaffian condition may be stated directly at the level of the completed family block. Let $J_N$ be the antiunitary neutral real structure on the finite neutral family space. The Pfaffian branch is the subcase in which

$${\mathcal{A}}_N=J_N{K}_N,{\mathcal{A}}_N^T=-{\mathcal{A}}_N,det{K}_N=Pf{\left({\mathcal{A}}_N\right)}^2.$$

(131)

Thus the half-flux reading in (130) is tied to a skew neutral family form, not to a square root of the primitive link line. The seesaw comparison is the usual heavy-denominator comparison [66]. If $R_N$ is invertible, the light active block is the Schur complement

$$K_{\nu }^{\mathrm{eff}}=A_L-D_{\nu }^T{\left(M_R{R}_N\right)}^{-1}{D}_{\nu }.$$

(132)

The charged-lepton direct block determines a unitary basis $U_e$, while the effective neutral block (132) determines a unitary basis $U_{\nu }$. The lepton mixing matrix is the relative basis

$$U_{\mathrm{PMNS}}=U_e^{†}{U}_{\nu }.$$

(133)

Thus PMNS is read from the comparison between a $B-L$ preserving direct Dirac basis and a $B-L$ breaking neutral Schur basis.

If the singlet Majorana denominator $R_N$ is invertible on the heavy neutral block, the active neutral operator is the Schur complement (132). Its family basis is determined by the charged-lepton direct Dirac basis and by the diagonalization of $K_{\nu }^{\mathrm{eff}}$.

In the minimal $B-L$ filtered branch, large PMNS angles may be denominator-driven: the direct Dirac motion is controlled by (125), whereas the neutral operator contains $R_N^{-1}$ through (132). Soft gaps in the neutral Majorana shape can therefore amplify neutral-family directions before diagonalization of $K_{\nu }^{\mathrm{eff}}$.

A useful diagnostic denominator on the central carrier is the circulant Majorana shape

$$R_N^{\mathrm{circ}}=m\mathbf{1}-r(S+S^{†}).$$

(134)

Its eigenvalues are $m-2r,m+r,m+r$. If $m=2r+\delta$ with $0<\delta \ll r$, then

$${\left(R_N^{\mathrm{circ}}\right)}^{-1}={\displaystyle \frac{1}{3r+\delta }}\mathbf{1}+{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{1}{\delta }}-{\displaystyle \frac{1}{3r+\delta }}\right)J,J_{ab}=1.$$

(135)

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 and the denominator changes scales without producing a relative PMNS basis. Large PMNS angles in this branch therefore 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 (134). 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

$${\displaystyle \frac{1}{\sqrt{3}}}(1,1,1),{\displaystyle \frac{1}{\sqrt{6}}}(2,-1,-1),{\displaystyle \frac{1}{\sqrt{2}}}(0,1,-1).$$

(136)

Thus the soft circulant denominator with a clock-even perturbation gives a tri-bimaximal-type leading neutral basis. Nonzero ${\theta }_{13}$ and the observed departures from this leading pattern belong to higher Schur corrections. The real clock-even perturbation also preserves the local Pfaffian orientation as long as the neutral denominator remains invertible.

5.7. Contact classes and finite Schur-Kuranishi equation

The baryon contact sector is separated from the weak bridge and from the neutral Majorana denominator. The weak bridge changes only weak exterior degree and preserves $B-L$. The singlet Majorana and active Majorana classes have $|\Delta (B-L\left)\right|=2$. The doubled top-form contact has bidegree $(6,4)=2(3,2)$ and is a determinant-sector contact class. Its coefficient is therefore a completed Schur-Kuranishi coefficient, not a consequence of the one-Higgs bridge.

In the minimal weak-bridge layer, the doubled top-form baryon contact coefficient is not generated by the finite algebra generated by the one-Higgs weak bridge. The weak bridge changes only the weak exterior degree and preserves the color degree, while the doubled top-form entry in Table 10 is a separate determinant-sector contact class in the finite Schur-Kuranishi equation.

A completed branch is described by a finite Schur-Kuranishi equation on the selected boundary cycle. Let b denote the finite branch coordinates after gauge, diffeomorphism, and unitary redundancies have been removed. The finite residual map has the form

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

(137)

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.

Definition 10

(Schur-phantom channel). Let $b=b_{\mathrm{red}}\oplus x$ be a local split of the finite branch coordinates in (137), after gauge, diffeomorphism, and unitary redundancies have been removed. The finite coordinate channel x is called Schur-phantom at b if ${\mathcal{F}}_{\mathrm{fin}}(b_{\mathrm{red}}\oplus x)={\mathcal{F}}_{\mathrm{fin}}\left(b_{\mathrm{red}}\right)$ to the retained Schur order, and if the deletion $x\mapsto 0$ preserves the boundary-sector projections, the $B-L$ degree, exterior bidegree, finite-shadow and torsor labels, and the rank and isolation of the Riesz low window. The channel is removable if this deletion strictly lowers a monotone finite Schur-coordinate complexity ${\mathcal{C}}_{\mathrm{Sch}}$.

Proposition 16

(Schur no-phantom reduction). Let b be a local zero of (137) in a fixed reconstruction class, after the entries preceding ${\mathfrak{E}}_{\mathrm{Schur}}$ in (16) have been closed. Let the reduced Schur-Kuranishi representative be chosen by minimizing ${\mathcal{C}}_{\mathrm{Sch}}$ on this constrained zero set. If x is a removable Schur-phantom channel in the sense of Definition 10, then $x=0$ in the reduced representative.

Proof. 

By Definition 10, the deletion $x\mapsto 0$ keeps the finite residual (137), the previously closed sector labels, and the Riesz low-window data unchanged to the retained Schur order. Hence $b_{\mathrm{red}}$ belongs to the same constrained zero set as b. Since the deletion strictly lowers ${\mathcal{C}}_{\mathrm{Sch}}$, a representative with $x\ne 0$ cannot be reduced. Thus $x=0$ in the reduced Schur-Kuranishi representative. □

Consequently, a completed numerical correction belongs to the reduced branch only if it is visible in $P_{\mathrm{obs}}$, protected by the $B-L$ or bidegree filtration, carried by the finite torsor labels, or forced by the gap-mediated Schur compression (104). The higher-moment terms of Proposition 13 are of the last type.

The completed branch is organized by the $B-L$-filtered Schur-Kuranishi map (137). For a Schur-admissible branch on the boundary cycle of Remark 2, the projected gauge fields are obtained from (101); the central response carrier is (53); a reduced primitive normal representative gives the rank-three low window (108); Dirac/Yukawa blocks are matrix elements of (103); the weak bridge is the color-singlet $B-L$ preserving tangent listed in Table 9; CKM tests Schur-visible non-circulant motion in the direct Dirac sector with detector (72) and bound (127); light neutrino masses are read from (132); and baryon contact coefficients belong to the separated doubled top-form channel recorded in Table 10.

This summary identifies the structural status of the completed layer. The finite carrier, determinant package, torsor complex, weak-bridge uniqueness, and channel filtration are structural once the boundary cycle has been fixed. The rank-three family reading follows for a reduced primitive normal representative under the hypotheses of Proposition 14. The determinant-shadow seed is a minimal-branch normalization. CKM tests Schur-visible non-circulant motion in direct Dirac blocks, with detector (72). PMNS and neutrino masses test the inverse $B-L$ breaking Majorana denominator. Contact coefficients test a separate doubled top-form block, with the one-bridge exclusion recorded after Table 10. Numerical reference values and decay comparison formulae are deferred to the quantitative branch tests of Section 6; the particle-data inputs are read from [32], while standard Higgs decay formulae and loop form factors are used only as comparison readings [67].

6. Quantitative Branch Tests and Minimal-Branch Benchmarks

The preceding section fixed the structural status of the completed low layer. The finite carrier, determinant package, full finite shadow, torsor complex, weak-bridge uniqueness, and $B-L$ filtration are structural once the primitive boundary cycle has been fixed. The reduced primitive normal representative fixes the primitive pre-torsor multiplicity before the projective-color factor is attached. The rank-three low-cluster reading follows under the hypotheses of Proposition 14 and the weak-parity isolation of Corollary 2. The entries in this section have a different status. They are zero-remainder readings, first-Schur readings, scale-free shape diagnostics, or falsification tests of a minimal completed branch. They are not used to prove the carrier theorem. Numerical reference values are taken from the particle-data compilation [32].

The minimal branch used below is the reduced completed branch in the sense of Proposition 16. Its numerical benchmarks still require four completed-branch readings: the primitive determinant-tangent normalization (112), the flat torsor trace reading (75), the minimal central path assignment for the first CKM magnitudes, and the Pfaffian half-flux reading (130). Changing any of these completed-branch readings may change the numerical table without changing the structural carrier. A change which can be realized only by a removable Schur channel is not a different reduced branch.

6.1. Neutral determinant-shadow cell

The neutral-cell input is the determinant-tangent scalar step (111) with the primitive unit (112). The leading neutral-cell weights are fixed by the selected low carrier. The weak-angular reservoir is $C\oplus W\oplus W^{*}$, while the radial scalar reservoir is the rank-five carrier $C\oplus W$. Hence $dimC+2dimW=7$ and $dimC+dimW=5$. With $dimC=3$ and $dimW=2$, the leading entries are

$$A_0={\displaystyle \frac{3}{7}},B_0={\displaystyle \frac{6}{49}},C_0={\displaystyle \frac{9}{35}}.$$

(138)

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

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

(139)

The determinant-scale contribution from the same rank split is

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

(140)

The factor in (140) is read as the usual one-loop unit with the projective-color suppression $1/dim{\left(C\right)}^2$ supplied by the selected rank split. The scale-free determinant-shadow angle is

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

(141)

With (112), this gives ${sin}^2{\theta }_{\mathrm{link}}=0.223184071$. The corresponding on-shell comparison is $1-M_W^2/M_Z^2$. With the PDG averages used here, this gives $0.22320949$, so the finite-link value differs by $-2.54\times {10}^{-5}$, about $0.10$ propagated standard deviations. The scheme-dependent effective leptonic weak angle is a separate observable.

The same finite Schur block has a primitive determinant-scale reading. With the non-reduced Planck normalization used in the determinant branch and the rank-split contribution (140), the zero-remainder scale is

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

(142)

This gives $v_{\mathrm{EW}}^{\mathrm{prim}}=246.2205\mathrm{GeV}$. The Fermi value $v_F={\left(\sqrt{2}{G}_F\right)}^{-1/2}$ is $246.2196719\mathrm{GeV}$. The difference is $8.3\times {10}^{-4}\mathrm{GeV}$, equivalently $3.4$ ppm. At the same zero-remainder scale, the neutral-cell mass reading is

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

(143)

The associated neutral-cell readings are listed in Table 11.

6.2. Central family seed and CKM path benchmarks

The benchmark in this subsection is the numerical reading of Definition 9. Assume that the projective-color low carrier is the reduced three-sector torsor carrier and that the leading direct Dirac family blocks have a common circulant baseline. If the primitive non-circulant Schur channels have valuations $w\left(V_1\right)=1$ and $w\left(V_2\right)=2$, and reduced closure excludes an independent direct $1\leftrightarrow 3$ edge of valuation 1 or 2, then the minimal connected Schur-torsor graph has

$$v_{12}=1,v_{23}=2,v_{13}=3.$$

(144)

With direct-sector denominators bounded away from zero in (125), and without an imposed up-down cancellation, the corresponding quark mixing orders are

$$s_{12}=O\left(ϵ\right),s_{23}=O\left({ϵ}^2\right),s_{13}=O\left({ϵ}^3\right),J_q=O\left({ϵ}^6\right)$$

(145)

for a central loop phase of order one. The last order is the usual product order of the three small CKM angles with the central phase supplied by (123).

The determinant shadow may also be read in the projective-color central sector. In the minimal flat torsor branch, the finite trace coefficient is (75). Thus the central Schur-Berry scaling is

$${\kappa }_{\mathrm{SB}}=1+{tr}_{\mathrm{cen}}\left({\Delta }_{\theta }\right){Tr}_{{\Lambda }^{\mathrm{even}}V}\left(Y^2\right)u_{\mathrm{prim}}=1+{\displaystyle \frac{20}{3}}{u}_{\mathrm{prim}},{\lambda }_{\mathrm{cen}}={\displaystyle \frac{{\kappa }_{\mathrm{SB}}B}{A+{\kappa }_{\mathrm{SB}}B}}.$$

(146)

The factor $20/3$ is the finite torsor trace contribution from (75). This determinant-shadow seed is read as a CKM seed only after the Schur-visible non-circulant direct-sector mechanism of Proposition 15 has been imposed.

With (112), (146) gives

$${\lambda }_{\mathrm{cen}}=0.2250096.$$

(147)

Solving (146) directly for the PDG central value of $s_{12}$ gives ${\kappa }_{\mathrm{CKM}}=1.0105566$, while (146) gives ${\kappa }_{\mathrm{SB}}=1.0105543$. The difference is $2.3\times {10}^{-6}$ at the level of the central scaling factor. This is the numerical consistency check of the common primitive unit and the torsor trace.

The minimal flat torsor normalization of (145) sets $ϵ={\lambda }_{\mathrm{cen}}$ and assigns the first three CKM magnitudes by central path length:

$$s_{12}={\lambda }_{\mathrm{cen}},s_{23}=\sqrt{{\displaystyle \frac{2}{3}}}{\lambda }_{\mathrm{cen}}^2,s_{13}={\displaystyle \frac{1}{3}}{\lambda }_{\mathrm{cen}}^3.$$

(148)

The corresponding closed-loop CP diagnostic is

$$J_q={\displaystyle \frac{1}{4}}{\lambda }_{\mathrm{cen}}^6.$$

(149)

Using (124), the same minimal assignment gives

$$sin{\delta }_{\mathrm{CKM}}^{\min }=0.9435538,{\delta }_{\mathrm{CKM}}^{\min }=70.{657}^{\circ },{\vartheta }_{\Gamma }^{\min }=23.{552}^{\circ }.$$

(150)

The conjugate orientation gives $109.{343}^{\circ }$ for the CKM phase. If the one-phase torsor identification is imposed, substitution of ${\vartheta }_{\Gamma }^{\min }$ into (63) gives the relative central response spectrum $(0.16661,2.22459,3.60880)$ in units of ${\nu }_{\Gamma }$. The comparison is shown in Table 12. The first line is the determinant-shadow seed. The remaining lines require the minimal central path assignment in (148) and (149). The standard CKM convention is the one of [28,29,30]; the CP-sensitive comparison uses [31].

The same minimal assignment gives the absolute-value CKM matrix in Table 13. In the standard Wolfenstein reading it gives $A=0.8164966$, $R_b=0.4082483$, and the barred unitarity-triangle coordinates $\overline{\rho }=0.1321$, $\overline{\eta }=0.3755$.

The CKM tables have a different status from the carrier theorem. The value (147) is fixed by the single primitive-unit determinant shadow and the torsor trace. The higher path readings in (148) and (149) are predictions of the completed direct branch only after Definition 9, the valuation (144), and the flat torsor normalization have been imposed. Proposition 16 does not fix the numerical path coefficients by itself. It excludes repairs by an observable-invisible direct $1\leftrightarrow 3$ channel or by an additional low family coordinate which leaves the finite Schur residual and the filtered sector labels unchanged. A completed branch with a different torsor-path normalization is a different reduced branch only if the change is visible to the non-circulant detector or protected by the finite channel filtration.

6.3. Charged-lepton balance diagnostic

The charged-lepton diagnostic evaluates the scale-free residual (115) 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 the reference values $m_e=0.51099895069\mathrm{MeV}$, $m_{\mu }=105.6583755\mathrm{MeV}$, and $m_{\tau }=1776.93\mathrm{MeV}$ from the numerical data compilation [32], with the torsor ordering $(\tau ,e,\mu )$, the coordinate extraction in (114) gives

$$a_{\ell }=0.707109118112537,{\varphi }_{\ell }=0.222224761894593,C_{\ell }=25.0538962437811{\mathrm{MeV}}^{1/2}.$$

(151)

The corresponding balance defect is

$${\Sigma }_{\ell }^{\mathrm{obs}}=6.60981393\times {10}^{-6},Q_{\ell }^{\mathrm{obs}}=0.6666644634.$$

(152)

Thus the exact Koide value is not imposed. In the notation of (116), the observed quotient corresponds to a small positive Schur-layer balance defect. The comparison with the zero-correction values is

$$a_{\ell }-{\displaystyle \frac{1}{\sqrt{2}}}=2.33692599\times {10}^{-6},{\varphi }_{\ell }-{\displaystyle \frac{2}{9}}=2.53967237\times {10}^{-6}.$$

(153)

The first number is the torsor-coordinate form of the logarithmic balance defect in (152). The second number is only a clock comparison. The balance residual controls the first shift through (116) once ${\Sigma }_{\ell }$ has been supplied; it does not fix the torsor angle.

The zero-correction balance ${\Sigma }_{\ell }=0$, combined with $m_e$ and $m_{\mu }$ as inputs, gives $m_{\tau }=1776.9690\mathrm{MeV}$. This is a useful charged-lepton shape test of the equal Hodge-norm split between the harmonic line and the torsor complement, but the observed nonzero value in (152) is the quantity to be matched by a completed charged-lepton Schur tensor. The diagnostic is summarized in Table 14.

6.4. Neutral Pfaffian, PMNS, radial, and decay diagnostics

The neutral Majorana reading gives an independent test. In the Pfaffian branch (130), $M_R=6.31\times {10}^{14}\mathrm{GeV}$. For a unit neutral Dirac Yukawa, $m_D=v_F/\sqrt{2}=174.10\mathrm{GeV}$, the type-I seesaw diagnostic gives $m_{\nu }^I=m_D^2/M_R=0.0480\mathrm{eV}$. This value is to be compared with the atmospheric neutrino scale, $m_3\simeq 0.05\mathrm{eV}$ in a normal hierarchical spectrum. Through (133), full PMNS data depend on the charged-lepton basis, the family shape $R_N$, the neutral Dirac bridge $D_{\nu }$, and the active term $A_L$ in (132). The usual seesaw comparison is represented by [66].

The leading neutral basis (136) is a zeroth degeneracy-resolution pattern for a soft neutral denominator. A nonzero ${\theta }_{13}$, the solar and atmospheric deviations, and CP-sensitive PMNS data require the completed neutral family shape in (132). The common-circulant subcase leaves no relative family basis and therefore does not produce large PMNS angles by itself.

The neutral bosonic cell gives a separate radial test. If the finite neutral cell is locked under radial scale variation, the standard $hWW$ and $hZZ$ normalizations are recovered at first order. Trace-neutral leakage gives one correlation between deviations in the two channels. In the trace-neutral parametrization $\delta A+\delta B+\delta C=0$, the first radial deviations may be written

$${\kappa }_W-1={\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta A}{A}},{\kappa }_Z-1={\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta A+\delta B}{A+B}}.$$

(154)

Thus a one-parameter leakage direction gives a correlated $hWW/hZZ$ test. This is a structural test of the completed neutral cell rather than an independent mass fit. 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 16. 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 (109). 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; it does not change the exterior channel.

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

$${\Delta }_{i\to F}=m_i-\sum _{a\in F}{m}_a.$$

(155)

A channel is open only when (155) 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 [67].

6.5. Contact scale and falsification tests

The doubled top-form baryon contact class is a completed-branch coefficient. It is separated from the weak bridge by exterior bidegree and from the Majorana class by $B-L$ filtration. Hence proton-contact diagnostics cannot be inferred from the one-Higgs weak bridge, and they need not share the Pfaffian Majorana scale unless the completed determinant branch imposes such an additional relation. Proposition 16 excludes a replacement of this separated contact channel by a removable low weak-bridge coordinate.

A direct tunnelling assignment $M_X=M_{\mathrm{Pl}}exp(-S_{\mathrm{prim}})$ would put the baryon 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 [68]. The doubled top-form assignment places the contact at the tower scale. For an order-one finite coefficient and $M_X\simeq M_{\mathrm{Pl}}$, the standard dimension-six estimate gives ${\tau }_p\sim {10}^{48}\mathrm{yr}$. With a two-top-form normalization suppression by ${\left(2\pi \right)}^{-4}$, one obtains $M_X\simeq 3\times {10}^{17}\mathrm{GeV}$ and ${\tau }_p\sim {10}^{41}\mathrm{yr}$. Thus an observed $p\to e^{+}{\pi }^0$ signal near ${10}^{35}\mathrm{yr}$ would rule out the minimal doubled-top-form contact assignment.

The generator-level falsification map is separated from the numerical branch tests. It records which residual closure would fail before a completed numerical comparison is made.

Table 15. Generator-level falsification map.

Generator entry	Structural test	Falsifier
${\mathfrak{E}}_{\mathrm{link}}$	primitive projective link and degree-one positive class	a positive degree $n>1$ gives an equally reduced first-threshold support
${\mathfrak{E}}_{\mathrm{Gauss}}$	two boundary-central principal charges of types $V_1$ and $V_2$	the principal charges are not central in finite local sector representations
${\mathfrak{E}}_{\mathrm{vis}}$	minimal separated Toeplitz support (26)	a smaller boundary-central support sees both Gauss-local channels
${\mathfrak{E}}_{\mathrm{red}}$	absence of unsourced low-visible integer types	a required low principal type $V_{\ell }$, $\ell \ge 3$, is present before Schur completion
${\mathfrak{E}}_{\det }$	split determinant carrier (32) and global form (33)	line-operator or determinant tests require a different compact global form
${\mathfrak{E}}_{\mathrm{fin}}$	full ${\mathbb{Z}}_6$ finite shadow and its projective-color projection	the finite coefficient shadow is not the generator used in (46)
${\mathfrak{E}}_{\mathrm{tor}}$	torsor-admissible projective-color edge complex with residual closure (59)	the transported boundary charges do not descend to unitary torsor edges, or the flat structural branch has $R_{\mathrm{tor}}^{\Gamma }\ne 0$
${\mathfrak{E}}_{\mathrm{gap}}$	stable Riesz low-sector projection	the Callias-Schur gap closes or a hidden pre-torsor multiplicity remains
${\mathfrak{E}}_{\mathrm{mix}}$	non-circulant Schur-visible family motion	quark mixing is generated by a common circulant family algebra
${\mathfrak{E}}_{\mathrm{bal}}^{\ell }$	charged-lepton balance residual (115)	a completed charged-lepton Schur tensor gives a balance defect incompatible with (152)
${\mathfrak{E}}_{\mathrm{Schur}}$	reduced Schur-Kuranishi completion, no-phantom condition, and minimal Schur-torsor valuation (144)	a removable low Schur channel is required to repair the completed data, or a direct $1\leftrightarrow 3$ edge of valuation 1 or 2 is required
${\mathfrak{E}}_{\mathrm{RG}}$	scale comparison of completed branch data	independent sector-by-sector thresholds are required for the same boundary cycle

Table 15. Generator-level falsification map.

Generator entry	Structural test	Falsifier
${\mathfrak{E}}_{\mathrm{link}}$	primitive projective link and degree-one positive class	a positive degree $n>1$ gives an equally reduced first-threshold support
${\mathfrak{E}}_{\mathrm{Gauss}}$	two boundary-central principal charges of types $V_1$ and $V_2$	the principal charges are not central in finite local sector representations
${\mathfrak{E}}_{\mathrm{vis}}$	minimal separated Toeplitz support (26)	a smaller boundary-central support sees both Gauss-local channels
${\mathfrak{E}}_{\mathrm{red}}$	absence of unsourced low-visible integer types	a required low principal type $V_{\ell }$, $\ell \ge 3$, is present before Schur completion
${\mathfrak{E}}_{\det }$	split determinant carrier (32) and global form (33)	line-operator or determinant tests require a different compact global form
${\mathfrak{E}}_{\mathrm{fin}}$	full ${\mathbb{Z}}_6$ finite shadow and its projective-color projection	the finite coefficient shadow is not the generator used in (46)
${\mathfrak{E}}_{\mathrm{tor}}$	torsor-admissible projective-color edge complex with residual closure (59)	the transported boundary charges do not descend to unitary torsor edges, or the flat structural branch has $R_{\mathrm{tor}}^{\Gamma }\ne 0$
${\mathfrak{E}}_{\mathrm{gap}}$	stable Riesz low-sector projection	the Callias-Schur gap closes or a hidden pre-torsor multiplicity remains
${\mathfrak{E}}_{\mathrm{mix}}$	non-circulant Schur-visible family motion	quark mixing is generated by a common circulant family algebra
${\mathfrak{E}}_{\mathrm{bal}}^{\ell }$	charged-lepton balance residual (115)	a completed charged-lepton Schur tensor gives a balance defect incompatible with (152)
${\mathfrak{E}}_{\mathrm{Schur}}$	reduced Schur-Kuranishi completion, no-phantom condition, and minimal Schur-torsor valuation (144)	a removable low Schur channel is required to repair the completed data, or a direct $1\leftrightarrow 3$ edge of valuation 1 or 2 is required
${\mathfrak{E}}_{\mathrm{RG}}$	scale comparison of completed branch data	independent sector-by-sector thresholds are required for the same boundary cycle

The falsification logic of the minimal $B-L$ filtered branch is summarized in Table 16. Numerical comparisons are to be made with [32] only after the relevant completed-branch hypothesis has been fixed.

The quantitative outputs are summarized in Table 17. The table separates fixed minimal-branch values from diagnostics that require an additional completed-branch choice. The experimentally falsifiable part of the reduced completed branch is read in the following operational sense. The CKM entries in (148) and (149) test the minimal direct branch once the central path completion has been fixed; a required direct $1\leftrightarrow 3$ edge of valuation 1 or 2, or a removable repair channel excluded by Proposition 16, falsifies that reduced branch. The Pfaffian Majorana entry tests the separated $|\Delta (B-L\left)\right|=2$ neutral denominator used in (130). The contact entry tests the doubled top-form assignment and the weak-bridge exclusion recorded after Table 10. The radial-leakage entry is a correlated Higgs diagnostic until a completed leakage direction has been fixed.

7. Conclusions

The closure defect of the self-reconstruction loop has been organized as a filtered residual system. Its lexicographic critical equations give the closure laws used in the local branch: the projective-link, Gauss-local, Toeplitz-visibility, determinant, finite-shadow, torsor, and Schur entries are fixed at different reconstruction depths.

A primitive self-reconstructing Codazzi defect has been analyzed as a local boundary-cycle reconstruction problem. Resolving the worldline defect gives a projective link, the primitive transverse class gives the degree-one line, and the positive link modes are the Borel-Weil blocks. After the scalar trace is separated, the natural transverse two-jet has the two principal non-scalar types recorded in (22). Boundary superselection makes these two Gauss-local charges central. Toeplitz visibility then selects the minimal separated carrier (26).

The determinant-compatible structural package is obtained after this carrier has been selected. The split top-form condition gives the compact carrier group (32) and its global Standard-Model form (33). The even exterior package (35) gives the one-generation representation package, with hypercharge and $B-L$ read from the exterior degrees in (36) and (38). Hypercharge gives the determinant direction, while $B-L$ gives the anomaly-free Schur-channel filtration recorded in Table 7. The full finite-coefficient shadow of the primitive class is the ${\mathbb{Z}}_6$ class (46). Its weak projection is compatible with the chirality/exterior-parity lock (102) and has the finite parity response of Lemma 5, while its projective-color projection gives the torsor (52).

The projective-color torsor supplies the finite central part of the boundary cycle. Its vertex space is the central carrier (53), and its edge resolution is the torsor complex (54). In a flat torsor trivialization, the adjacent central response is the torsor Laplacian (61). The Wilson holonomy is the finite circulation datum, the torsor cohomology (66) gives the flat-branch obstruction, and the finite trace contribution to the determinant-shadow seed is (75). The Schur-visible non-circulant detector is the mixed torsor curvature norm (72). Thus the projective-color shadow supplies a finite boundary complex rather than only a three-dimensional label set.

A compact-leaf current-residual source model and an Alena-Codazzi realization criterion have been supplied. The residual scalar, conserved translational current, Codazzi multiplier, compact-leaf source collar, thin-core limit, moment-resolved two-channel source, and relative Gauss-local surface charges give the data of Definition 2 under the hypotheses of Theorem 4. The same collar supplies the continuum vorticity response whose quantized projective-color boundary shadow is (56). Under Schur-admissibility, the selected support is represented by an isolated low-sector bundle which persists under the allowed perturbations. Under torsor-admissible boundary transport, the torsor-cycle realization realizes the projective-color torsor complex on the transported boundary charge pair; the flat torsor branch is selected by (59).

The completed low layer is organized by the $B-L$-filtered Schur-Kuranishi summary. Projected gauge fields are obtained from the moving low-mode connection. The projective-color carrier appears as the central factor of the locked low window under the Callias-Schur gap condition. The complementary weak-parity mode must be fixed or gapped as in Corollary 2. Dirac and Yukawa blocks are compressed weak-bridge matrix elements of the locked low operator. The weak bridge is the color-singlet $B-L$ preserving tangent to the chirality/exterior lock. The charged-lepton direct block admits the scale-free balance residual (115); its zero-correction limit gives the Koide-type quotient (117), while the observed defect is a target for the finite Schur tensor in (118). CKM mixing is read as the relative non-circulant Hodge-Schur curvature of the two $B-L$ preserving direct Dirac blocks in the sense of Definition 9. The finite detector is (72), with the perturbative bound (127). Relative $V_1/V_2$ charge transport supplies the clock degree used in the minimal path assignment of (144).

The neutral and contact sectors are separated by the same $B-L$ filtration. Light neutrino masses are obtained from the $B-L$ breaking neutral Schur complement (132). A soft neutral denominator with a clock-even perturbation gives the leading neutral basis (136), while the observed departures from that leading pattern belong to higher Schur corrections. The Pfaffian Majorana branch requires a neutral Pfaffian structure of the family operator. The doubled top-form contact belongs to a separate $B-L$ preserving Schur-Kuranishi channel and is excluded from the finite weak-bridge layer by the channel separation after Table 10.

The minimal numerical branch gives a finite set of tests. The neutral determinant-shadow cell gives the link-angle benchmark and the zero-remainder electroweak mass readings in Table 11. The central determinant-shadow seed and the minimal central path completion give the CKM benchmarks in Table 12. The charged-lepton balance diagnostic in Table 14 records the observed nonzero Hodge-balance defect of the direct charged-lepton block. The Pfaffian half-flux branch gives a seesaw denominator in the atmospheric neutrino range. The doubled top-form contact assignment gives a proton-contact reach estimate and excludes the direct primitive tunnelling assignment as a proton-decay reading. The status of the quantitative outputs is summarized in Table 17, and the falsification conditions are listed in Table 16.

The resulting reconstruction has the form of a faithful minimal closure chain. First, the fidelity entry of (14) keeps the two principal boundary charges in independent finite charge sectors. Second, separated Toeplitz visibility gives the rank-five carrier (26). Third, the complexity entry gives the degree-one line in Proposition 3. Fourth, the split determinant package gives the global carrier group, the even exterior module, anomaly cancellation, and the normalization (41). Fifth, full finite-shadow conservation keeps the ${\mathbb{Z}}_6$ class before its projective-color and weak-parity readings. Sixth, the unique color-singlet $B-L$ preserving odd lock tangent is the weak bridge $W\oplus W^{*}$. Seventh, a reduced primitive normal representative removes charge-invisible pre-torsor multiplicity, and the projective-color torsor gives the rank-three low cluster (108) under the hypotheses of Proposition 14 and Corollary 2. Eighth, Proposition 16 removes finite Schur channels which are invisible to the completed obstruction coordinates and to the already fixed sector labels. CKM mixing is read as Schur-Berry motion generated by non-circulant torsor-visible corrections, with detector (72). The minimal non-circulant Hodge-Schur branch of Definition 9 selects the central-path CKM readings in (148) and (149); in the one-phase central-loop reduction, the CP-sensitive part is fixed by (124), with the minimal numerical phase recorded in (150). Higher integer source moments do not reselect the primitive carrier and enter the low operator through the Schur-suppressed contribution (105). In this form, the Standard-Model carrier is obtained as the minimal determinant-compatible, $B-L$-filtered, torsor-stabilized Schur-Berry closure of a reduced primitive Codazzi defect. The numerical Yukawa, PMNS, running, and contact coefficients remain completed-branch data on this closure.

8. Discussion

The construction gives a local route from a four-dimensional codimension-three defect to a finite determinant-oriented boundary cycle, ordered by the filtered self-reconstruction residual. Its first part is structural. The primitive projective link, the positive line, and the two Gauss-local boundary charges determine the boundary-central Borel-Weil support through Theorem 1. The degree-one line is obtained from reduced primitive visibility in Proposition 3. The carrier group, the determinant-compatible exterior package, the hypercharge degree, the $B-L$ degree, the local anomaly checks, the normalization (41), and the ${\mathbb{Z}}_6$ finite-coefficient shadow are read from the selected carrier in Remark 1. The projective-color projection then gives the torsor (52), the central response carrier (53), and the finite torsor complex (54). For a reduced primitive normal representative satisfying the hypotheses of Proposition 14, this central carrier becomes the rank-three low cluster (108).

The reduced condition has two roles. Algebraically, it removes unsourced integer Toeplitz-visible types from the positive link support and removes charge-invisible pre-torsor multiplicity from the primitive low window. Analytically, it is applied after the multiplicity-one thin-core component has supplied the primitive worldline and before the projective-color torsor is attached. It fixes the primitive multiplicity; the Callias-Schur gap then preserves the corresponding low window under the completed perturbation. Thus the torsor factor is the finite-coefficient central response, while an additional pre-torsor multiplicity would be a completed-representative excess.

The Alena-Codazzi collar is used as a compact-leaf source model and as a realization criterion for the primitive Gauss-local hypotheses. The carrier is selected by the singular boundary-charge algebra, while the smooth compact-leaf background belongs to the realization layer, as recorded in Section 4. Under the torsor-admissible boundary transport of Definition 7, the torsor-cycle realization realizes the finite torsor cycle on the transported boundary charge pair. The flat branch is the residual closure (59).

This places the result between several familiar geometric mechanisms. Kaluza-Klein-type constructions encode gauge variables by enlarging the geometric structure [1]; Eisenhart-Duval-type constructions rewrite forced motion as geodesic motion [2]; Randers and Finsler models encode charged trajectories geometrically [3]. The present construction uses a local boundary operation: the finite carrier and the torsor complex are reconstructed from the resolved link of the defect after the primitive Gauss-local source data have been fixed.

The twistor input is limited to the projective-line reduction. The optical Codazzi branch supplies the projective-spinor reading of the link in the standard spinor convention of [7]. The broader twistor comparison class remains the one of [8], with Standard-Model-oriented twistor comparisons represented by [9]. Only the rank-one projective line attached to the defect is used here. The internal carrier is then selected by the Borel-Weil tower and the ${\mathbb{CP}}^1$ Toeplitz cutoff, with the standard finite-mode comparison supplied by [14,15].

The relation to finite internal-space models is correspondingly precise. In almost-commutative geometry the finite internal algebra is part of the spectral input, as in [69], and the spectral action gives the corresponding dynamical principle [70]. Recent spectral-geometric refinements remain a natural comparison class [71]. In the present construction the finite module and the torsor cycle are reconstructed before the spectral completion is solved. The projective-color torsor is then organized as the projective-color finite Hodge-spectral module (56), with graph-Hodge, vector-bundle Laplacian, and finite spectral-module comparisons represented by [59,60,61]. Its Laplacian spectrum records the central torsor response and the Wilson obstruction; in the Alena-Codazzi realization this is the quantized boundary shadow of the vorticity sector. Fermion masses and mixing matrices remain Schur-completed low-sector data. The exterior package agrees with the familiar $SU\left(5\right)$ and $Spin\left(10\right)$ representation organization [16], and it is comparable to Clifford-ideal descriptions [17], but it is applied only after the rank-five support has been selected from the link.

The same organization has a K-theoretic reading. The boundary torsor complex (54) is the finite cochain skeleton of the projective-color shadow. The locked low sector is the finite even module on the selected carrier, with the odd weak bridge and the completed Schur operator playing the role of the finite odd part. The determinant, Pfaffian, and doubled top-form readings belong to determinant-line operations on the corresponding family. This is an organizing interpretation here. The differential K-theory comparison is the one used for determinant families [34], while Callias-type refinements provide the analytic comparison class [72].

The analytic layer has a separate role. The boundary-admissible Dirac-Callias representative isolates the low sector, while the Riesz projection and the Schur complement produce the completed finite operator. The comparison with Callias theory uses [22,23]; the perturbation step uses the spectral-projection language of [24]. Under the gap hypothesis of Proposition 14, the projective-color carrier becomes a factor of the locked low window. This is a rank and bundle statement; the effective matrix on that window may still contain central Schur terms. The projected connection is the Berry-Wilczek-Zee connection on the moving low-mode bundle [25,26]. Determinant-line and regular-family readings belong to the Bismut-Freed comparison class [27].

The main organizational point of the completed finite layer is the $B-L$ filtration. Hypercharge gives the determinant direction, while $B-L$ separates the Schur-Kuranishi channels and is locally anomaly-free on the same exterior package by Table 7. Table 9 identifies $W\oplus W^{*}$ as the unique color-singlet $B-L$ preserving odd tangent to the chirality/exterior lock. This weak bridge gives the direct Dirac/Yukawa blocks. The singlet Majorana channel has $|\Delta (B-L\left)\right|=2$ and supplies the neutral Schur denominator. The doubled top-form contact is a separate $B-L$ preserving contact channel and is excluded from the finite weak-bridge layer by the channel separation after Table 10. This filtration is summarized in Table 10 and keeps CKM, PMNS, Majorana, and proton-contact diagnostics in distinct finite channels.

The charged-lepton balance diagnostic is confined to the color-singlet direct Dirac block. It uses only the real decomposition of the projective-color torsor carrier into the family singlet and its real torsor complement. The residual (115) gives the Koide relation as the zero-correction limit, but the measured quotient is represented by the nonzero defect in (152). Thus the diagnostic becomes predictive only after the finite charged-lepton Schur tensor in (118) has been supplied.

The CKM mechanism is a direct Dirac mechanism. The projective-color projection gives the central carrier (53). A common circulant family response lies in the kernel of (73) and gives only a common Fourier basis. In the vortex-shadow reading, this is the flat circulant boundary transport. Nontrivial CKM data require the Schur-visible non-circulant correction described by Proposition 15. The finite detector for this correction is the family spectral seminorm (73), equivalently the mixed torsor curvature norm (72), and its first-order size is bounded by (127). In this status, the central determinant-shadow seed is the flux-independent trace output of the minimal branch, while the higher central path readings are predictions only after the minimal path completion has been imposed. The standard comparison convention is the CKM language of [28,29,30]; CP-sensitive finite invariants are compared with [31].

The neutrino mechanism is a second Schur complement. The charged-lepton basis is a direct $B-L$ preserving Dirac basis, whereas the active neutrino basis is read from the $B-L$ breaking neutral Schur complement. The PMNS matrix is the relative-basis reading (133). Thus large PMNS angles may be denominator-driven through the neutral Schur denominator (132). The Pfaffian half-flux Majorana branch gives the heavy denominator used in the seesaw comparison [66]. The matrix itself depends on the neutral Dirac bridge, the Majorana family shape, the active Majorana correction, and the degeneracy resolution of the neutral denominator.

The numerical results of Section 6 are therefore branch diagnostics. The neutral-cell benchmark, the primitive determinant-scale reading, the CKM seed, the minimal central-path CKM entries, the Pfaffian seesaw scale, and the doubled-top-form contact estimate have different logical status. The torsor spectrum in (63) belongs to the finite Hodge-spectral boundary cycle; masses and mixing entries belong to the completed Schur layer. Numerical comparisons use the standard data compilation [32]. The proton-contact comparison is made in the usual effective-operator convention [68].

The remaining comparison classes are recorded because they delimit the completed problem. In the geometric branch, the classical Rainich starting point is [73], the aligned Einstein-Maxwell comparison is represented by [74], and generalized stress-energy classifications are represented by [75]. Plebanski-type substructure separation [76] and the pure-connection or gauge-gravity formulations of [77,78] provide the four-dimensional gauge-geometric comparison. The spinorial and form-based extensions used for orientation are those of [79,80,81]. The self-dual-form background is represented by [82,83,84]. Standard differential-geometric conventions follow [85,86]. Codazzi comparison results outside the local collar are represented by [40,87,88].

The spin and index comparison is also fixed. Twistor and spin-geometry conventions beyond the projective-line reduction are represented by [89,90,91]. The index-theoretic background is the standard one of [92,93]. The two-dimensional magnetic zero-mode comparison is represented by [94], while homogeneous-vector-bundle harmonic analysis is represented by [95]. The differential K-theory determinant comparison uses [34].

The finite internal comparison remains broad but subordinate to the carrier theorem and to the torsor cycle. Lorentzian noncommutative Standard-Model extensions are represented by [96]. No-doubling spectral geometries and electroweak theta-term variants are represented by [97,98]. General spectral noncommutative reviews and constructions are represented by [99]. Division-algebraic one-generation and family constructions are represented by [100,101,102].

The gauge-field comparison is used at the level of projected connections and completed branch dynamics. Dynamical principal-bundle formulations are represented by [103]. Kaluza-Klein internal-symmetry and test-particle comparisons are represented by [104,105]. The relational internal-space viewpoint is represented by [106]. The local gauge-theory convention is the one of [107], and coupled Yang-Mills-Higgs-Dirac analysis is represented by [108].

The analytic boundary and Callias comparison is used to delimit the Schur-admissible class. Boundary elliptic estimates are represented by [109,110,111]. Pseudodifferential boundary calculus and conic degeneration are represented by [112,113,114]. The APS and Dirac-boundary comparison is represented by [115,116,117]. Callias-type refinements used as comparison points are represented by [72,118,119]. The generalized Dirac-Schrodinger comparison is represented by [120]. The qualitative PDE convention is represented by [121].

The completed quantitative comparisons are kept as branch diagnostics. Line-operator constraints in the Standard Model are represented by [122]. One-loop matching and background-field heavy-field elimination are represented by [123,124]. The Higgs-sector numerical convention is represented by [67]. Non-Abelian Wilson-loop and Wilczek-Zee measurements are represented by [125]. Geometric matter models provide a separate soliton comparison class [126].

Several limitations remain part of the formulation. The faithful minimal reconstruction action (14) selects the finite structural carrier inside a fixed local reconstruction class after the relevant support entries have been closed; it is not an existence theorem for all current-residual collars. The support theorem does not determine numerical Yukawa singular values. The Alena-Codazzi realization criterion gives a non-empty compact-leaf source model and a conditional current-residual realization class, not a classification of all primitive current-residual collars. Torsor-admissible transport is an additional finite boundary condition on the transported charge pair, while flat torsor closure is the residual condition (59). The finite ${\mathbb{Z}}_6$ shadow fixes the congruence and the projective-color torsor, but the spectral low cluster is isolated by the boundary-admissible Callias-Schur gap. The locked factorization of Proposition 14 is a statement about the transported low window and its rank, while the completed effective operator may contain central Schur terms.

The primitive unit (112) is used as the minimal-branch normalization of the one-dimensional determinant-tangent correction, with the normalization status stated in Remark 3. The full ${\mathbb{Z}}_6$ shadow is represented on the master carrier (48), while its completed phase lift is constrained by (50). The one-phase CP reading (124) is a completed central-loop reduction, not a new carrier selection. The charged-lepton balance residual fixes only the scale-free Hodge-balance quotient up to the finite defect ${\Sigma }_{\ell }$; it does not determine the torsor angle or the overall charged-lepton Yukawa scale. The Pfaffian Majorana reading requires the neutral Pfaffian structure (131); it is not a square root of the primitive line $L_{\Gamma }$. The exact microscopic boundary deformation complex whose finite quotient is (54) remains part of the analytic completion. The $B-L$-filtered Schur completion gives a finite predictive module, but the completed low operator, threshold conversion, running-coupling matching, the charged-lepton Schur tensor, and the full PMNS family shape still require the corresponding closed-branch data. The role of the present result is to reduce those questions to a finite carrier, a determinant-oriented torsor cycle, a channel filtration, and a set of falsification tests.