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 current-residual branch first supplies a local trace-adjusted Codazzi closure, and a four-dimensional anisotropic Lorentzian branch with the corresponding primitive optical defect is used to reconstruct a finite internal carrier from the resolved projective link of the defect.

The organizing principle is that physical configurations are 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 closure defect is measured by

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

(2)

where $\mathfrak{M}$ denotes the corresponding quotient configuration space. An exactly closed theory has $D\left[\Phi \right]=0$ on physical configurations. In a local representation in which the closure defect is the variational gradient of an action, the condition $D\left[\Phi \right]=0$ gives the Euler-Lagrange equations. If the defect is the Einstein residual, the same form gives the Einstein equation. If the defect is a curvature-compatibility condition, the corresponding Bianchi-type closure is obtained. In the present work only the primitive local link reduction of (1) is used.

The same local object has four readings. The geometric reading gives an optical Codazzi link. The topological reading gives the primitive class of the twisting line. The representation-theoretic reading gives the Borel-Weil carrier and the Toeplitz visibility thresholds. The spectral reading gives a Dirac-Callias representative, a Riesz low-sector bundle, and the Schur-Berry data used in the completed branch. These readings are kept attached to one defect $\Gamma$ throughout the paper. The structural chain is therefore current-Codazzi closure, thin-core worldline, primitive threshold, two-channel boundary charge, Toeplitz carrier, determinant package, central torsor, and Schur-Berry completion.

Table 1. Four readings of the same primitive defect.

Reading	Local datum	Output	Main language
Geometric	Optical Codazzi branch near a worldline defect.	Resolved link $S_{\Gamma }^2\simeq {\mathbb{CP}}_{\Gamma }^1$.	Rainich-Codazzi geometry
Topological	Primitive positive transverse class.	Twisting line $L_{\Gamma }\simeq \mathcal{O}\left(1\right)$ and the filling obstruction.	Chern class and line bundles
Quantized link	Scalar-sector two-jet with Codazzi-Gauss link charges.	Minimal boundary-central carrier $E_3\oplus E_2$.	Borel-Weil and Toeplitz quantization
Spectral completion	Boundary-admissible normal representative.	Projected gauge field, Schur-compressed finite operators, and branch diagnostics.	Dirac-Callias and Schur-Berry theory

Table 1. Four readings of the same primitive defect.

Reading	Local datum	Output	Main language
Geometric	Optical Codazzi branch near a worldline defect.	Resolved link $S_{\Gamma }^2\simeq {\mathbb{CP}}_{\Gamma }^1$.	Rainich-Codazzi geometry
Topological	Primitive positive transverse class.	Twisting line $L_{\Gamma }\simeq \mathcal{O}\left(1\right)$ and the filling obstruction.	Chern class and line bundles
Quantized link	Scalar-sector two-jet with Codazzi-Gauss link charges.	Minimal boundary-central carrier $E_3\oplus E_2$.	Borel-Weil and Toeplitz quantization
Spectral completion	Boundary-admissible normal representative.	Projected gauge field, Schur-compressed finite operators, and branch diagnostics.	Dirac-Callias and Schur-Berry theory

The geometric input belongs to the non-null Rainich-Codazzi class. The Rainich stress reading follows the classical electromagnetic reconstruction problem [5]; type-D and aligned non-null comparisons are represented by [6]. The optical consequence of the two-eigenvalue Codazzi closure is standard in the corresponding Codazzi geometry [7]. In four spacetime dimensions, a worldline defect has link $S^2$, and the optical projective-spinor reading identifies this link with ${\mathbb{CP}}^1$ in the standard spinor language [8]. The broader twistor comparison class is represented by [9], while the Standard-Model twistor direction gives a representation-theoretic comparison [10].

The quantized link part is elementary. The primitive line is read as $\mathcal{O}\left(1\right)$ on ${\mathbb{CP}}^1$, and its positive holomorphic sections form the Borel-Weil tower. The Borel-Weil theorem is used in its standard homogeneous-bundle form [11], with the $SU\left(2\right)$ representation conventions of [12]. Equivalently, the same spaces may be compared with monopole zero modes on $S^2$ [13] and with the Taub-NUT Dirac comparison class [14]. The finite visibility rule is the ${\mathbb{CP}}^1$ Berezin-Toeplitz, or fuzzy-sphere, cutoff [15]. The fuzzy-sphere and finite-matrix comparisons are represented by [16,17].

The source hypothesis used by the carrier theorem is deliberately small. A natural scalar-sector normal source of transverse order at most two has, after the scalar singlet is separated, only two non-scalar associated-graded components. They are the $V_1$ phase-current channel and the $V_2$ trace-free Codazzi-gap channel. In the Alena-Codazzi realization these coefficients have a principal Codazzi-Gauss reading as two link charges, hence as boundary superselection data. In the abstract support theorem this input is kept as separated central implementability. The Toeplitz thresholds then give the rank-five block $E_3\oplus E_2$, with $E_3$ carrying the $V_2$ channel and $E_2$ carrying the separated $V_1$ channel. The mixed $E_2$-$E_3$ blocks have only half-integer $SU\left(2\right)$ content, so the integer source type acts diagonally on the selected low support.

After the carrier has been selected, the split determinant reduction is read on the selected Hermitian blocks. Before this reduction, the natural compact split basis group is $U\left(3\right)\times U\left(2\right)$. The finite determinant-obstruction count has the same local kernel as the stabilizer of the split top exterior form, and the admissible carrier-basis changes are reduced to $S\left(U\right(3)\times U(2\left)\right)$. Transformations mixing the two blocks belong to the unsplit rank-five comparison class and to the mixed half-integer sector of the low support. The determinant-compatible even exterior package on the selected rank-five carrier is then the local one-generation representation module. Its exterior bidegree also carries a separate $B-L$ operator and the associated $Spin\left(10\right)$-type $U{\left(1\right)}_X$ grading. This is close to the familiar $SU\left(5\right)$ and $Spin\left(10\right)$ organization [18]. Clifford-ideal realizations give a related finite-algebraic comparison [19]; modern Clifford variants remain useful comparison points [20]. Almost-commutative geometry gives a separate reference class in which the finite internal algebra is part of the input [21], while the spectral action provides the dynamical principle [22]. Lorentzian noncommutative refinements are represented by [23].

The determinant global form also gives the central family layer. Its ${\mathbb{Z}}_6$ shadow has a projective-color projection to a ${\mathbb{Z}}_3$ torsor and a weak parity projection used in the locked exterior package. The central response space is the corresponding regular three-state space. This gives the mod-three family-response factor. The interpretation as exactly three physical families requires, in addition, the simple primitive locked-kernel condition in the completed spectral problem. In this sense the torsor fixes the central factor, while the final family multiplicity is read after the branch operator has been completed.

The spectral completion is formulated by a boundary-admissible Dirac-Callias normal representative. The Callias mechanism [24] and its geometric form [25] isolate the low spectral window. Perturbation theory is used in the sense of [26]. The Riesz projection defines the moving low-sector bundle, and the projected connection is the Berry connection [27]. Its non-Abelian version is the Berry-Wilczek-Zee connection [28]. The determinant-line and family-index comparisons belong to the usual regular-family language of Dirac-type operators [29]; differential K-theoretic refinements give the corresponding index framework [30].

The local perturbative field-theory reading is obtained after the carrier and the isolated low-sector representative have been fixed. The projected connection gives the gauge-field reading on the selected bundle, with standard Yang-Mills-Higgs-Dirac conventions as in [31,32]. The exterior odd bridge gives the finite Dirac channels. Feshbach-Schur compression gives the effective sectoral operators. Vacuum normalization, projected bosonic curvature, Higgs-coupling and decay readings, fermionic mass matrices, Yukawa operators, and CKM/PMNS diagnostics are therefore completed-branch readings. The CKM comparison uses the usual quark-sector mixing language of [33,34,35]. The CP-sensitive invariant is compared with the Jarlskog invariant [36]; numerical reference values are taken from the standard particle-data compilation [37].

The Alena-type residual collar is used as a concrete realization of the current-Codazzi source hypotheses. It is motivated by the Alena Tensor identification [38]. In the present role it supplies the residual density, the translational-current coefficient, and the vorticity terms used in the branch stress response [39]. The continuum, variational, and Higgs-like branch-potential inputs are those of [40,41,42]. In the closed split-conserved sector, the residual scalar and the translational coefficient are fixed by the Codazzi multiplier and current-compatibility equations, and the source separates the phase-current and vorticity/Codazzi coefficients before the Toeplitz visibility rule is applied.

The paper is organized as follows. Section 2 records the current-Codazzi closure, the Alena-Codazzi source reduction, the two-jet statement, and the principal Codazzi-Gauss reading of the two non-scalar channels. Section 3 gives the projective link, the primitive line, the Borel-Weil tower, and the minimal carrier selected by Toeplitz visibility and boundary-charge centrality. Section 4 records the split determinant reduction, the even exterior package, the anomaly count, and the ${\mathbb{Z}}_3$ family layer. Section 5 gives the Dirac-Callias completion, the projected gauge-field reading, the charge-compatible Schur-Kuranishi reduction, and the quantitative vacuum, bosonic, fermionic, mass, decay, and mixing diagnostics. Section 6 collects the main results and conclusions, including the status of each structural and completed-branch statement. Section 7 places the construction among the geometric, representation-theoretic, noncommutative, and spectral frameworks used for comparison.

2. Alena-Codazzi Source and Two-Jet Reduction

The link quantization used below requires only a primitive optical Codazzi defect and a separated scalar-sector source of transverse order at most two. The Alena-type collar is used as a sufficient current-residual mechanism producing this datum on a regular collar. The residual scalar supplies the scalar branch coefficient, the translational current supplies the order-one phase-current coefficient, and the vorticity/Codazzi response supplies the trace-free order-two coefficient. The detailed normalization of the Alena sector is not used in the carrier calculation; only the current-Codazzi closure, the induced associated-graded source, and the coefficient separation enter the support theorem.

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

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

(3)

Here $\zeta$ is the amplitude of the phase-current variable, ${\mu }_{\zeta }={\mu }_{\zeta }\left({\rho }_{\zeta }\right)$ is positive, and $R_{\omega }$ is the normalized vorticity response. The scalar $p_{\Lambda }$ is used in this section as the Alena branch scalar. Its projected gauge-field reading is part of the spectral completion in Section 5. The Alena Tensor identification supplies the branch stress interpretation [38], while the current and vortex terms used here are the residual-collar data of [39]. The continuum, variational, and branch-potential inputs are those of [40,41,42].

The associated translational current is

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

(4)

The frozen amplitude block is assumed non-degenerate:

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

(5)

This condition is used only as a local persistence hypothesis for the scalar collar. The thin-core compactness assumptions used for the current-residual realization are stated below; the full closed-branch compactness belongs to the completed branch problem.

The product-rule Hilbert response of (3) is assumed to have a frozen local split

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

(6)

where ${\Xi }_{\mu \nu }$ denotes the variation of the residual scalar density and $Y_{\mu \nu }$ denotes the branch-response tensor coming from the variation of $p_{\Lambda }$. The density contribution is included in $Y_{\mu \nu }$. The collar is called closed split-conserved when the two summands in (6) are separately k-conserved on the frozen collar:

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

(7)

This split is the coefficient separation used later by the Toeplitz support rule. The first conserved coefficient is assigned to the phase-current channel. The trace-adjusted Codazzi coefficient is assigned to the vorticity/Codazzi-gap channel before the normal trace-free $V_2$ representative is taken.

Put $\tau =k^{\mu \nu }{Y}_{\mu \nu }$ and $B_{\mu \nu }=Y_{\mu \nu }-{\displaystyle \frac{1}{3}}\tau k_{\mu \nu }$. This is the four-dimensional trace-adjusted tensor used in the multiplier equation. It is not the normal trace-free $V_2$ representative. Its k-trace is $-\tau /3$. If an auxiliary flat reference metric $\eta$ is used, then ${tr}_{\eta }B=Y_{\eta }-K\tau /3$, with $Y_{\eta }={\eta }^{\mu \nu }{Y}_{\mu \nu }$ and $K={\eta }^{\mu \nu }{k}_{\mu \nu }$. The trace-free coefficient used by the link charge is the associated-graded normal component appearing in (12). The scalar $\phi$ is called an admissible Codazzi multiplier on a non-degenerate collar set when $A_{\mu \nu }=\phi B_{\mu \nu }$ satisfies the trace-adjusted Codazzi condition on the punctured collar. With $C_{\alpha \mu \nu }^B={\nabla }_{\alpha }^{\left(k\right)}{B}_{\mu \nu }-{\nabla }_{\mu }^{\left(k\right)}{B}_{\alpha \nu }$ and $\theta =dlog\left|\phi \right|$, this condition is

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

(8)

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

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

(9)

A nonzero local multiplier exists when the full multiplier equation (8) holds with $\theta ={\theta }^B$ and ${\theta }^B$ is exact. In that case $\phi$ is fixed up to one nonzero constant factor. The possible periods and singular strata of B are closed-branch data.

If ${\theta }^B=d{f}_B$, then the scalar part of (3) gives

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

(10)

Together with (4), this gives the remaining current compatibility

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

(11)

Equations (8)-(11) give the local current-Codazzi closure of the residual collar. On an exact non-degenerate connected collar, the pair $(\phi ,{\zeta }^2)$ is fixed by $B_{\mu \nu }$ up to the single multiplier constant and the remaining transport compatibility (11). These relations do not enter the Borel-Weil support calculation except through the existence of the separated principal coefficients.

Proposition 1 

(Current-Codazzi closure). Let ${\Omega }_0$ be a connected non-degenerate frozen collar set on which $B_{\mu \nu }$ is invertible. Assume the closed split-conserved condition (7). Then the trace-adjusted Codazzi part of the residual collar can be closed by the scalar multiplier in (3) if and only if (8) holds with $\theta ={\theta }^B$ and the one-form (9) is exact. In that case (10) is the local scalar reduction, up to one nonzero multiplier constant, and (11) is the remaining current equation. Conversely, a nonzero residual scalar satisfying (8) and the conserved current condition (4) gives (10)-(11).

Proof. 

Substitution of $A_{\mu \nu }=\phi B_{\mu \nu }$ into the trace-adjusted Codazzi condition gives (8). Since the collar is four-dimensional and $B_{\mu \nu }$ is invertible on ${\Omega }_0$, contraction with ${\beta }^{\mu \nu }$ gives (9); hence the multiplier one-form is fixed by $B_{\mu \nu }$. A nonzero scalar multiplier exists precisely when the full multiplier equation holds with this one-form and when it is exact. The scalar reduction then follows from (3). Applying (4) to the reduced scalar coefficient gives (11). The converse follows by the same substitutions. □

An Alena-type collar component satisfying Proposition 1 will be called current-Codazzi closed. A current-Codazzi closed component will be called primitive and non-degenerate when its transverse degree is one with positive orientation and when the frozen collar has a nonzero Codazzi gap. The same gap is used as the local optical splitting, the support-selection barrier, and the zeroth-order input in the Callias isolation. It is called two-channel generic when both coefficients in (12) are nonzero. It is called Schur-admissible when a boundary-admissible normal representative exists and the Callias-Schur gap assumptions used in Section 5 hold on the parameter domain.

In the penalty-dominant thin-core use of a current-Codazzi closed collar, the concentrated current-residual core is the compact object, while Proposition 1 supplies the Codazzi closure on the punctured collar. A finite-energy family with controlled boundary charge has integral-current compactness in the standard sense of [43,44]. The vortex-concentration comparison is the one of [45,46]. The realization input used below is recorded separately.

Proposition 2 

(Thin-core current-defect limit). Let a family of current-Codazzi closed collars have uniformly bounded core energy, uniformly bounded concentrated current mass, controlled nonzero boundary charge, and a penalty-dominant Codazzi residual on the punctured collar. Assume that the concentrated current-residual core is represented by integral 1-currents with no interior boundary in the source-free collar. Then, after passage to a subsequence, the cores converge as integral currents to an integral 1-current with no interior boundary in the source-free collar. On any primitive multiplicity-one regular component of this limit with nonzero frozen gap, the limit supplies the worldline Γ, the degree-one transverse class, and the Codazzi-closed optical link.

Proof. 

The current conservation gives no interior endpoint in the source-free punctured collar. The current-mass and boundary-charge bounds give compactness of the concentrated 1-currents in the integral-current sense. The boundary condition gives the limiting boundary charge. The penalty-dominant Codazzi residual, together with Proposition 1, gives the Codazzi closure on the complement of the limiting core. On a multiplicity-one primitive regular component the controlled boundary charge fixes the degree-one transverse class, and the nonzero frozen gap gives the optical two-eigenvalue splitting. □

Proposition 2 is used only as the realization side of the source hypotheses; the carrier theorem uses only the primitive degree and the associated-graded source below.

Let $\Gamma$ be a primitive regular collar component and let $N_{\Gamma }$ be the oriented transverse normal fiber. A scalar-sector source of transverse order at most two has associated-graded normal terms in ${Sym}^0\left(N_{\Gamma }^{*}\right)\oplus {Sym}^1\left(N_{\Gamma }^{*}\right)\oplus {Sym}^2\left(N_{\Gamma }^{*}\right)$. On the oriented rank-three normal fiber, these have the $SU\left(2\right)\simeq Spin\left(3\right)$ types $V_0$, $V_1$, and $V_0\oplus V_2$. After the scalar trace has been separated, only $V_1$ and $V_2$ remain. The $V_1$ term is the linear phase-current response. The $V_2$ term is the trace-free quadratic vorticity/Codazzi response. No type $V_{\ell }$ with $\ell \ge 3$ occurs at transverse order at most two.

The following statement is the source input used in Section 3.

Proposition 3 

(Alena-Codazzi two-jet reduction). Let a current-Codazzi closed Alena-type collar be restricted to a primitive regular component. Let ${\nu }_{\Gamma }^{\perp }$ be the degree-one transverse trace on the resolved normal link. Then the non-scalar associated-graded normal two-jet has the form

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

(12)

The first summand has type $V_1$, and the trace-free quadratic summand has type $V_2$. On the locus $c_1\left(\phi \right)c_2\left(\phi \right)\ne 0$, the two principal non-scalar channels are present and separated.

Proof. 

Only the associated-graded normal two-jet is used. The order-zero part gives the scalar type $V_0$. The order-one normal response gives the standard vector type $V_1$. The order-two symmetric response decomposes as $V_0\oplus V_2$, and its trace-free part gives the $V_2$ component. The current-Codazzi closed condition assigns the current trace and the residual Codazzi trace to distinct frozen coefficients and restricts them without changing their associated-graded $SU\left(2\right)$ type. The primitive transverse trace gives (12). The nonzero coefficient condition gives the stated two-channel genericity. □

A moment-resolved reading gives the same source datum. The first moment $m\in N_{\Gamma }$ gives the $V_1$ link component, while the trace-free second moment $M_0\in {Sym}_0^2\left(N_{\Gamma }\right)$ gives the $V_2$ component. The assignment $(m,M_0)\mapsto ({\xi }_{V_1},{\xi }_{V_2})$ is the standard $SU\left(2\right)$ module identification for an oriented rank-three normal fiber. Hence the two-channel condition is open and dense in the model moment space.

The same split has a principal Gauss reading on the punctured normal fiber. This reading is used only at the associated-graded level. It does not add a field equation to the collar.

Lemma 1 

(Principal Codazzi-Gauss charges). On the principal normal model $N_{\Gamma }\setminus \left\{0\right\}\simeq {\mathbb{R}}^3\setminus \left\{0\right\}$, let $A_{ij}$ be the trace-free Codazzi representative of the $V_2$ coefficient obtained from (8). Then on the punctured principal normal model there is a harmonic function u such that $A_{ij}={\partial }_i{\partial }_{j}u$ and $\Delta u=0$. The singular $\ell =2$ coefficient of u is a surface charge $Q_2\in {Sym}_0^2\left(N_{\Gamma }\right)\simeq V_2$, independent of the linking radius in a source-free annulus. The $V_1$ coefficient is the first normal moment of the conserved current in (4).

Proof. 

Only the principal flat normal chart is used. For fixed j, put ${\alpha }_j=A_{ij}d{x}^i$. The 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 of A gives ${\partial }_i{v}_j={\partial }_j{v}_i$, hence $v_j={\partial }_{j}u$ on the same punctured chart. The trace-free condition gives $\Delta u=0$.

For $M\in {Sym}_0^2\left(N_{\Gamma }\right)$ set $H_M\left(x\right)=M_{ij}{x}^i{x}^j$. Then $H_M$ is harmonic. Green’s identity on a source-free annulus gives the equality of

$${\int }_{S_r^2}\left(u{\partial }_n{H}_M-H_M{\partial }_{n}u\right)dS$$

(13)

for the two boundary radii. This surface pairing extracts the $\ell =2$ singular coefficient of u, equivalently a vector in $V_2^{*}\simeq V_2$. The $V_1$ statement follows from (4) by the corresponding first-moment pairing. The multipole convention is the one used in [47]; the harmonic Green-pairing convention is standard elliptic theory [48]. □

Corollary 1 

(Principal boundary-charge centrality). Let ${\mathcal{A}}_{\mathrm{eff}}$ be the local effective observable algebra on the source-free punctured collar, after gauge and boundary data have been fixed. If ${\mathcal{A}}_{\mathrm{eff}}$ preserves the link charges of Lemma 1, then the sector projections of the $V_1$ and $V_2$ coefficients lie in $Z\left({\mathcal{A}}_{\mathrm{eff}}\right)$.

Proof. 

The statement is made in a fixed boundary-charge sector. The charges of Lemma 1 are surface readings on a linking sphere. A local observable supported in a source-free annulus cannot change such a reading without producing a nonzero charge difference between the two boundary spheres. Hence admissible local observables commute with the two charge operators. Their sector projections are finite spectral functions of these charges and therefore commute with ${\mathcal{A}}_{\mathrm{eff}}$. This is the standard superselection reading of boundary charges [49]. □

The lemma and corollary give only the principal charge reading. On a regular curved collar the same charge is transported by the frozen collar connection and by the adjoint link modes.

Lemma 2 

(Curved Codazzi-Gauss transport). Let ${\Omega }_{r_0,r_1}$ be a source-free punctured frozen collar between two linking spheres. Let A denote the trace-free $V_2$ representative associated with a trace-adjusted Codazzi coefficient, and let $\mathfrak{C}$ denote the corresponding first-order Codazzi operator on trace-free symmetric coefficients. Assume $\mathfrak{C}A=0$ on ${\Omega }_{r_0,r_1}$. If the transported $V_2$ adjoint modes are regular and satisfy ${\mathfrak{C}}^{*}\Psi =0$, then the boundary pairing ${\mathcal{B}}_{S_r}(A,\Psi )$ is independent of r, after the adjoint mode has been transported along the collar. Hence the $V_2$ link charge is covariantly constant on the regular source-free collar. The $V_1$ current charge is transported in the same sense by (4).

Proof. 

The Green identity for $\mathfrak{C}$ on ${\Omega }_{r_0,r_1}$ gives

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

(14)

Both volume terms vanish by the equations for A and $\Psi$. Therefore the two boundary pairings agree. The transported adjoint modes identify the boundary $V_2$ spaces on the two linking spheres, so the corresponding link charge is covariantly constant. For the $V_1$ channel the same statement is Stokes’ theorem applied to the conserved current in (4). In the principal flat chart the adjoint $V_2$ modes reduce to the quadratic harmonic tests used in Lemma 1. □

The carrier calculation below uses only the associated-graded charge types $V_1$ and $V_2$.

The degree-one condition on ${\nu }_{\Gamma }^{\perp }$ is the primitive sector used in the link construction. After the real blow-up and the positive orientation convention, it gives the primitive twisting line used in Section 3. Higher transverse degrees define non-primitive link quantization data. The Toeplitz support calculation is independent of the special Alena normalization in (3); it uses only the primitive link and the separated source type (12).

3. ${\mathbb{CP}}^1$ Geometric Quantization and the Minimal Carrier

The carrier theorem uses only the primitive projective link, the line $L_{\Gamma }\simeq \mathcal{O}\left(1\right)$, and the separated source type obtained in Proposition 3. The analytic representative and the Schur-Berry completion are not needed for the support calculation. They are used later to attach the selected finite carrier to a closed spectral branch.

3.1. Projective Link and Primitive Line

Let k be the branch metric and let A be the trace-adjusted branch tensor. On the non-degenerate set the non-null Rainich type determines an orthogonal splitting

$${TM|}_U=E\oplus F,dimE=2,dimF=2,$$

(15)

where E is Lorentzian and F is spacelike. The differential closure is the Codazzi condition

$${\nabla }_{\alpha }^{\left(k\right)}{A}_{\mu \nu }={\nabla }_{\mu }^{\left(k\right)}{A}_{\alpha \nu }.$$

(16)

The algebraic Rainich input is the standard non-null stress reconstruction [5]. Generic perturbations of a Rainich algebraic stratum are constrained in the sense of [50]. Self-dual and conformal-metric comparison classes are represented by [51,52,53].

If $A^{♯}$ has two distinct eigenvalues M and P with eigenbundles E and F, the component form of (16) gives

$${\left({\nabla }_{X_1}^{\left(k\right)}{X}_2\right)}_F={\displaystyle \frac{k(X_1,X_2)}{M-P}}{\left({\nabla }^{\left(k\right)}M\right)}_F,X_1,X_2\in \Gamma \left(E\right),$$

(17)

and

$${\left({\nabla }_{V_1}^{\left(k\right)}{V}_2\right)}_E={\displaystyle \frac{k(V_1,V_2)}{P-M}}{\left({\nabla }^{\left(k\right)}P\right)}_E,V_1,V_2\in \Gamma \left(F\right).$$

(18)

The two null directions in E are therefore geodesic and shear-free. The identities are the standard two-eigenvalue Codazzi relations [7]. Lorentzian Codazzi space-times are treated in [54], while homogeneous Codazzi fields give a separate comparison class [55]. Related $2+2$ optical geometries occur in [56,57]. The divergence-free Maxwell comparison remains in the Rainich class [6].

The worldline core $\Gamma$ is obtained by following the compact transverse source along the Lorentzian principal plane. The defect is resolved by real blow-up:

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

(19)

For a worldline in a Lorentzian D-manifold the link is $S^{D-2}$. Hence the projective-line link used here is specific to $D=4$. The optical spinor reading identifies the boundary fiber in (19) with the projective spinor sphere:

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

(20)

The spinor convention is the standard one of [8,58]. The curved-twistor comparison is represented by [59].

Locally, the complement of a worldline has the codimension-three topology

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

(21)

The primitive transverse class occupies the generator in (21). Equivalently, if the oriented transverse frame is resolved across the defect, its north-south patching on an equatorial collar is described by a clutching map $g_{NS}:S^1\to SO\left(2\right)\simeq U\left(1\right)$ of degree one in the simple positive sector. The removable-singularity and local elliptic background are standard [48,60]. The associated line on the projective link is therefore

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

(22)

A complex line over a punctured normal ball extends smoothly over the filled ball only with trivial Chern number on the boundary sphere. Thus (22) is the local filling obstruction carried by the primitive defect.

A worldline defect $\Gamma$ will be called a primitive optical Codazzi defect when (16) holds on the punctured collar, the non-degenerate two-eigenvalue splitting is optical, the real blow-up has the projective link (20), and the resolved oriented transverse frame has simple positive degree one. With a scalar-sector source of transverse order at most two, the defect is called filtered primitive. The word primitive refers to the degree-one class in (22).

3.2. Borel-Weil Tower and Toeplitz Visibility

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

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

(23)

The Borel-Weil identification is used in the usual homogeneous-bundle form [11], with $SU\left(2\right)$ conventions as in [12]. The same zero-mode count is the monopole-harmonic count on $S^2$ [13] and appears in the Taub-NUT Dirac comparison class [14]. In the $spi{n}^c$ reading, (23) is the positive twisted link index of the canonical projective-line Dirac operator. The Atiyah-Singer framework [61] and the spinorial K-orientation of [62] give the stable index language, while the support calculation below uses only (23).

The source side is supplied by Section 2. After the scalar singlet is separated, the associated-graded source type used on the link is

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

(24)

The $V_1$ component is the phase-current coefficient and the $V_2$ component is the trace-free vorticity/Codazzi coefficient. The spherical tensor conventions are those of [63]; the multipole language is the one used in [47].

On the Borel-Weil block $E_q$, Toeplitz observables have the traceless endomorphism content

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

(25)

This is the Clebsch-Gordan decomposition of the spin $(q-1)/2$ representation. In the quantization language it is the finite-mode Berezin-Toeplitz visibility rule on ${\mathbb{CP}}^1$ [15]. The fuzzy-sphere cutoff gives the same finite matrix content [16], and finite-matrix brane models give a broader comparison class [17].

A support assignment is called separated centrally implementable when the two nonzero principal coefficients in (24) are represented by two independent central projections in the $SU\left(2\right)$-equivariant commutant of the chosen finite support. In the Alena-Codazzi realization, Lemma 1 supplies these coefficients as principal boundary charges, and Corollary 1 supplies their centrality. It is Toeplitz-visible when the corresponding $SU\left(2\right)$ types occur in (25). The selected primitive carrier is

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

(26)

If $S_{\Gamma }:=E_2$, then $E_3\simeq {Sym}^2{S}_{\Gamma }$. Hence (26) is the first spinor block together with its symmetric square.

Theorem 1 

(Primitive centrally separated carrier). Let Γ be a primitive optical Codazzi defect with projective link (20) and primitive line (22). Let the scalar-sector source have transverse order at most two and assume that the two non-scalar principal coefficients in (24) are nonzero and separated. Among separated centrally implementable Toeplitz-visible supports, the minimal support is the carrier (26).

Proof. 

The tower is (23) and the visibility rule is (25). Each $E_q$ is an irreducible $SU\left(2\right)$-module, so the center of its equivariant commutant is $\mathbb{C}$. In a separated centrally implementable support, the two separated source coefficients therefore occupy two central projections. By (25), the $V_1$ channel is visible first on $E_2$, while the $V_2$ channel is visible first on $E_3$. The least total rank of two blocks carrying these two channels is $2+3$, and the assignment of the $V_2$ channel to $E_3$ is forced because $V_2$ is absent from ${End}_0\left(E_2\right)$. This gives (26). □

The use of two blocks is forced by central separation. A single $E_3$ block has Toeplitz visibility for both integer types, but its $SU\left(2\right)$-equivariant irreducible support has only one central block and does not implement the two independent boundary-charge sectors.

Corollary 2 

(Codazzi-Gauss central carrier). If the separated source type (24) is supplied by an Alena-Codazzi collar satisfying Lemma 1 and Corollary 1, then the central implementability hypothesis in Theorem 1 is supplied by the two principal link charges. The minimal Toeplitz-visible support is therefore (26).

Proof. 

The two principal coefficients in (24) are boundary-charge sectors by Lemma 1. Their sector projections are central by Corollary 1. Theorem 1 applies. □

The same Clebsch-Gordan count gives the mixed-block selection rule. Since $E_2$ has spin $1/2$ and $E_3$ has spin 1, the mixed block $Hom(E_2,E_3)$ has type $V_{1/2}\oplus V_{3/2}$, and the opposite mixed block has the same half-integer content. Thus the low mixed blocks carry only half-integer $SU\left(2\right)$ types. An integer principal source of type (24) has no equivariant mixed component on (26). In the Codazzi-Gauss reading this is the principal low-sector form of charge-sector diagonality.

The degree-one class is also the source of the threshold separation. If $L_{\Gamma }$ is replaced formally by a degree-n line $\mathcal{O}\left(n\right)$, the q-th positive block is $H^0({\mathbb{CP}}^1,\mathcal{O}\left(n(q-1)\right))$. The first block on which $V_{\ell }$ is visible is

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

(27)

Thus the primitive degree-one case is the only positive degree for which the two channels have the separated first thresholds $V_1\mapsto E_2$ and $V_2\mapsto E_3$. For $n\ge 2$, both channels occur in the same first nontrivial block of the modified tower.

Corollary 3 

(Primitive separated-charge threshold). Among positive transverse degrees, the primitive degree-one line is the unique case in which the two principal Codazzi-Gauss charge types have distinct first Toeplitz thresholds. For $n\ge 2$, the first-threshold separation of the $V_1$ and $V_2$ charge sectors is lost.

Proof. 

This is the specialization of (27) to $\ell =1,2$. □

Non-primitive positive degrees are different filtered link data. They are not excluded by the local topology, but they do not realize the separated first-threshold mechanism used in Theorem 1.

The local negative control is obtained from the same $SU\left(2\right)$ rule [12]. Let $P_{23}$ be the projection onto $E_2\oplus E_3$ in the positive tower. The diagonal low blocks contain $V_1$ for $E_2$ and $V_1\oplus V_2$ for $E_3$, and the mixed low blocks have only half-integer content. Hence no principal integer multipole $V_{\ell }$ with $\ell \ge 3$ is seen in the compressed low support $P_{23}(·)P_{23}$. In the low-high blocks, $V_{\ell }$ first occurs through $Hom(E_3,E_{2\ell -1})$ and $Hom(E_2,E_{2\ell })$. Thus the first $V_3$ couplings start only through $E_5$ and $E_6$. Higher transverse moments are therefore Schur-completion data. Under the Callias-Schur gap and the subcritical low-high bound of Section 5, they can deform the effective low operator but do not reselect the primitive carrier (26). This gives the local negative control: the Standard-Model carrier selection is sensitive to the primitive separated-threshold case, while higher moments remain in the completed operator. Charge-compatible Schur completion additionally preserves the Codazzi-Gauss sector projections.

The carrier theorem is independent of the special residual normalization in (3). The Alena-Codazzi collar produces the separated source type (24) and its principal boundary-charge reading; the primitive projective link produces (23); the carrier follows from (25) after central implementability has been supplied. Any current-residual, geometric, or analytic realization with the same primitive filtered link datum and the same separated centrally implementable $V_1,V_2$ principal channels gives the same local carrier. The finite representation package is attached only after (26) has been selected.

4. Standard-Model Carrier and the Central Family Layer

The support theorem has fixed the finite carrier (26). The structures recorded in this section are attached after that selection has been made. The split determinant carrier group is read as the stabilizer of the selected top exterior line. The finite determinant-obstruction count below selects the same infinitesimal kernel. The even exterior package then gives the local one-generation representation module. The central family layer is the finite-coefficient reading of the same primitive class after the determinant global form has been fixed.

Set

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

(28)

By (26), $dimC=3$ and $dimW=2$. Each selected Borel-Weil block carries its standard $SU\left(2\right)$-invariant Hermitian structure. At the finite carrier level the allowed split unitary basis changes are initially $U\left(C\right)\times U\left(W\right)$. The determinant reduction fixes the carrier top form on ${\Lambda }^{5}V$. Thus admissible split basis changes satisfy

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

(29)

A split unitary map acts on ${\Lambda }^{5}V$ by multiplication by the left-hand side of (29). Hence the compact carrier-basis group is

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

(30)

Equivalently, this is the compact split determinant Levi group of the rank-five carrier. The induced global form is

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

(31)

The full compact group $SU\left(V\right)$ is an unsplit rank-five comparison group. Its off-diagonal infinitesimal part belongs to the mixed low blocks described after Theorem 1. The link-equivariant commutant of the irreducible Borel-Weil blocks remains the corresponding Schur commutant. The $V_2$ source channel selects $C=E_3$ and hence the color block; the $V_1$ channel selects $W=E_2$ and hence the weak block.

The determinant-compatible even exterior package is

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

(32)

It is the chiral exterior package of the selected rank-five carrier, restricted to (30). Its dimension is $1+\left({\displaystyle \genfrac{}{}{0pt}{}{5}{2}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{5}{4}}\right)=16$. The representation-theoretic comparison with the usual $SU\left(5\right)$ and $Spin\left(10\right)$ organization is standard [18]. Related Clifford-ideal realizations are discussed in [19,20]. Symmetry-breaking and roadmap comparisons are represented by [64,65]. The octonionic ladder and internal-space comparisons are represented by [66,67]. Differential-form realizations of fermions provide another exterior-algebra comparison class [68].

Let $N_C$ and $N_W$ be the C- and W-degree operators on ${\Lambda }^{•}V$. The hypercharge convention is the degree-counting normalization compatible with (29):

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

(33)

The electric charge is $Q=T_3+Y$, with $T_3$ acting on the weak factor. Weak-hypercharge superselection gives a useful algebraic comparison [69].

The exterior bidegree also gives the independent charge

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

(34)

It is independent of (33), since it depends only on the color degree. On the six summands of Table 2, (34) gives the usual one-generation $B-L$ values. The associated $Spin\left(10\right)$-type grading is

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

(35)

The same finite package gives the hypercharge-square trace

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

(36)

Table 3. $B-L$ and $U{\left(1\right)}_X$ degree readings on the even exterior package.

Summand in ${\mathit{\Lambda }}^{\mathbf{even}}\mathit{V}$	Local reading	$\mathit{B}-\mathit{L}$	${\mathit{X}}_{\mathbf{Spin}}$
${\Lambda }^{0}V$	neutral singlet	1	5
${\Lambda }^{2}C$	up-type conjugate	$-{\displaystyle \frac{1}{3}}$	1
$C\otimes W$	quark doublet	${\displaystyle \frac{1}{3}}$	1
${\Lambda }^{2}W$	charged singlet	1	1
${\Lambda }^{2}C\otimes {\Lambda }^{2}W$	down-type conjugate	$-{\displaystyle \frac{1}{3}}$	$-3$
${\Lambda }^{3}C\otimes W$	lepton doublet	$-1$	$-3$

Table 3. $B-L$ and $U{\left(1\right)}_X$ degree readings on the even exterior package.

Summand in ${\mathit{\Lambda }}^{\mathbf{even}}\mathit{V}$	Local reading	$\mathit{B}-\mathit{L}$	${\mathit{X}}_{\mathbf{Spin}}$
${\Lambda }^{0}V$	neutral singlet	1	5
${\Lambda }^{2}C$	up-type conjugate	$-{\displaystyle \frac{1}{3}}$	1
$C\otimes W$	quark doublet	${\displaystyle \frac{1}{3}}$	1
${\Lambda }^{2}W$	charged singlet	1	1
${\Lambda }^{2}C\otimes {\Lambda }^{2}W$	down-type conjugate	$-{\displaystyle \frac{1}{3}}$	$-3$
${\Lambda }^{3}C\otimes W$	lepton doublet	$-1$	$-3$

The identifications in Table 2 use the top-form trivialization determined by (29). Thus ${\Lambda }^{2}C\otimes {\Lambda }^{2}W$ is identified with $C^{*}$, and ${\Lambda }^{3}C\otimes W$ is identified with $W^{*}$. The same top-form constraint reduces the projected low-sector connection from $\mathfrak{u}\left(C\right)\oplus \mathfrak{u}\left(W\right)$ to the infinitesimal algebra of (30).

The finite determinant-obstruction count is a degree count on (32). For

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

(37)

the infinitesimal form of (29) is

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

(38)

The local degree factors of ${\Lambda }^{\mathrm{even}}V$ are

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

(39)

The quadratic factor in the last row is positive definite. Hence the common local kernel of (39) is exactly (38). This is the infinitesimal stabilizer of the top exterior line in (29). Thus (33) is the normalized degree-counting representative of the determinant-line anomaly-free class.

Corollary 4 

(Top-form determinant kernel). For the selected carrier (28), the infinitesimal stabilizer of the split top exterior line is the common local kernel of the determinant-obstruction factors in (39). Hence the normalized representative (33) is fixed by the same condition as the determinant-preserving split basis group.

Proof. 

The infinitesimal top-form condition is (38). The degree factors in (39) have the same common kernel, since the remaining quadratic factor in the last row is positive definite. The normalized degree-counting representative is therefore (33). □

Corollary 5 

(Local determinant carrier package). The Codazzi-Gauss central carrier (26), with the determinant constraint (29), gives the compact split basis group (30). The even exterior package (32) gives the local one-generation module recorded in Table 2, with the degree readings (33), (34), and (35).

Proof. 

The carrier and its block dimensions are fixed by (26). The determinant constraint gives (30). The exterior module is (32). The degree readings are (33), (34), and (35), with the representation content recorded in Table 2. □

In the determinant-sensitive family reading, the finite factor in (39) is the local part seen by the Bismut-Freed determinant form for a boundary-admissible Callias-APS family [29,70]. The primitive boundary degree in (22) supplies the local multiplicity. The finite calculation therefore identifies the determinant-preserving subalgebra. Flat or torsion determinant holonomies, global inflow, and cobordism refinements of the resolved worldline defect belong to the global branch problem. Principal-bundle characteristic classes give the standard topological language [71], while secondary classes give the comparison of [72]. Line-operator detection of the global form is the comparison used in [73].

Proposition 4 

(Local determinant flatness). On the determinant kernel (38), the local Bismut-Freed determinant-curvature contribution of the selected finite package vanishes. The remaining determinant information is therefore flat or torsion and belongs to the global branch problem.

Proof. 

The local degree factors of the selected package are listed in (39). Their common kernel is (38). The local curvature of the determinant connection is represented by the corresponding finite anomaly factor in the Bismut-Freed family reading. Hence its local contribution vanishes on the determinant kernel. The remaining flat or torsion holonomy is not detected by the local degree count. □

The odd exterior sector is used as the structural companion of (32):

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

(40)

Every element of $V\oplus V^{*}$ acts by Clifford multiplication on ${\Lambda }^{•}V$ and defines an odd map from (32) to (40). The local weak bridge is the restriction to $W\oplus W^{*}$. Together with the projected connection of Section 5, this gives the finite superconnection-type reading of the weak insertion in the sense of [74]. Defect-localized zero modes provide the classical comparison [75], while domain-wall fermions give the lattice comparison class [76].

The local one-Higgs channels used later are the neutral weak-bridge channels generated by this odd insertion. Their finite content is summarized in Table 4. The bridge restricted to $W\oplus W^{*}$ does not generate a local diquark one-Higgs channel. In exterior bidegree, a $QQ$ bilinear has degree $(2,2)$, while the split top-form convention has degree $(3,2)$; the missing factor is color-odd of degree $(1,0)$, whereas the elementary weak bridge has degree $(0,\pm 1)$. Baryon-violating classes may occur as finite contact classes in the completed package, but they are not produced by the elementary weak bridge.

The same bidegree separates the elementary weak bridge from the contact classes. The weak bridge has bidegree $(0,\pm 1)$ and gives the one-Higgs channels of Table 4. A baryon-violating local contact, if present in the Schur-Kuranishi image, is represented at the doubled top-form bidegree $(6,4)=2(3,2)$. It preserves (34). The singlet Majorana class is separated by $\Delta (B-L)=\pm 2$.

Table 5. Contact-class selection by exterior bidegree.

Class	Bidegree	Reading	$\mathit{B}-\mathit{L}$ response
Weak bridge	$(0,\pm 1)$	one-Higgs Dirac and Yukawa channels	preserved
Doubled top-form contact	$(6,4)$	baryon-violating Schur-Kuranishi contact	preserved
Singlet Majorana class	effective singlet	neutral $B-L$-breaking contact	changed by two units

Table 5. Contact-class selection by exterior bidegree.

Class	Bidegree	Reading	$\mathit{B}-\mathit{L}$ response
Weak bridge	$(0,\pm 1)$	one-Higgs Dirac and Yukawa channels	preserved
Doubled top-form contact	$(6,4)$	baryon-violating Schur-Kuranishi contact	preserved
Singlet Majorana class	effective singlet	neutral $B-L$-breaking contact	changed by two units

When the branch spinor factor is included, exterior parity is locked diagonally with branch chirality. With $N=N_C+N_W$ and ${\Gamma }_V={(-1)}^N$, the primitive product-type locking is

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

(41)

The mirror complement is an analytic sector. In a locked Fredholm realization it is separated by a gap condition of the form

$${\Pi }_{\mathrm{mir}}{K}_{\mathrm{lock}}{\Pi }_{\mathrm{mir}}\ge m_{\Gamma }{\Pi }_{\mathrm{mir}},m_{\Gamma }>0.$$

(42)

This condition belongs to the closed branch operator and does not change the finite representation content selected in Theorem 1. Fermionic statistics belong to the quantization of the corresponding branch spinor field.

The neutral local channels are fixed by Table 2. The active lepton doublet sits in ${\Lambda }^{3}C\otimes W\simeq W^{*}$, while the neutral singlet sits in ${\Lambda }^{0}V$. With (33), the one-Higgs Dirac channel formed by the lepton doublet, the weak factor, and the singlet is neutral. The active Majorana bilinear has its first neutral active class in the two-weak-factor channel. A singlet Majorana term is allowed by the finite representation content and belongs to the global effective-operator problem. The singlet and active two-weak-factor classes are in the standard seesaw comparison range [77]. Baryon-violating contact classes belong to the usual proton-decay comparison range [78]; in the present local reading they are separated from the weak-bridge generated one-Higgs channels.

4.1. Central Family Torsor

The primitive class also has a finite-coefficient reading. Integrally it has already been used to obtain the primitive line (22) and the Borel-Weil tower (23). After the split determinant global form (31) has been fixed, the associated ${\mathbb{Z}}_6$ central shadow has two local projections. The projective-color projection is used below for the ${\mathbb{Z}}_3$ family torsor, while the weak parity projection is the finite shadow compatible with the locked exterior parity (41). The reduction used in the central reading is

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

(43)

Thus the same primitive class is read integrally in the Borel-Weil tower and modulo three in the determinant-center layer. Since closed local color observables are singlet-valued, the color block is naturally projective at this level. Algebraic three-generation models with $S_3$ family symmetry give a separate comparison class [79]; recent Clifford-family variants provide another comparison [80].

Let ${\mathfrak{L}}_{\Gamma }$ be the affine ${\mathbb{Z}}_3$ torsor determined by (43). The corresponding central response space is

$${\mathcal{H}}_{\mathrm{cen}}={\ell }^2\left({\mathfrak{L}}_{\Gamma }\right).$$

(44)

After an auxiliary origin has been chosen, (44) is identified with the regular representation of ${\mathbb{Z}}_3$. With $\omega =exp(2\pi i/3)$, the clock and shift generators are

$$Z{e}_a={\omega }^a{e}_a,S{e}_a=e_{a+1},a\in {\mathbb{Z}}_3,ZS=\omega SZ.$$

(45)

The clock records the central grading. The adjacent shift records nearest-neighbour transport on the torsor. A finite Dirichlet response is

$${\mathcal{E}}_{\mathrm{cen}}\left(\psi \right)={\nu }_{\Gamma }\sum _{a\in {\mathbb{Z}}_3}{|{\psi }_{a+1}-e^{i{\theta }_{\Gamma }}{\psi }_a|}^2,{\nu }_{\Gamma }\ge 0,$$

(46)

with ${\theta }_{\Gamma }$ the discrete holonomy. Thus (46) is the quadratic form of the magnetic graph Laplacian on the affine three-cycle. Its quadratic operator is

$$Y_{\mathrm{cen}}=2{\nu }_{\Gamma }\mathbf{1}-{\nu }_{\Gamma }{e}^{-i{\theta }_{\Gamma }}S-{\nu }_{\Gamma }{e}^{i{\theta }_{\Gamma }}{S}^{†}.$$

(47)

The central response is called unblocked when ${\nu }_{\Gamma }>0$. In the Fourier-character basis, the eigenvalues of (47) are

$$y_k=2{\nu }_{\Gamma }\left[1-cos\left({\theta }_{\Gamma }+{\displaystyle \frac{2\pi k}{3}}\right)\right],k\in {\mathbb{Z}}_3.$$

(48)

Away from the finite degeneracy locus of (48), the lowest eigenspace is one-dimensional and is selected by the corresponding Fourier-character idempotent.

The finite Heisenberg algebra generated by Z and S is $End\left({\mathcal{H}}_{\mathrm{cen}}\right)\simeq M_3\left(\mathbb{C}\right)$. The corresponding traceless anti-Hermitian closure is $\mathfrak{su}\left(3\right)$. This is an algebraic statement about the central response space (44); no gauge-color interpretation is assigned to this central closure. Physical generation splitting is read only after the sectoral finite operator has been supplied by the completed branch.

In a local central coordinate of ${\mathbb{Z}}_3$ degree one, the quadratic scalar is radial, while the first oriented scalar carrying the central phase is cubic. The first oriented adjacent closed cycle on the torsor is therefore

$$C_3={Tr}_{H_0}\left(V_{02}{V}_{21}{V}_{10}\right),$$

(49)

where $V_{10}:H_0\to H_1$, $V_{21}:H_1\to H_2$, and $V_{02}:H_2\to H_0$ are adjacent central response maps. For the unblocked minimal response (46),

$$C_3=-{\nu }_{\Gamma }^3{e}^{-3i{\theta }_{\Gamma }}.$$

(50)

The phase of (50) is a closed-cycle datum of the torsor. Under independent unitary changes of the three torsor fibers, the adjacent maps in (49) transform contragrediently around the loop, so the trace and its phase are frame-independent. In a completed branch, the corresponding cubic Kuranishi coefficient may be generated by Schur elimination of adjacent central detector blocks. Vanishing of the completed coefficient is then a spectral cancellation condition.

Proposition 5 

(Central family count). If the primitive locked kernel over the central torsor has rank r, then the family-response multiplicity is

$$N_{\mathrm{fam}}=3r.$$

(51)

In the simple primitive locked-kernel case one obtains $N_{\mathrm{fam}}=3$.

Proof. 

The torsor (44) has three affine labels. By assumption the primitive locked kernel has the same rank r over each label. The total rank is therefore $3r$. If $r=1$, this gives three family-response states. □

The torsor fixes the factor 3. The rank r and the sectoral eigenvalue splitting are completed-branch data. This is the finite count associated with the unblocked central response away from the degeneracy locus of (48). In this reading, the three family labels are the finite magnetic-cycle labels of (46), while the physical splitting is supplied only by the completed sectoral operator. Leading family operators supported only in the circulant algebra $\mathbb{C}\left[S\right]$ are simultaneously diagonalized by the finite Fourier transform on (44). Hence physical mixing requires a completed-branch contribution outside $\mathbb{C}\left[S\right]$. The corresponding Schur-Berry diagnostics are formulated in Section 5.

5. Spectral Completion and Quantitative Checks

The support theorem is representation-theoretic. The present section records the spectral reading attached to the same primitive defect. The selected carrier (26) is represented by an isolated low spectral sector of a gauge-fixed normal family. The Callias gap isolates the low window, the Riesz projection gives the moving low-sector bundle, the Feshbach-Schur compression eliminates the high sector, and the remaining finite equations are organized by a Kuranishi map. Thus the completed branch is treated here as a finite charge-compatible Schur-Kuranishi problem on the selected carrier. The Callias mechanism [24] and its geometric formulation [25] give the comparison class. Perturbation theory is used in the sense of [26].

At a closed branch ${\Phi }_0$, let $L_{{\Phi }_0}=D{\mathcal{C}}_{{\Phi }_0}$ be the gauge-fixed linearization of the self-reconstruction defect in a local slice. The closed normal representative is the self-adjoint Dirac-Callias closure

$${\mathcal{N}}_{\mathrm{cl}}=\left(\begin{array}{cc}0& L_{{\Phi }_0}^{*}\\ L_{{\Phi }_0}& 0\end{array}\right)+{\Psi }_{\mathrm{Callias}}+{\Psi }_{\mathrm{ext}}+{\Psi }_{\mathrm{cen}}+{\Psi }_{\mathrm{hol}}.$$

(52)

The first summand carries the principal normal symbol. The four zeroth-order slots record, respectively, the Callias isolation, the exterior odd bridge, the central torsor response, and the low-sector holonomy response. Scalar renormalizations, gauge changes, and high-sector Schur remainders are read as completed-branch data. The determinant-line and family-index comparisons are the standard ones for regular Dirac-type families [29]; the corresponding differential K-theoretic language is represented by [30].

The principal normal symbol of $L_{{\Phi }_0}$ is the gauge-fixed Codazzi-Calderon symbol. Algebraic Rainich terms, frozen response terms, holonomy terms, and gap terms are lower order in the normal variable. After the real blow-up, the boundary operator is of Hodge-Dirac type. The representative (52) is called boundary-admissible when the optical middle-spinor projection of this boundary symbol lies in the elliptic homotopy class of Clifford multiplication on ${\mathbb{CP}}^1$. In that case it represents the same boundary class as the primitive twisted link Dirac operator associated with (22).

Let B be a local parameter domain in the completed branch problem, and let $N_{+}\left(b\right)$ be the positive gauge-fixed normal family obtained from (52). The family is called Schur-admissible on B when the Codazzi-Callias gap separates the selected low window from the high sector and the corresponding low-high coupling is subcritical. It is called charge-compatible when the Schur-compressed low operator preserves the sector projections supplied by Corollary 1. For $b\in B$, the Riesz projection onto the selected low window is

$$P_b={\displaystyle \frac{1}{2\pi i}}{\int }_{\gamma }{(z-N_{+}\left(b\right))}^{-1}dz,$$

(53)

where $\gamma$ encloses the low cluster and no other spectrum. The finite-rank bundle ${\mathcal{E}}_{\mathrm{low}}\to B$ is defined by ${\mathcal{E}}_{\mathrm{low},b}=P_b\mathcal{H}$.

The finite completed-branch datum consists of the transported Codazzi-Gauss sector projections, the determinant remainder, the central torsor response, the projected low-sector connection, the effective Schur operator, and the finite bridge or contact coefficients. These data are evaluated on the carrier fixed by (26); they do not reselect the carrier while the Callias-Schur gap remains open. The structural entries are fixed before the finite closure equation is solved. The remaining entries are Schur-Berry coefficients of the completed branch. When Assumption 4 is imposed, this structural-soft split is detected by the finite stiffness operator.

5.1. Low-Sector Connection and Projected Gauge Field

The selected carrier is represented analytically when the low cluster has rank $dim(E_3\oplus E_2)$ and carries the same separated block labels as (26).

Proposition 6 

(Callias-Schur low-sector representative). Let Γ satisfy the hypotheses of Theorem 1. Assume that the gauge-fixed normal family $N_{+}\left(b\right)$ is obtained from a boundary-admissible representative and satisfies the Codazzi-Callias gap and the subcritical Schur bound on B. Then (26) is represented by an isolated finite-rank bundle ${\mathcal{E}}_{\mathrm{low}}\to B$. Higher-sector data enter the effective low operator through the Schur complement and do not change the selected support while the gap and the subcritical bound remain open. If the family is charge-compatible, the Schur-compressed low operator also preserves the Codazzi-Gauss sector projections.

Proof. 

The gap gives the spectral separation of the low window from the high sector. Hence (53) is well-defined and varies smoothly with b. The principal support is the one selected in Theorem 1. High-sector perturbations contribute to the effective low operator by the Schur complement. Under the subcritical bound this correction is relatively bounded and cannot create a new principal low block. If charge-compatibility is imposed, the Schur-compressed low operator preserves the sector projections by definition. □

Let ${\nabla }^0$ be the trivial connection on the ambient Hilbert bundle over B. The projected Berry-Wilczek-Zee connection on the isolated low-sector bundle is

$${\nabla }^{\mathrm{BWZ}}=P{\nabla }^{0}P.$$

(54)

Its curvature is

$$F^{\mathrm{BWZ}}=P\left(dP\right)\wedge \left(dP\right)P.$$

(55)

The comparison with Berry phase [27] and its non-Abelian form [28] is direct. Wilson-loop realizations of the Wilczek-Zee phase give a physical comparison [81].

On the full selected carrier, (54) is read as the projected gauge connection of the moving low-sector bundle. In a local orthonormal frame $e_i\left(b\right)$ of ${\mathcal{E}}_{\mathrm{low}}$, its connection matrix is

$${\left(A_{\mu }^P\right)}_{ij}=\langle e_i,{\partial }_{\mu }{e}_j\rangle .$$

(56)

Under a local change of low-sector frame $u\left(b\right)$, the usual gauge transformation law is obtained:

$$A_{\mu }^P⟼u^{-1}{A}_{\mu }^{P}u+u^{-1}{\partial }_{\mu }u.$$

(57)

After the determinant reduction (29), the structure algebra is

$${\mathfrak{g}}_{\mathrm{low}}=\mathfrak{s}(\mathfrak{u}\left(C\right)\oplus \mathfrak{u}\left(W\right))\simeq \mathfrak{s}(\mathfrak{u}\left(3\right)\oplus \mathfrak{u}\left(2\right)),$$

(58)

where $\mathfrak{s}$ denotes the trace-zero determinant direction fixed by the split top form. With the sign convention fixed by the chosen Lorentzian action, the local quadratic invariant of the projected curvature is

$$p_{\Lambda }^P={\displaystyle \frac{1}{4}}\sum _a{ϵ}_a{g}_a^{-2}{tr}_a\left(F_{\mu \nu }^{P,a}{F}^{P,a\mu \nu }\right),$$

(59)

where a runs over the simple and central factors of (58). The constants $g_a$ and threshold normalizations are completed spectral data. The standard Yang-Mills-Higgs-Dirac conventions used for the local field-theory reading are those of [31,32].

The projected curvature stress of (59) is the bosonic comparison object used in the Alena branch reading. Let ${{\rm Y}}_{\mu \nu }^P$ denote the gauge-side Hilbert stress obtained from (59) by metric variation with respect to the reference metric $\eta$. To avoid conflict with the hypercharge notation, the Alena branch-response tensor is denoted in this paragraph by $Y_{\mu \nu }^{\mathrm{br}}\left[k\right]$.

A projected gauge sector is called Alena-compatible on a connected patch when ${{\rm Y}}_{\mu \nu }^P$ is symmetric, traceless, and of non-null Rainich type,

$${{\rm Y}}^{P\mu }{}_{\mu }=0,{{\rm Y}}^{P\mu }{}_{\alpha }{\rm Y}^{P\alpha }{}_{\nu }={\rho }^2{\delta }^{\mu }{}_{\nu },\rho >0,$$

(60)

the associated $2+2$ splitting is non-degenerate, and the finite-anisotropy ratio $r=u/p_{\Lambda }^P$ satisfies $\left|r\right|<1$, where $u={{\rm Y}}_{UU}^P$ in an adapted orthonormal tetrad. The local Alena normalization is fixed by requiring $p_0=p_{\Lambda }^P{(1-r)}^2/16$ to be constant on the patch. In this sector the scalar $p_{\Lambda }$ in (3) is read as $p_{\Lambda }^P$ through (59).

The adapted Rainich splitting and the constant $p_0$ determine an anisotropic Lorentzian branch metric $k_{\mu \nu }$, uniquely up to the adapted $2+2$ frame freedom, such that

$$p_{\Lambda }^P=p_0{K}^2,Y_{\mu \nu }^{\mathrm{br}}\left[k\right]=-{{\rm Y}}_{\mu \nu }^P,$$

(61)

where $K={\eta }^{\mu \nu }{k}_{\mu \nu }$ and

$$Y_{\mu \nu }^{\mathrm{br}}\left[k\right]=p_0{K}^2\left({\displaystyle \frac{4}{K}}{k}_{\mu \nu }-{\eta }_{\mu \nu }\right).$$

(62)

Indeed, in an adapted $\eta$-orthonormal tetrad $(U,N,W,S)$, the non-null Rainich stress has canonical components ${{\rm Y}}_{UU}^P=u$, ${{\rm Y}}_{NN}^P=-u$, and ${{\rm Y}}_{WW}^P={{\rm Y}}_{SS}^P=u$. With $tanh\chi =r$, define

$$k_{\mu \nu }=U_{\mu }{U}_{\nu }-N_{\mu }{N}_{\nu }-e^{2\chi }\left(W_{\mu }{W}_{\nu }+S_{\mu }{S}_{\nu }\right).$$

(63)

Then $K=2+2e^{2\chi }=4/(1-r)$, and the normalization above gives $p_0{K}^2=p_{\Lambda }^P$. Since $4/K=1-r$ and $e^{2\chi }=(1+r)/(1-r)$, substitution in (62) gives $Y_{UU}^{\mathrm{br}}=-u$, $Y_{NN}^{\mathrm{br}}=u$, and $Y_{WW}^{\mathrm{br}}=Y_{SS}^{\mathrm{br}}=-u$. Thus (61) holds tensorially. The sign in $Y^{\mathrm{br}}=-{{\rm Y}}^P$ is therefore the Alena-Rainich sign convention induced by moving the projected curvature stress to the branch-response side of the reconstruction.

The corresponding bosonic checks are finite completed-branch checks. They are listed in Table 6. The carrier theorem fixes the space on which they are evaluated; the numerical normalization is fixed only after the projected connection and the determinant scalar have been completed.

5.2. Schur Compression and Sectoral Operators

The Riesz variation is controlled by the low-high detector. For a tangent direction $X\in T_{b}B$, put

$$V_X=(1-P_b)\left({\partial }_X{N}_{+}\left(b\right)\right)P_b.$$

(64)

The first variation of $P_b$ is obtained by solving the Sylvester equation between the low and high spectral blocks. Hence nonzero detector components give nonzero low-high variations as long as the gap remains open. Substitution into (55) gives the corresponding non-Abelian low-sector curvature component.

The central torsor of Section 4.1 supplies the algebraic family-response space. The analytic question is whether the completed branch produces Schur-visible detector components outside the circulant algebra $\mathbb{C}\left[S\right]$. Let ${\Pi }_{\mathrm{circ}}$ denote the Hilbert-Schmidt orthogonal projection from $M_3\left(\mathbb{C}\right)$ onto $\mathbb{C}\left[S\right]$. Since the Weyl monomials $Z^m{S}^n$ are Hilbert-Schmidt orthogonal, every monomial with nonzero clock degree has zero circulant projection.

In a one-gap central model with gap $\Delta >0$, Schur-visible detector components $V_X=aZ$ and $V_Y=bS$ give

$$F_{XY}^{\mathrm{cen}}={\Delta }^{-2}\left(\overline{a}b{Z}^{†}S-\overline{b}a{S}^{†}Z\right),$$

(65)

and

$${\Pi }_{\mathrm{circ}}\left(F_{XY}^{\mathrm{cen}}\right)=0,\parallel F_{XY}^{\mathrm{cen}}{\parallel }_{\mathrm{HS}}^2=6{\left|ab\right|}^2{\Delta }^{-4}.$$

(66)

Thus a Schur-visible phase direction together with a Schur-visible adjacent direction gives a genuine non-circulant central curvature component. For a multi-gap high sector, the coefficient of the same Weyl term is replaced by a finite Schur sum over high gaps. Its vanishing is a spectral cancellation condition.

Leading central family operators supported only in the circulant algebra have the form

$$Y_f^{\left(0\right)}=A_f\mathbf{1}+{\beta }_{f}S+{\overline{\beta }}_f{S}^{†}.$$

(67)

They are simultaneously diagonalized by the finite Fourier transform on (44). Hence a family sector whose completed response remains inside $\mathbb{C}\left[S\right]$ has no physical mixing from this central layer. The central torsor supplies the family labels, while physical family mixing requires a component outside $\mathbb{C}\left[S\right]$. If one leading sector has simple spectrum, such a component gives the sectoral basis rotation. A CP-sensitive invariant also requires a non-removable complex component in the same non-circulant Schur-Berry correction. Thus family enumeration, family mixing, and CP sensitivity occupy different finite parts of the completed branch.

Proposition 7 

(Central mixing criterion). If the completed family operators of two sectors remain in the circulant algebra $\mathbb{C}\left[S\right]$, the central layer gives no relative mixing between these sectors. If one sector has simple leading central spectrum and the Schur-Berry correction contains a nonzero component orthogonal to $\mathbb{C}\left[S\right]$, then the corresponding central-sector eigenbasis is rotated. A CP-sensitive invariant additionally requires a non-removable closed-cycle phase carried by a non-circulant correction and sectoral non-commutation.

Proof. 

Operators in $\mathbb{C}\left[S\right]$ are functions of the same shift and are diagonalized by the Fourier basis on (44). Hence their relative central eigenbasis is unchanged. A component orthogonal to $\mathbb{C}\left[S\right]$, such as the Schur-Berry component detected in (65)-(66), gives off-diagonal central entries in the Fourier basis when the leading central spectrum is simple. The sectoral basis rotation then follows by first-order perturbation theory. A closed-cycle phase is frame-independent by (49)-(50); it gives a CP-sensitive scalar only when it appears in non-commuting sectoral data. □

Under Assumption 3, the complex adjacent-loop row of Table 7 has a single non-removable phase source. Then a CP-sensitive sectoral invariant tests the central torsor holonomy rather than an independent family phase. In this reading the torsor supplies the candidate phase, and CP sensitivity appears only when that phase is carried by a non-circulant Schur-Berry component. If the completed Schur-Berry graph contains an additional non-trivial phase cycle, the corresponding branch is outside the single-holonomy class.

The closed-branch finite operators are obtained by Feshbach-Schur compression. If K is a closed-branch operator with low-high block decomposition $K_{\alpha \beta }$, and if ${\Pi }_f$ denotes the finite projector onto a sector f, the corresponding Yukawa-type operator is

$$Y_f={\Pi }_f{K}_{\mathrm{eff}}{\Pi }_f,K_{\mathrm{eff}}=K_{LL}-K_{LH}{K}_{HH}^{-1}{K}_{HL}+\dots ,$$

(68)

whenever the Schur expansion is defined in the admissible window. The high block is invertible under the Callias-Schur gap and the subcritical bound. The remaining finite variables are constrained by the local Kuranishi equation of the completed branch. Equation (68) gives the structural form of the sectoral operator; its numerical eigenvalues are not fixed by the carrier theorem.

Proposition 8 

(Charge-neutral Schur compression). Assume that the low-high splitting commutes with the Codazzi-Gauss sector projections and that the closed-branch operator K preserves these projections. Then, whenever the Schur expansion in (68) is defined, the Schur-compressed low operator preserves the induced low-sector projections.

Proof. 

Let ${\Pi }_a^L$ and ${\Pi }_a^H$ be the low and high parts of a sector projection. Since K preserves the full sector projections, the diagonal blocks preserve sectors and the off-diagonal blocks intertwine the corresponding low and high sector parts. The high block inverse has the same sector decomposition on the admissible Schur window. Hence the Schur term in (68) has no off-sector low component. The compressed operator therefore commutes with every ${\Pi }_a^L$. □

Proposition 9 

(Finite Kuranishi reconstruction map). Assume the Callias-Riesz low-high splitting and the invertibility of the high block in the completed branch. Then the local closure equation in a gauge slice reduces to a finite Kuranishi map κ on the low variables. Its zeros parametrize the local closed branches in the slice. After a finite metric has been chosen, the map ${\mathcal{R}}_{\mathrm{fin}}\left(x\right)=x-{\mathbf{G}}^{-1}\kappa \left(x\right)$ has the same fixed points. At a zero $x_{*}$, the finite stiffness operator is $D\kappa {\left(x_{*}\right)}^{*}D\kappa \left(x_{*}\right)$.

Proof. 

The high equation has invertible linearization by the Schur-admissible gap. The implicit-function theorem solves the high variables as functions of the low variables. Substitution gives the finite obstruction map $\kappa$. Its zeros are exactly the solutions of the full local equation in the chosen slice. The fixed-point statement follows from the invertibility of $\mathbf{G}$. The quadratic part of the reduced closure defect at a zero is represented by $D{\kappa }^{*}D\kappa$. □

The following completed-branch conditions are used only to organize the finite diagnostics below. They are not inputs to Theorem 1.

Hypothesis 1 

(Determinant torsion-triviality). The flat or torsion determinant holonomy left after Proposition 4 is trivial on the resolved worldline class used by the completed branch.

Hypothesis 2 

(Oriented Pfaffian Majorana branch). The completed neutral singlet Majorana channel is represented by an oriented Pfaffian line whose square is the determinant line of the corresponding complexified neutral branch.

Hypothesis 3 

(Single-holonomy Schur-Berry branch). After charge-sector reduction and determinant torsion removal in the sense of Assumption 1, the phase part of the finite Schur-Berry detector graph is generated by the adjacent central torsor holonomy ${\theta }_{\Gamma }$, whose oriented closed cubic is recorded in (49). For homology-pure sectoral coefficients, all non-removable phases are then read as integral multiples of ${\theta }_{\Gamma }$.

Hypothesis 4 

(Structural stiffness gap). At the completed closed point, the finite stiffness operator of Proposition 9 has a positive gap on the structural charge, determinant, central, and, when Assumption 2 is imposed, Pfaffian directions after residual moduli have been removed. The remaining soft directions are the Schur-Berry spectral coefficients.

The local weak bridge is the odd Clifford insertion from $W\oplus W^{*}$ between (32) and (40). It supplies the one-Higgs Dirac channels on the finite module. The Majorana and contact classes described after (41) are therefore read as sectoral closed-branch operators of the form (68). The finite representation content fixes which channels are neutral. The completed Schur-Kuranishi operator fixes their coefficients.

The local gauge equation is recovered from the same projected scalar. For a compactly supported variation of the projected connection, $\delta F_{\mu \nu }^P=D_{\mu }\delta A_{\nu }^P-D_{\nu }\delta A_{\mu }^P$, one obtains

$$\delta \int p_{\Lambda }^{P}d{\mathrm{vol}}_{\eta }=-\int \sum _a{ϵ}_a{g}_a^{-2}{tr}_a\left(D_{\mu }{F}^{P,a\mu \nu }\delta A_{\nu }^P\right)d{\mathrm{vol}}_{\eta }+\mathrm{boundary}.$$

(69)

Thus the source-free local gauge equation is $D_{\mu }{F}^{P,\mu \nu }=0$, with the standard right-hand side after external or matter currents have been included.

On the branch spinor bundle tensored with (32), the completed low-sector covariant derivative contains the branch spin connection and the projected gauge connection:

$$D_{\mu }^{\mathrm{low}}={\partial }_{\mu }+{\omega }_{\mu }^{\mathrm{branch}}+A_{\mu }^P.$$

(70)

The corresponding first-order Dirac term is the local fermionic kinetic term on the reconstructed carrier. Masses, Yukawa coefficients, mixing, running, and threshold conversion remain completed spectral data.

5.3. Vacuum, Mass, Mixing, and Decay Diagnostics

The determinant line of (52) is the natural place for scalar normalizations, eta data, phase holonomies, and the relative determinant scale reading. The neutral Schur cell is used only after the carrier, the determinant convention, and the hypercharge convention have been fixed. It is a diagnostic of the completed branch, not an input to Theorem 1. A numerical entry is called a branch prediction only when the determinant target, the Schur-Berry matching coefficient, or the bridge/contact matrix element entering it has been fixed independently by the same closed branch. Otherwise it is recorded as a diagnostic or a correlation test. The entries below have three statuses: fixed carrier weights, conditional completed-branch readings, and numerical Schur-Kuranishi diagnostics. They probe different blocks of the same finite Kuranishi closure problem: determinant normalization, scalar leakage, vortex-Yukawa deformation, Pfaffian neutral mass, non-circulant family mixing, phase holonomy, and contact-channel opening.

Let A, B, and C be the reduced neutral-cell entries for the weak-angular, determinant, and scalar radial directions. The constants used in the neutral-cell diagnostic are summarized in Table 8. The entries A, B, and C are completed-branch finite coefficients, while $C_0$, $u_{\mathrm{prim}}$, and $S_{\mathrm{prim}}$ fix the primitive benchmark normalization used below. None of these data is part of the carrier theorem.

The primitive neutral-cell weights are fixed by the selected low carrier. The weak-angular reservoir sees $C\oplus W\oplus W^{*}$, while the radial scalar direction sees the rank-five carrier $C\oplus W$. Hence, with $dimC=3$ and $dimW=2$,

$$A_0={\displaystyle \frac{dimC}{dimC+2dimW}}={\displaystyle \frac{3}{7}},B_0=A_0{\displaystyle \frac{dimW}{dimC+2dimW}}={\displaystyle \frac{6}{49}},C_0=A_0{\displaystyle \frac{dimC}{dimC+dimW}}={\displaystyle \frac{9}{35}}.$$

(71)

The constant-curvature representative in the primitive class (22) is normalized by

$${\displaystyle \frac{1}{2\pi }}{\int }_{{\mathbb{CP}}^1}{F}_{\Gamma }=1,{\Phi }_{\Gamma }={\int }_{{\mathbb{CP}}^1}{F}_{\Gamma }=2\pi .$$

(72)

The primitive determinant scalar used below is the product-link Chern-Weil normalization of the same primitive flux:

$$S_{\mathrm{prim}}={\int }_{{\mathbb{CP}}_{\Gamma }^1\times {\mathbb{CP}}_{\Gamma }^1}{\pi }_1^{*}{F}_{\Gamma }\wedge {\pi }_2^{*}{F}_{\Gamma }={\Phi }_{\Gamma }^2=4{\pi }^2.$$

(73)

Here ${\pi }_1$ and ${\pi }_2$ are the two projections from ${\mathbb{CP}}_{\Gamma }^1\times {\mathbb{CP}}_{\Gamma }^1$. Thus the factor $4{\pi }^2$ is the ordinary product integral of the primitive Chern flux. Since $dim{\Lambda }^{\mathrm{even}}V=16$, the scalar unit used in the first neutral Schur step is $u_{\mathrm{prim}}=1/\left(16S_{\mathrm{prim}}\right)=1/\left(64{\pi }^2\right)$.

The first scalar step is read additively in the Schur Hessian. If the scalar step is split-invariant and preserves the determinant-line tangent condition on $C\oplus W$, then it has the form

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

(74)

Indeed, ${Tr}_{C\oplus W}\delta S_{\mathrm{sc}}=0$ gives $3u_{\mathrm{prim}}+2\delta A=0$. Thus

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

(75)

The coefficient $-3/2$ is the dimension ratio $-dimC/dimW$. The correction is an additive transfer in the Schur gap entries, not a logarithmic determinant constraint on the eigenvalues.

The scale-free link angle is defined by

$${sin}^2{\theta }_{\mathrm{link}}={\displaystyle \frac{B}{A+B}}.$$

(76)

In the primitive neutral-cell reading used here, (76) gives

$${sin}^2{\theta }_{\mathrm{link}}=0.223184\dots .$$

(77)

The comparison value extracted from the on-shell masses is

$${sin}^2{\theta }_W^{\mathrm{on}-\mathrm{shell}}=1-{\displaystyle \frac{M_W^2}{M_Z^2}}.$$

(78)

Using the values quoted in [37], (78) gives ${sin}^2{\theta }_W^{\mathrm{on}-\mathrm{shell}}=0.22320949$. Hence the difference between (77) and (78) is $-2.54\times {10}^{-5}$, or about $0.10$ propagated standard deviations when the quoted $M_W$ and $M_Z$ errors are used. The effective leptonic mixing angle is a scheme-dependent observable and is not identified with (78).

The same finite Schur block can be read with a primitive collar scale. The rank-split scalar determinant contribution is ${(dimW/dimC)}^2{u}_{\mathrm{prim}}=1/\left(144{\pi }^2\right)$. With the spin-vorticity normalization $g=1/2$ fixed in [39], the zero-remainder relative determinant scale is

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

(79)

A finite scalar determinant remainder may be recorded by $v_{\mathrm{EW}}=e^{-{\delta }_{\mathrm{sc}}}{v}_{\mathrm{EW}}^{\mathrm{prim}}$. Here $M_{\mathrm{Pl}}$ denotes the non-reduced Planck mass used in the determinant-scale normalization. The radial determinant ratio is a normalization of the scalar direction, not an additional neutral-cell coefficient. In the canonical primitive determinant normalization, the finite target in (79) is fixed by (73) and by the rank-split scalar unit above; outside this normalization it remains a completed-branch determinant diagnostic.

With the same central constants as above, (79) gives $v_{\mathrm{EW}}^{\mathrm{prim}}=246.2205\mathrm{GeV}$. The Fermi value $v_F={\left(\sqrt{2}{G}_F\right)}^{-1/2}$, with $G_F$ taken from [37], is $246.2196719\mathrm{GeV}$. The zero-remainder primitive value differs from $v_F$ by $8.3\times {10}^{-4}\mathrm{GeV}$, equivalently by $3.4$ ppm. In the determinant-scale reading this is recorded as the finite scalar remainder ${\delta }_{\mathrm{sc}}\simeq 3.36\times {10}^{-6}$. The associated $G_F$ shift is fixed by the same scale relation and is not counted as a separate neutral-cell pull.

At the zero-remainder primitive scale, the neutral cell gives the diagnostic mass reading

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

(80)

Substitution into (80) gives

$$m_{\gamma }^{\mathrm{prim}}=0,m_W^{\mathrm{prim}}=80.3710\mathrm{GeV},m_Z^{\mathrm{prim}}=91.1885\mathrm{GeV},m_h^{\mathrm{prim}}=125.2403\mathrm{GeV}.$$

(81)

Relative to the current PDG averages [37], the zero-remainder values in (81) differ by $0.13$, $0.26$, and $0.37$ quoted standard deviations, respectively. If the scalar determinant remainder is fixed by $v_{\mathrm{EW}}=v_F$, the corresponding mass readings differ by $0.11$, $0.11$, and $0.36$ quoted standard deviations. The comparison is a neutral-cell diagnostic after the scalar-scale convention has been specified.

The Higgs mode is the radial reading of the same gap. If the primitive neutral cell is locked under the radial scale variation, ${\partial }_{logv}A={\partial }_{logv}B={\partial }_{logv}C=0$, the $hWW$ and $hZZ$ couplings are obtained by differentiating (80) with respect to v:

$$g_{hWW}={\displaystyle \frac{\partial m_W^2}{\partial v}}={\displaystyle \frac{2m_W^2}{v}},g_{hZZ}={\displaystyle \frac{\partial m_Z^2}{\partial v}}={\displaystyle \frac{2m_Z^2}{v}}.$$

(82)

Thus the Standard-Model normalization of the gauge-boson Higgs couplings is recovered as locked gap scaling. A radial leakage operator $L_{\mathrm{rad}}={\partial }_{logv}{S}_{\mathrm{eff}}$ gives

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

(83)

For a trace-neutral scalar leakage with ${\ell }_B=0$ one obtains the correlation

$${\kappa }_Z-1={\displaystyle \frac{A}{A+B}}({\kappa }_W-1).$$

(84)

This is a finite neutral-cell test. A nonzero deviation from (84) measures radial determinant leakage.

The fermionic reading is compared with the spin-vorticity vortex relation of [39],

$$y_f=cosh{\phi }_f-1,m_f={\displaystyle \frac{v}{\sqrt{2}}}{y}_f.$$

(85)

If the vortex angle is locked, ${\partial }_{logv}{\phi }_f=0$, then

$$g_{hff}={\displaystyle \frac{\partial m_f}{\partial v}}={\displaystyle \frac{m_f}{v}}.$$

(86)

More generally,

$${\kappa }_f-1=coth\left({\displaystyle \frac{{\phi }_f}{2}}\right){\partial }_{logv}{\phi }_f.$$

(87)

The finite test is therefore $Q_f=({\kappa }_f-1)tanh({\phi }_f/2)$. The locked vortex branch gives $Q_f=0$. A nonzero value would measure a radial deformation of the vortex angle rather than a new carrier coefficient.

The singlet Majorana channel of Table 4 has a separate half-flux diagnostic. Under Assumption 2, the determinant normalization (73) assigns the $B-L$-breaking neutral singlet pair the charge-$1/2$ flux and hence the action

$$S_M={\displaystyle \frac{1}{4}}{S}_{\mathrm{prim}}={\pi }^2,M_R=M_{\mathrm{Pl}}exp(-S_M).$$

(88)

With the same Planck normalization as in (79), this gives $M_R=6.31\times {10}^{14}\mathrm{GeV}$. For a unit neutral Dirac Yukawa in (85), $m_D=v_F/\sqrt{2}=174.10\mathrm{GeV}$, and the type-I seesaw diagnostic gives

$$m_{\nu }^{\mathrm{I}}={\displaystyle \frac{m_D^2}{M_R}}=0.0480\mathrm{eV}.$$

(89)

Thus the half-flux identification is the oriented Pfaffian square-root reading of the completed neutral determinant line in the Pfaffian class. In this form the seesaw estimate is a test of the Pfaffian square-root class of the neutral branch.

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

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

(90)

A channel is open only when (90) is non-negative and the corresponding exterior or weak-bridge matrix element is nonzero. For the Higgs scalar, the locked vortex reading gives the tree-level fermionic widths

$$\Gamma (h\to f\overline{f})=N_c{\displaystyle \frac{m_h{m}_f^2}{8\pi v^2}}{\left(1-{\displaystyle \frac{4m_f^2}{m_h^2}}\right)}^{3/2}.$$

(91)

Using running quark masses at the Higgs scale for the quark channels, as in the standard Higgs-decay treatment [82], the primitive gap reading gives the hierarchy recorded in Table 9. The values are intended as a zero-remainder diagnostic; higher-order QCD and electroweak corrections are not part of the carrier theorem.

The full on-shell thresholds $h\to WW$ and $h\to ZZ$ are closed at (81), since $m_h-2m_W=-35.50\mathrm{GeV}$ and $m_h-2m_Z=-57.14\mathrm{GeV}$. The channels $W{W}^{*}$ and $Z{Z}^{*}$ are therefore off-shell tails of the weak resolvent. The loop channels $gg$, $\gamma \gamma$, and $Z\gamma$ are determinant derivatives of bridge Hessians rather than elementary tree punctures. With the entries of Table 9, the summed Higgs width is approximately $4.0\mathrm{MeV}$, consistent with the Standard-Model Higgs-width scale quoted in [37].

The ordinary weak decays have the same threshold form. For $t\to bW$, using the masses in [37] and (81), one has ${\Delta }_{t\to bW}\simeq 88\mathrm{GeV}$. With the standard charged-current normalization this gives ${\Gamma }_t\simeq 1.35\mathrm{GeV}$ after the usual QCD correction. The muon and tau decays are lower-island weak-bridge transitions, while the electron is the lowest charged-lepton island. The absence of a local diquark one-Higgs bridge is the local reason why the weak bridge in Table 4 is not a proton-decay channel.

The baryon-violating contact reading is constrained by Table 5. A tunnelling assignment $M_X=M_{\mathrm{Pl}}exp(-S_{\mathrm{prim}})$ would put the contact scale at $O\left({10}^2\right)\mathrm{GeV}$ and is therefore excluded as a proton-decay reading. The doubled top-form assignment places the contact at the full tower scale. For an order-one finite coefficient and $M_X\simeq M_{\mathrm{Pl}}$, the standard dimension-six estimate gives ${\tau }_p\sim {10}^{48}\mathrm{yr}$. A two-top-form normalization suppression by ${\left(2\pi \right)}^{-4}$ gives $M_X\simeq 3\times {10}^{17}\mathrm{GeV}$ and ${\tau }_p\sim {10}^{41}\mathrm{yr}$. In this minimal contact-scale branch, the proton-decay contact is outside the reach of present and next-generation lifetime searches; an observed $e^{+}{\pi }^0$ signal around ${10}^{35}\mathrm{yr}$ would rule out this assignment.

For the family sector, write the Schur-completed non-circulant Weyl coefficient as

$${\Xi }_f=\sum _r{\overline{a}}_{f,r}{b}_{f,r}{\Delta }_{f,r}^{-2}.$$

(92)

If the non-circulant correction is represented at first order by ${\lambda }_{f}Z+{\overline{\lambda }}_f{Z}^{†}$, with ${\lambda }_f$ proportional to (92), and if the leading circulant eigenvalues are $y_{f,k}$, then the first change of the sectoral eigenbasis has adjacent entries

$${\left({\Theta }_f\right)}_{k+1,k}={\displaystyle \frac{{\lambda }_f}{y_{f,k}-y_{f,k+1}}}.$$

(93)

For two sectors $f,g$, the corresponding relative mixing is, to first order,

$$V_{fg}=\mathbf{1}+{\Theta }_g-{\Theta }_f+O\left({\Theta }^2\right).$$

(94)

Thus the completed branch must determine both the non-circulant coefficients and the sectoral circulant gaps. A PMNS analogue requires the corresponding lepton-sector coefficients and, for neutral leptons, the completed Dirac/Majorana selection data. Large lepton-sector angles may arise either from larger non-circulant Schur-Berry coefficients or from smaller neutral-sector family gaps in (93).

The CKM matrix is used as a quark-sector diagnostic in the standard three-angle parametrization [33,34]. The Wolfenstein hierarchy gives the conventional small-parameter comparison [35]. A central matching of the determinant shadow is recorded by

$${\lambda }_{\mathrm{F}}\left(\kappa \right)={\displaystyle \frac{\kappa B}{A+\kappa B}}.$$

(95)

The value $\kappa =1$ corresponds to a common normalization of the neutral and central shadows of the primitive class. The primitive Schur-Berry normalization is read from the even-package hypercharge-square trace and the scalar unit:

$${\kappa }_{\mathrm{SB}}=1+2{Tr}_{{\Lambda }^{\mathrm{even}}V}\left(Y^2\right)u_{\mathrm{prim}}=1+{\displaystyle \frac{5}{48{\pi }^2}}=1.0105542899\dots .$$

(96)

A value of $\kappa$ differing from (96) is read as an additional finite threshold correction in the completed branch. The Schur-length diagnostic is

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

(97)

The phase-sensitive invariant is compared with the closed central cubic (49). In the same normalization,

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

(98)

The invariant J is the usual rephasing invariant of the CKM matrix [36].

Substitution of (96) into (95), (97), and (98) gives

$$s_{12}^q=0.2250095997\dots ,s_{23}^q=0.0413386667\dots ,s_{13}^q=0.0037973610\dots ,J_q^{\mathrm{diag}}=3.24449\times {10}^{-5}.$$

(99)

For comparison, matching (95) to the PDG value ${\lambda }_{\mathrm{CKM}}=0.22501$ quoted in [37] gives

$${\kappa }_{\mathrm{CKM}}=1.0105566095\dots ,{\kappa }_{\mathrm{CKM}}-{\kappa }_{\mathrm{SB}}=2.3\times {10}^{-6}.$$

(100)

Table 10. Primitive Schur-Berry central matching comparison. The reference values are quoted from [37]. Correlations in the global fit are not used in the pull estimates.

Quantity	Diagnostic value	Reference value	Comment
$s_{12}$	$0.225010$	$0.22501\pm 0.00068$	from (96)
$s_{23}$	$0.041339$	$0.{04183}_{-0.00069}^{+0.00079}$	within $1\sigma$
$s_{13}$	$0.003797$	$0.{003732}_{-0.000085}^{+0.000090}$	within $1\sigma$
$A_{\mathrm{CKM}}$	$\sqrt{2/3}=0.81650$	$0.{826}_{-0.015}^{+0.016}$	within $1\sigma$
$J_q$	$3.2445\times {10}^{-5}$	$(3.{12}_{-0.12}^{+0.13})\times {10}^{-5}$	within $1\sigma$

Table 10. Primitive Schur-Berry central matching comparison. The reference values are quoted from [37]. Correlations in the global fit are not used in the pull estimates.

Quantity	Diagnostic value	Reference value	Comment
$s_{12}$	$0.225010$	$0.22501\pm 0.00068$	from (96)
$s_{23}$	$0.041339$	$0.{04183}_{-0.00069}^{+0.00079}$	within $1\sigma$
$s_{13}$	$0.003797$	$0.{003732}_{-0.000085}^{+0.000090}$	within $1\sigma$
$A_{\mathrm{CKM}}$	$\sqrt{2/3}=0.81650$	$0.{826}_{-0.015}^{+0.016}$	within $1\sigma$
$J_q$	$3.2445\times {10}^{-5}$	$(3.{12}_{-0.12}^{+0.13})\times {10}^{-5}$	within $1\sigma$

Using the standard CKM convention and (98), one obtains ${\delta }_q^{\mathrm{diag}}=1.2332\mathrm{rad}$, or $70.{66}^{\circ }$. The corresponding moduli are

$$|V_{\mathrm{CKM}}^{\mathrm{diag}}|=\left(\begin{array}{ccc}0.974349& 0.225008& 0.003797\\ 0.224868& 0.973512& 0.041339\\ 0.008798& 0.040570& 0.999138\end{array}\right).$$

(101)

The associated unitarity-triangle angles are

$${\alpha }_{\mathrm{diag}}=85.{99}^{\circ },{\beta }_{\mathrm{diag}}=23.{39}^{\circ },{\gamma }_{\mathrm{diag}}=70.{62}^{\circ }.$$

(102)

The coefficient (96) is the primitive Schur-Berry normalization used in this diagnostic. The matched value in (100) is retained only as a comparison with the quark-sector reference value. Additional deviations would measure completed threshold corrections in the non-circulant sectoral data.

The lowest cyclic CP-odd test is obtained from the adjacent sectoral amplitudes. If $q_k$ denotes the amplitude entering the $k\mapsto k+1$ step, then rephasing of the three family basis vectors sends $q_k$ to $e^{i({\varphi }_{k+1}-{\varphi }_k)}{q}_k$. Hence the cyclic product is rephasing-invariant, and the first loop-type CP-odd scalar is

$$J_{fg}^{\mathrm{loop}}=\Im \left(q_0{q}_1{q}_2\right).$$

(103)

In the one-gap minimal central model, (103) is proportional to the imaginary part of the Schur-completed central cubic. Physical CP still requires sectoral non-commutation; a common removable phase gives no CP-odd invariant.

The multiplicity count of the central factor is the one already recorded in (51). The simple primitive locked-kernel case has $r=1$ and gives three families. The scalar commutant, the rank r, the sectoral eigenvalue splitting, the Yukawa coefficients, the threshold conversion, and the running-coupling matching remain completed-branch data.

6. Main Results and Conclusions

The local reconstruction can now be stated as one result. The hypotheses are separated according to their role. In the Alena-Codazzi realization, Proposition 1 supplies the local current-Codazzi closure on the punctured collar, and Proposition 2 supplies the integral-current core and the thin-core worldline realization on a primitive regular component. The primitive optical Codazzi defect supplies the link and the primitive line. The scalar-sector two-jet supplies the two non-scalar source channels. Lemma 1 and Corollary 1 supply the corresponding boundary-charge centrality; in the abstract statement this is retained as separated central implementability. The finite determinant count supplies the local determinant kernel. The central torsor supplies the mod-three family factor, and the Schur-Berry completion supplies the mixing and CP diagnostics.

Theorem 2 

(Primitive self-reconstructing carrier). Let Γ be a primitive optical Codazzi defect in a four-dimensional Lorentzian branch. In the Alena-Codazzi realization this input is supplied on a primitive regular component of the limit in Proposition 2. Assume that the resolved link is the projective link (20), that the primitive positive transverse class gives (22), and that the scalar-sector source has transverse order at most two with nonzero separated non-scalar channels as in (24). Then the positive link quantization is the Borel-Weil tower (23). Among separated centrally implementable Toeplitz-visible supports, the minimal support is the carrier (26).

The finite determinant-obstruction kernel is (38). Equivalently, the selected carrier has compact determinant-preserving basis group (30). The determinant-compatible even exterior package is (32), with hypercharge (33), and gives the local one-generation representation content recorded in Table 2.

The determinant global form gives the central reduction (43). The corresponding central response space is (44). If the central response is unblocked, the clock-shift algebra generated by (45) acts irreducibly on this three-dimensional space, and the closed central cubic is (50). The family multiplicity has the form (51). Exact three-family multiplicity is obtained in the simple primitive locked-kernel case.

If a boundary-admissible normal representative satisfies the Codazzi-Callias gap and the subcritical Schur bound, then the selected support is represented by the isolated low-sector bundle defined by (53). The projected connection is (54); after the determinant reduction its structure algebra is (58). In the Alena-compatible non-null Rainich sector, the projected curvature stress reconstructs the branch response by (61). The completed branch supplies the Schur-compressed sectoral operators (68), the local gauge equation (69), the low-sector Dirac covariant derivative (70), and the vacuum, bosonic, fermionic, mass, and mixing diagnostics of Section 5.

Proof. 

The projective link and the primitive line are the content of (20) and (22). In the Alena-Codazzi realization the thin-core source of the worldline is Proposition 2; in the abstract theorem the primitive optical defect is retained as input. The Borel-Weil tower is (23). The source type is supplied by Proposition 3 in the Alena-Codazzi realization, or by the abstract scalar-sector two-jet hypothesis. In the concrete collar realization, Lemma 1 and Corollary 1 supply the boundary-central source, and Corollary 2 gives the carrier. In the abstract formulation the support statement is Theorem 1.

The determinant group, exterior package, and hypercharge are constructed in Section 4. The determinant kernel is the finite degree count (39), equivalently Corollary 4. The central torsor, clock-shift response, closed cubic, and family multiplicity are (44)-(51), with the count recorded in Proposition 5. The analytic low-sector representative is Proposition 6. The projected connection and Schur-compressed sectoral operators are (54) and (68), and the central mixing criterion is Proposition 7. □

Corollary 6 

(Alena-type current-residual realization). Let the collar be a primitive non-degenerate current-Codazzi closed Alena-type collar with residual scalar (3). Assume that its thin-core realization satisfies Proposition 2, and assume two-channel genericity in the sense of Proposition 3. Then the collar supplies the source, primitive-defect, and boundary-charge centrality hypotheses of Theorem 2 through Proposition 1, Proposition 2, Lemma 1, and Corollary 1. If the collar is Schur-admissible, the selected carrier is represented by the isolated low-sector bundle of Proposition 6. If the completion is charge-compatible, the Schur-compressed low operator preserves the same charge-sector split.

Proof. 

The Alena residual scalar gives the current-residual collar. Proposition 1 gives the trace-adjusted Codazzi closure and fixes the scalar and translational coefficient up to the multiplier constant and the remaining current compatibility. Proposition 2 gives the integral-current limit and its primitive worldline realization on the regular component used here. Proposition 3 gives the separated non-scalar associated-graded source, Lemma 1 gives its principal boundary-charge reading, and Corollary 1 gives centrality in the effective local algebra. The primitive transverse class gives the projective link and the primitive line used in Section 3. Theorem 2 then applies. Schur-admissibility gives the analytic representative by Proposition 6. □

The status of the ingredients is summarized in Table 11. The entries are separated into structural local results, completed-branch reductions, and numerical or conditional diagnostics. This separation is useful because the carrier theorem, the determinant package, the family torsor, the Schur-Kuranishi map, and the phenomenological readings enter at different levels of the construction.

The finite local reconstruction has the following form. In the Alena-Codazzi realization, a current-Codazzi closed thin-core collar supplies an integral-current limit, and its primitive regular component supplies the worldline defect. The defect defines the projective link and the primitive line. The scalar-sector two-jet defines the non-scalar source types. In the Alena-Codazzi realization these types are read as the two principal Codazzi-Gauss link charges of Lemma 1, with centrality supplied by Corollary 1. On regular curved collars the same charge reading is transported by Lemma 2. The Borel-Weil tower and Toeplitz visibility then select the rank-five block $E_3\oplus E_2$. The degree-one case is the unique separated-charge threshold by Corollary 3. The finite determinant-obstruction kernel gives the split determinant basis group; the even exterior package then gives the one-generation module. Proposition 4 identifies the corresponding local determinant flatness. The same determinant global form supplies the central mod-three response.

The analytic representative gives the passage to local field theory. The low spectral sector carries the projected Berry-Wilczek-Zee connection, and this connection is read as the local gauge field on the reconstructed carrier. Its curvature-square scalar gives the projected bosonic invariant entering the Alena branch comparison. The odd exterior bridge gives the finite Dirac channels. The high sector is eliminated by the Schur complement. Charge-neutral closed representatives give charge-compatible Schur compression by Proposition 8. The completed local closure is organized by the finite Kuranishi map of Proposition 9. Its stiffness operator separates structural directions from the Schur-Berry spectral coefficients when Assumption 4 holds. The sectoral finite operators, Yukawa matrices, threshold masses, decay constants, CKM/PMNS matrices, and CP-sensitive loop invariants are completed-branch quantities.

The quantities collected in Section 5.3 are therefore diagnostic outputs of the completed spectral layer. The normalization $S_{\mathrm{prim}}$ used there is the product-link Chern-Weil integral (73) of the primitive flux, not an independent numerical fit. The neutral-cell angle, the mass readings, the primitive electroweak scale, the Higgs-coupling and decay readings, the proton-decay contact estimate, the CKM matching coefficient, and the loop-type CP invariant are finite tests of a completed branch. They probe different blocks of the same finite Kuranishi closure problem. The half-flux seesaw value (89) becomes a Pfaffian square-root determinant reading under Assumption 2. A single CP phase is obtained only in the single-holonomy class of Assumption 3. The numerical use of these diagnostics requires the finite determinant target, the Schur-Berry matching operator, the bridge matrix elements, and the sectoral Yukawa or vortex data. The local carrier theorem fixes the representation support on which those quantities are evaluated.

The exterior bidegree refines the finite module by the structural charge (34). The weak bridge, the doubled top-form contact, and the Majorana class then occupy separated Schur-Kuranishi channels. The weak bridge gives the local Dirac and Yukawa sector, the doubled top-form contact gives the high-scale baryon-violating contact class, and the oriented Pfaffian condition gives the $B-L$-breaking neutral singlet sector. These channels are evaluated on the same selected carrier but belong to different finite blocks.

The construction also gives a compact form of the self-reconstruction loop (1). The same defect carries the geometric link, the topological class, the quantized representation support, and the spectral low-sector representative. The closure condition (2) is then read in the finite sector through the Kuranishi reduction of Proposition 9: the Schur-compressed operators, determinant data, central response, projected connection, and finite bridge coefficients must reconstruct the same branch data from which the defect was obtained. The Alena-type collar gives one explicit current-residual realization of this local loop.

Several data are intentionally left in the completed branch problem. These include the rank r in (51), the scalar determinant remainder beyond the canonical primitive target, the high-sector Schur sums, the sectoral Yukawa eigenvalues, the non-circulant family corrections beyond the primitive central matching, and the running-coupling matching. They are not required for the carrier reconstruction. They are required for a completed phenomenological model on the selected carrier.

7. Discussion

The construction is local and finite. It starts from one primitive defect and keeps the geometric, topological, representation-theoretic, and spectral readings attached to that defect. This is the main difference between the present reconstruction and mechanisms in which an internal space, a finite algebra, or a gauge group is supplied as external structure. The comparison with Kaluza-Klein geometry is therefore limited to the common use of bundle and connection data [1]. Dynamical principal-bundle formulations give a closer comparison at the level of moving gauge variables [83]. Standard principal-bundle and connection conventions are those of [84].

The completed branch has a finite form in the present setting. The carrier, determinant kernel, exterior module, and central torsor are structural data of the primitive defect. The remaining vacuum, mass, mixing, CP, Majorana, and contact coefficients are finite Schur-Kuranishi data on that support. This gives a structural-soft split: the structural entries are fixed by the local reconstruction, while the soft entries are completed-branch coefficients. The local determinant calculation leaves only a flat or torsion determinant obstruction, so the corresponding global problem is reduced to a finite holonomy check on the completed branch. The determinant, Pfaffian, single-holonomy, and stiffness conditions used in the phenomenological tests are imposed at this level.

The family layer also has a separated finite reading. The central torsor gives the family labels and the factor in Proposition 5. Circulant family operators give no central-layer mixing. A non-circulant Schur-Berry correction gives the sectoral basis rotation, as recorded in Proposition 7. A CP-sensitive invariant requires the same non-circulant correction to carry a non-removable phase. In the single-holonomy class this phase is the central torsor holonomy.

The Lorentzian input belongs to the Rainich-Codazzi side rather than to a higher-dimensional metric ansatz. The background Lorentzian conventions are those of [85]. The Codazzi part uses the two-eigenvalue closure discussed in Section 3.1; the Riemannian comparison class is represented by [86]. The algebraic non-null stress reading follows the Rainich line [5], with perturbative restrictions of Rainich algebraic types as in [50]. The aligned type-D and shear-free null-congruence comparisons are those of [6,57].

The projective link is the four-dimensional feature of the worldline defect. Twistor and spinor geometry provide the natural language for this identification. The spinor conventions used here are the standard ones of [8,58]; curved-twistor geometry is represented by [59]. The self-dual and pure-connection comparison classes include the Plebanski and Urbantke descriptions [51,53]. Modern pure-connection formulations are represented by [87,88], while spinorial higher-spin comparisons are represented by [89]. Twistor-space Standard-Model comparisons are those of [10,90].

The carrier selection itself is the ${\mathbb{CP}}^1$ quantization step. The role of K-theory is mainly to provide the stable index language for the analytic representative. The Borel-Weil part is the standard homogeneous-bundle construction [11]. The representation count is the elementary $SU\left(2\right)$ Clebsch-Gordan count in the conventions of [12]. Spherical analysis on homogeneous bundles gives the broader harmonic-analysis language [91]. The Toeplitz reading uses the Berezin-Toeplitz quantization framework [15]; the fuzzy-sphere cutoff of [16] and the finite-matrix brane models of [17] are comparison realizations of the same finite-mode principle.

The Dirac index language is used in a local form. The Atiyah-Singer theorem [61] and the Atiyah-Bott-Shapiro spinorial orientation [62] provide the stable reference. Standard spin geometry is represented by [92]. The self-dual Dirac comparison of [93] and the Clifford-spinor conventions of [94] give related spinorial settings. The Aharonov-Casher formula gives the two-dimensional zero-mode comparison [95], while the monopole harmonics of [13] and the Taub-NUT Dirac model of [14] give concrete spectral comparisons.

The boundary and conic analytic machinery is kept in the background because the carrier theorem does not require it. It becomes relevant when the normal representative (52) is treated as a boundary problem. The elliptic boundary estimates of [96,97], together with the non-homogeneous boundary theory of [98], give the standard analytic setting. Seeley calculus for singular integrals and boundary problems is represented by [99]; the conic degeneration comparison is represented by [100]. The functional-calculus background of [101] is the corresponding operator-theoretic language.

The Callias-Schur completion lies in the class of Fredholm Dirac problems with a mass-type endomorphism. The original Callias mechanism [24] and the geometric theorem of [25] are used for the low-sector isolation. Perturbed Callias operators are treated in [102]. Pseudodifferential Callias theory is represented by [103]; recent Callias index formulations are represented by [104]. APS and Callias boundary comparisons are given by [70,105]. The Cauchy-data and boundary-Dirac comparison is represented by [106,107].

The thin-core and finite-energy aspects of the Alena collar have standard geometric-measure and vortex analogues. Integral-current compactness is represented by [43]; the geometric-measure background is that of [44]. Ginzburg-Landau compactness and vortex concentration give a useful comparison [45]; the Jacobian-current description is represented by [46]. The removable-singularity and elliptic estimates used in the local collar analysis are the standard ones of [48,60]. The PDE conventions are compatible with [108].

The finite Standard-Model package is close to the usual grand-unified exterior-algebra organization. The $SU\left(5\right)$ and $Spin\left(10\right)$ comparison is represented by [18]. Clifford-ideal models give a related algebraic language [19]; further Clifford and ideal-based Standard-Model variants are represented by [20]. Octonionic ladder-operator and internal-space constructions provide another comparison class [66,67]. Recent algebraic unification and symmetry-breaking comparisons include [64,65,109].

Almost-commutative geometry is a useful contrast because its finite internal algebra is usually part of the input. The Connes-Lott model gives the classical comparison [21]. The spectral action provides the corresponding dynamical principle [22], and recent refinements are represented by [110]. Lorentzian refinements are represented by [23]. No-doubling and internal finite-space questions are discussed in [111,112]. Further noncommutative and internal-space comparisons are given by [113,114,115,116].

The finite labels are treated here as sector labels of the reconstructed carrier. In the Alena-Codazzi realization, the $V_1,V_2$ labels have the principal boundary-charge reading of Lemma 1, and their effective centrality is Corollary 1. This is compatible with the superselection viewpoint of [49]. The line-operator sensitivity to global form is the comparison used in [117]. Characteristic classes and secondary classes enter through the determinant and central readings [71,72]. The $S_3$ family models of [79] and the Clifford-family variants of [80] provide algebraic comparison points for the central torsor, while the torsor itself is read here from the projective-color reduction of the determinant global form.

The weak bridge and the finite Dirac channels have several standard analogues. Quillen superconnections give the structural comparison for the odd exterior insertion [74]. Defect-localized zero modes are represented by the Jackiw-Rebbi mechanism [75]. Domain-wall fermions give the lattice comparison [76]. Differential-form fermion models are represented by [68]. The seesaw comparison is the usual one [77], and the baryon-violating contact comparison is the standard proton-decay one [78].

The quantitative part of Section 5.3 is a completed-branch diagnostic. The Alena-Rainich reconstruction (61), the bosonic checks of Table 6, the weak-bridge channels of Table 4, the decay readings of Table 9, and the family criterion of Table 7 are finite tests of the same completed low-sector operator. Effective-operator matching and covariant derivative expansion methods give the corresponding field-theoretic comparison [118]. Precision electroweak matching and scheme dependence are compared with [119]. Standard Higgs decay formulae and loop form factors are used only as comparison readings of the bridge-resolvent calculation [82]. The CKM comparison uses the usual quark mixing structure of [33,34,35]. The CP-sensitive invariant is compared with [36], and the numerical reference values are those of [37].

The local support mechanism also has a minimal analogue reading. The structural test is the separation of a dipole link charge and a trace-free quadrupole link charge, followed by the Toeplitz thresholds of Section 3. The degree-n threshold rule (27) supplies the corresponding negative control: non-primitive positive degrees define different filtered link data and lose the separated first-threshold mechanism.

The geometric-matter viewpoint is also relevant. Finite-dimensional geometric models of matter give one comparison class [120]. The present construction differs in its local use of a resolved defect link and in the Codazzi-Gauss source of the central carrier. The finite carrier is obtained before the completed spectral branch is solved, while masses, mixing, and running data are read only after the Schur-Berry completion. This separation is the reason for the status table in Section 6.

The main limitation is structural and has been kept explicit. In the Alena-Codazzi realization, the primitive worldline is obtained through the thin-core current-defect limit, while the abstract carrier theorem retains the primitive optical defect as input. The carrier $E_3\oplus E_2$ follows from the primitive ${\mathbb{CP}}^1$ link, the degree-one line, and the separated $V_1,V_2$ source with central implementability. In the Alena-Codazzi realization the current-residual closure is Proposition 1, while centrality is supplied at the principal level by the Codazzi-Gauss boundary charges and Corollary 1; in the abstract carrier theorem central implementability remains a support hypothesis. Corollary 3 gives the local non-primitive threshold control. The split group $S\left(U\right(3)\times U(2\left)\right)$ is obtained from the local finite determinant kernel and the top-form stabilizer. Residual flat or torsion determinant phases remain part of the completed global branch problem. The central ${\mathbb{Z}}_3$ layer uses the determinant global form and its projective-color reduction, and its unblocked response is the finite magnetic three-cycle (46). The weak ${\mathbb{Z}}_2$ shadow is used only through the exterior parity locking (41). The equality $N_{\mathrm{fam}}=3$ uses the simple primitive locked-kernel condition. The numerical mass, vacuum, Higgs-coupling, decay, CKM/PMNS, and CP diagnostics use the completed Schur-Berry branch, with charge-compatible Schur compression when the Codazzi-Gauss sector split is retained.

The finite completion questions are correspondingly explicit. A boundary-admissible normal representative has to be constructed in closed branch charts with the primitive boundary class of (22). The completed determinant norm has to fix the finite determinant target entering (79), or the corresponding scalar remainder has to be retained. The bridge-resolvent matrix elements entering the decay readings of Table 9 have to be obtained from the same completed low-sector operator. The sectoral Schur-Weyl coefficients in (92) have to be computed for the quark and lepton sectors, including the Dirac/Majorana choice in the neutral lepton sector. The relation between the local ${\mathbb{Z}}_3$ torsor and the full determinant global form has to be checked beyond the local projective-color reduction. The finite contact classes separated from the elementary weak bridge have to be tested inside the Schur-Kuranishi subimage. Finally, the loop invariant (103) has to be compared with the Schur-completed version of (49) after sectoral non-commutation has been fixed.

The completed branch is not used to change the primitive carrier, the local determinant kernel, the exterior representation package, or the local anomaly degree count. Its admissible finite data are the determinant remainder, the locked-kernel rank, the sectoral Schur coefficients, the bridge-resolvent matrix elements, the finite contact classes, and the Schur-compressed central operators obtained from the same boundary-admissible normal representative. Failure to construct such data, loss of the Callias-Schur gap, a locked-kernel rank different from the simple primitive value, or incompatible sectoral coefficients would invalidate the corresponding completed branch.

The sharpest finite tests are correlation tests. The scalar determinant remainder may fix the absolute electroweak scale, while the neutral-cell ratios $m_W/m_Z=\sqrt{A/(A+B)}$ and $m_h/m_Z=2\sqrt{C/(A+B)}$ remain scale-free. The locked-cell Higgs couplings in (82), the leakage relation (84), and the decay hierarchy in Table 9 test the same Callias-Schur gap reading. Similarly, the primitive central matching (96) fixes (95), and then the quantities in (97) and (98) are fixed within the central Schur-Berry diagnostic. A completed branch which requires independent corrections to these correlated ratios, or incompatible values of the same Schur-compressed coefficients in the bosonic, decay, family, and CP sectors, fails the finite reconstruction test.