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 four-dimensional anisotropic Lorentzian branch with a primitive optical Codazzi 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 \(\widehat\Phi\): \[\Phi \longmapsto \left( \mathcal{A}_\Phi, \delta_\Phi, \operatorname{Sec}_{\rm iso}(\mathcal{A}_\Phi), \nabla_\Phi \right) \longmapsto \widehat\Phi . \label{eq:intro-self-reconstruction-loop}\tag{1}\] The equality is understood on the physical quotient, after gauge, diffeomorphism, and unitary equivalences have been removed. The closure defect is measured by \[D[\Phi] = \operatorname{dist}_{\mathfrak M}^2 \left( [\Phi], [\widehat\Phi] \right), \label{eq:intro-closure-defect}\tag{2}\] where \(\mathfrak M\) denotes the corresponding quotient configuration space. An exactly closed theory has \(D[\Phi]=0\) on physical configurations. In a local representation in which the closure defect is the variational gradient of an action, the condition \(D[\Phi]=0\) 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 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 \(\mathcal{O}(1)\) 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 \(SU(2)\) 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] and [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 support theorem, the two principal coefficients are kept separated and are required to be represented by independent central projections in the equivariant commutant. 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 \(SU(2)\) 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 \(U(3)\times U(2)\). 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 \(S(U(3)\times U(2))\). 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. This is close to the familiar \(SU(5)\) and \(\operatorname{Spin}(10)\) 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] and [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], and [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 two-channel 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], and [42]. In the closed split-conserved sector, the residual 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 Alena-Codazzi source reduction and the two-jet statement. Section 3 gives the projective link, the primitive line, the Borel-Weil tower, and the minimal centrally implemented carrier. 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, 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 realization of this source datum. 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 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 \[\mathcal{L}_{\rm cr} = \varphi p_\Lambda, \qquad \varphi = 1-\zeta^2-\mu_\zeta R_\omega . \label{eq:alena-residual-lagrangian}\tag{3}\] Here \(\zeta\) is the amplitude of the phase-current variable, \(\mu_\zeta=\mu_\zeta(\rho_\zeta)\) 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], and [42].
 
The associated translational current is \[J^\mu_{\rm tr} = p_\Lambda \zeta^2 U^\mu, \qquad \nabla^{(k)}_\mu J^\mu_{\rm tr}=0 . \label{eq:alena-current-conservation}\tag{4}\] The frozen amplitude block is assumed non-degenerate: \[V''_\zeta(\rho_\zeta)+R_\omega\mu''_\zeta(\rho_\zeta)>0 . \label{eq:alena-amplitude-nondegeneracy}\tag{5}\] This condition is used only as a local persistence hypothesis for the scalar collar. The thin-core compactness and the corresponding penalty-dominant Codazzi limit belong to the completed branch problem.
 
The product-rule Hilbert response of 3 is assumed to have a frozen local split \[T^{\rm cr}_{\mu\nu} = \Xi_{\mu\nu} + \varphi Y_{\mu\nu}, \label{eq:alena-split-stress}\tag{6}\] 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: \[\nabla^{(k)}_\mu\Xi^{\mu\nu}=0, \qquad \nabla^{(k)}_\mu(\varphi Y^{\mu\nu})=0 . \label{eq:alena-split-conservation}\tag{7}\] 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-free Codazzi coefficient is assigned to the vorticity/Codazzi-gap channel.
 
Put \(\tau=k^{\mu\nu}Y_{\mu\nu}\) and \(B_{\mu\nu}=Y_{\mu\nu}-\frac{1}{3}\tau k_{\mu\nu}\). The scalar \(\varphi\) is called an admissible Codazzi multiplier on a non-degenerate collar set when \(A_{\mu\nu}=\varphi B_{\mu\nu}\) satisfies the trace-adjusted Codazzi condition on the punctured collar. With \(C^B_{\alpha\mu\nu}=\nabla^{(k)}_\alpha B_{\mu\nu}-\nabla^{(k)}_\mu B_{\alpha\nu}\) and \(\theta=d\log|\varphi|\), this condition is \[C^B_{\alpha\mu\nu} + \theta_\alpha B_{\mu\nu} - \theta_\mu B_{\alpha\nu} = 0 . \label{eq:alena-codazzi-multiplier-condition}\tag{8}\] On a connected set where \(B_{\mu\nu}\) is invertible, with inverse \(\beta^{\mu\nu}\), the multiplier one-form is fixed by \(B\): \[\theta^B_\alpha = -\frac{1}{3}\beta^{\mu\nu}C^B_{\alpha\mu\nu}. \label{eq:alena-theta-b}\tag{9}\] A nonzero local multiplier exists when 8 holds with \(\theta=\theta^B\) and \(\theta^B\) is exact. In that case \(\varphi\) is fixed up to one nonzero constant factor. The possible periods and singular strata of \(B\) are closed-branch data.
 
If \(\theta^B=df_B\), then the scalar part of 3 gives \[\varphi=C_\varphi e^{f_B}, \qquad \zeta^2 = 1-\mu_\zeta R_\omega-C_\varphi e^{f_B}. \label{eq:alena-scalar-reduction}\tag{10}\] Together with 4 , this gives the remaining current compatibility \[U(\mu_\zeta R_\omega) + C_\varphi e^{f_B}\theta^B(U) = \left( 1-\mu_\zeta R_\omega-C_\varphi e^{f_B} \right) \left( \nabla^{(k)}_\mu U^\mu + U(\log p_\Lambda) \right). \label{eq:alena-current-compatibility}\tag{11}\] Equations 8 11 are used only to constrain the residual scalar and the current coefficient. On an exact non-degenerate connected collar, the pair \((\varphi,\zeta^2)\) 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.
 
A regular Alena-type collar component will be called primitive and non-degenerate when its transverse degree is one with positive orientation, when the residual sector is closed split-conserved in the sense of 7 , when the residual scalar is an admissible Codazzi multiplier in the sense of 8 , 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 ?? 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 the collar, the current-residual core is taken to converge to a Codazzi-closed optical branch with a nonzero frozen gap. 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]. On a primitive multiplicity-one regular component, the limit supplies the worldline \(\Gamma\), the degree-one transverse class, and the Codazzi-closed optical link. This statement 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 \(\operatorname{Sym}^0(N_\Gamma^*)\oplus\operatorname{Sym}^1(N_\Gamma^*)\oplus\operatorname{Sym}^2(N_\Gamma^*)\). On the oriented rank-three normal fiber, these have the \(SU(2)\simeq\operatorname{Spin}(3)\) 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 \(\ell\geq 3\) occurs at transverse order at most two.
 
The following statement is the source input used in Section 3.

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 \(M_0\in\operatorname{Sym}^2_0(N_\Gamma)\) gives the \(V_2\) component. The assignment \((m,M_0)\mapsto(\xi_{V_1},\xi_{V_2})\) is the standard \(SU(2)\) module identification for an oriented rank-three normal fiber. Hence the two-channel condition is open and dense in the model moment space.
 
The degree-one condition on \(\nu^\perp_\Gamma\) 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 ?? .

3 \(\mathbb{CP}^1\) Geometric Quantization and the Minimal Carrier↩︎

The carrier theorem uses only the primitive projective link, the line \(L_\Gamma\simeq\mathcal{O}(1)\), and the separated source type obtained in Proposition 1. 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 \[TM|_U = E\oplus F, \qquad \dim E=2, \qquad \dim F=2, \label{eq:cp1-rainich-splitting}\tag{12}\] where \(E\) is Lorentzian and \(F\) is spacelike. The differential closure is the Codazzi condition \[\nabla^{(k)}_\alpha A_{\mu\nu} = \nabla^{(k)}_\mu A_{\alpha\nu}. \label{eq:cp1-codazzi-condition}\tag{13}\] 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 [47]. Self-dual and conformal-metric comparison classes are represented by [48], [49], and [50].
 
If \(A^\sharp\) has two distinct eigenvalues \(M\) and \(P\) with eigenbundles \(E\) and \(F\), the component form of 13 gives \[(\nabla^{(k)}_{X_1}X_2)_F = \frac{k(X_1,X_2)}{M-P} (\nabla^{(k)}M)_F, \qquad X_1,X_2\in\Gamma(E), \label{eq:cp1-codazzi-E}\tag{14}\] and \[(\nabla^{(k)}_{V_1}V_2)_E = \frac{k(V_1,V_2)}{P-M} (\nabla^{(k)}P)_E, \qquad V_1,V_2\in\Gamma(F). \label{eq:cp1-codazzi-F}\tag{15}\] 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 [51], while homogeneous Codazzi fields give a separate comparison class [52]. Related \(2+2\) optical geometries occur in [53] and [54]. 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: \[\widetilde{X}=[M;\Gamma], \qquad \partial\widetilde{X}=S(N\Gamma), \qquad S(N\Gamma)_t\simeq S^2_\Gamma . \label{eq:cp1-real-blowup}\tag{16}\] 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 16 with the projective spinor sphere: \[S^2_\Gamma \simeq \mathbb{CP}^1_\Gamma . \label{eq:cp1-projective-link}\tag{17}\] The spinor convention is the standard one of [8] and [55]. The curved-twistor comparison is represented by [56].
 
Locally, the complement of a worldline has the codimension-three topology \[\mathbb{R}^4\setminus\mathbb{R} \simeq \mathbb{R}\times(\mathbb{R}^3\setminus\{0\}), \qquad H^2(\mathbb{R}^4\setminus\mathbb{R};\mathbb{Z})\simeq\mathbb{Z} . \label{eq:cp1-worldline-complement}\tag{18}\] The primitive transverse class occupies the generator in 18 . 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 \(g_{NS}:S^1\to SO(2)\simeq U(1)\) of degree one in the simple positive sector. The removable-singularity and local elliptic background are standard [57], [58]. The associated line on the projective link is therefore \[c_1(L_\Gamma)=1, \qquad L_\Gamma\simeq\mathcal{O}(1). \label{eq:cp1-primitive-line}\tag{19}\] A complex line over a punctured normal ball extends smoothly over the filled ball only with trivial Chern number on the boundary sphere. Thus 19 is the local filling obstruction carried by the primitive defect.
 
A worldline defect \(\Gamma\) will be called a primitive optical Codazzi defect when 13 holds on the punctured collar, the non-degenerate two-eigenvalue splitting is optical, the real blow-up has the projective link 17 , 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 19 .

3.2 Borel-Weil tower and Toeplitz visibility↩︎

The positive quantization of the primitive link is the Borel-Weil tower \[E_q = H^0(\mathbb{CP}^1_\Gamma,L_\Gamma^{\,q-1}) \simeq H^0(\mathbb{CP}^1,\mathcal{O}(q-1)), \qquad q\geq1 . \label{eq:cp1-borel-weil-tower}\tag{20}\] The Borel-Weil identification is used in the usual homogeneous-bundle form [11], with \(SU(2)\) 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 \(spin^c\) reading, 20 is the positive twisted link index of the canonical projective-line Dirac operator. The Atiyah-Singer framework [59] and the spinorial \(K\)-orientation of [60] give the stable index language, while the support calculation below uses only 20 .
 
The source side is supplied by Section 2. After the scalar singlet is separated, the associated-graded source type used on the link is \[\left( \operatorname{gr}J^{\leq2}_\perp \right)_{\rm ns} = V_1\oplus V_2 . \label{eq:cp1-filtered-source-type}\tag{21}\] 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 [61]; the multipole language is the one used in [62].
 
On the Borel-Weil block \(E_q\), Toeplitz observables have the traceless endomorphism content \[\operatorname{End}_0(E_q) = \bigoplus_{\ell=1}^{q-1}V_\ell . \label{eq:cp1-endomorphism-visibility}\tag{22}\] 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 21 are represented by two independent central projections in the \(SU(2)\)-equivariant commutant of the chosen finite support. It is Toeplitz-visible when the corresponding \(SU(2)\) types occur in 22 . The selected primitive carrier is \[V_\Gamma = C_\Gamma\oplus W_\Gamma, \qquad C_\Gamma=E_3, \qquad W_\Gamma=E_2 . \label{eq:cp1-minimal-carrier}\tag{23}\] If \(S_\Gamma:=E_2\), then \(E_3\simeq\operatorname{Sym}^2S_\Gamma\). Hence 23 is the first spinor block together with its symmetric square.

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 \(\operatorname{Hom}(E_2,E_3)\) 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 \(SU(2)\) types. An integer principal source of type 21 has no equivariant mixed component on 23 .
 
The degree-one class is also the source of the threshold separation. If \(L_\Gamma\) is replaced formally by a degree-\(n\) line \(\mathcal{O}(n)\), the \(q\)-th positive block is \(H^0(\mathbb{CP}^1,\mathcal{O}(n(q-1)))\). The first block on which \(V_\ell\) is visible is \[q_\ell(n) = 1+ \left\lceil\frac{\ell}{n}\right\rceil . \label{eq:cp1-degree-n-threshold}\tag{24}\] 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 \(n\geq2\), both channels occur in the same first nontrivial block of the modified tower.
 
The local negative control is obtained from the same \(SU(2)\) 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 \(\ell\geq3\) is seen in the compressed low support \(P_{23}(\cdot)P_{23}\). In the low-high blocks, \(V_\ell\) first occurs through \(\operatorname{Hom}(E_3,E_{2\ell-1})\) and \(\operatorname{Hom}(E_2,E_{2\ell})\). Thus the first \(V_3\) couplings start only through \(E_5\) and \(E_6\). Higher transverse moments therefore belong to the high-sector completion. 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 23 .
 
The carrier theorem is independent of the special residual normalization in 3 . The Alena-Codazzi collar produces the separated source type 21 ; the primitive projective link produces 20 ; the carrier follows from 22 . Any current-residual, geometric, or analytic realization with the same primitive filtered link datum and the same nonzero separated \(V_1,V_2\) principal channels gives the same local carrier. The finite representation package is attached only after 23 has been selected.

4 Standard-Model Carrier and the Central Family Layer↩︎

The support theorem has fixed the finite carrier 23 . 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 \[C:=C_\Gamma, \qquad W:=W_\Gamma, \qquad V:=C\oplus W . \label{eq:sm-carrier-decomposition}\tag{25}\] By 23 , \(\dim C=3\) and \(\dim W=2\). Each selected Borel-Weil block carries its standard \(SU(2)\)-invariant Hermitian structure. At the finite carrier level the allowed split unitary basis changes are initially \(U(C)\times U(W)\). The determinant reduction fixes the carrier top form on \(\Lambda^5V\). Thus admissible split basis changes satisfy \[\det(g_C)\det(g_W)=1 . \label{eq:sm-top-form-constraint}\tag{26}\] A split unitary map acts on \(\Lambda^5V\) by multiplication by the left-hand side of 26 . Hence the compact carrier-basis group is \[G_\Gamma = S(U(C)\times U(W)) \simeq S(U(3)\times U(2)). \label{eq:sm-split-unimodular-group}\tag{27}\] Equivalently, this is the compact split determinant Levi group of the rank-five carrier. The induced global form is \[S(U(3)\times U(2)) \simeq \frac{SU(3)_c\times SU(2)_L\times U(1)_Y}{\mathbb{Z}_6}. \label{eq:sm-global-form}\tag{28}\] The full compact group \(SU(V)\) is an unsplit rank-five comparison group. Its off-diagonal infinitesimal part belongs to the mixed low blocks described after Theorem 2. 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 \[F^{\rm even}_\Gamma = \Lambda^{\rm even}V . \label{eq:sm-even-package}\tag{29}\] It is the chiral exterior package of the selected rank-five carrier, restricted to 27 . Its dimension is \(1+\binom52+\binom54=16\). The representation-theoretic comparison with the usual \(SU(5)\) and \(\operatorname{Spin}(10)\) organization is standard [18]. Related Clifford-ideal realizations are discussed in [19] and [20]. Symmetry-breaking and roadmap comparisons are represented by [63] and [64]. The octonionic ladder and internal-space comparisons are represented by [65] and [66]. Differential-form realizations of fermions provide another exterior-algebra comparison class [67].
 
Let \(N_C\) and \(N_W\) be the \(C\)- and \(W\)-degree operators on \(\Lambda^\bullet V\). The hypercharge convention is the degree-counting normalization compatible with 26 : \[Y = -\frac{1}{3} N_C+\frac{1}{2} N_W . \label{eq:sm-hypercharge}\tag{30}\] The electric charge is \(Q=T_3+Y\), with \(T_3\) acting on the weak factor. Weak-hypercharge superselection gives a useful algebraic comparison [68].

The identifications in Table 2 use the top-form trivialization determined by 26 . Thus \(\Lambda^2C\otimes\Lambda^2W\) is identified with \(C^*\), and \(\Lambda^3C\otimes W\) is identified with \(W^*\). The same top-form constraint reduces the projected low-sector connection from \(\mathfrak u(C)\oplus\mathfrak u(W)\) to the infinitesimal algebra of 27 .
 
The finite determinant-obstruction count is a degree count on 29 . For \[Y_{a,b} = aN_C+bN_W , \label{eq:sm-general-hypercharge}\tag{31}\] the infinitesimal form of 26 is \[3a+2b=0 . \label{eq:sm-unimodular-linear-condition}\tag{32}\] The local degree factors of \(\Lambda^{\rm even}V\) are \[\begin{array}{c|c} \text{local anomaly factor} & \text{degree factor}\\ \hline SU(3)^2U(1) & 3a+2b\\ SU(2)^2U(1) & 3a+2b\\ \mathrm{grav}^2U(1) & 8(3a+2b)\\ U(1)^3 & 4(3a+2b)(9a^2+6ab+5b^2) \end{array} \label{eq:sm-anomaly-degree-factors}\tag{33}\] The quadratic factor in the last row is positive definite. Hence the common local kernel of 33 is exactly 32 . This is the infinitesimal stabilizer of the top exterior line in 26 . Thus 30 is the normalized degree-counting representative of the determinant-line anomaly-free class.
 
In the determinant-sensitive family reading, the finite factor in 33 is the local part seen by the Bismut-Freed determinant form for a boundary-admissible Callias-APS family [29], [69]. The primitive boundary degree in 19 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 [70], while secondary classes give the comparison of [71]. Line-operator detection of the global form is the comparison used in [72].
 
The odd exterior sector is used as the structural companion of 29 : \[F^{\rm odd}_\Gamma = \Lambda^{\rm odd}V . \label{eq:sm-odd-package}\tag{34}\] Every element of \(V\oplus V^*\) acts by Clifford multiplication on \(\Lambda^\bullet V\) and defines an odd map from 29 to 34 . 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 [73]. Defect-localized zero modes provide the classical comparison [74], while domain-wall fermions give the lattice comparison class [75].
 
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 3. 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 \((0,\pm1)\). Baryon-violating classes may occur as finite contact classes in the completed package, but they are not produced by the elementary weak bridge.

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 \[\Gamma_{\rm lock} = \gamma^5_{\rm branch}\otimes\Gamma_V . \label{eq:sm-parity-locking}\tag{35}\] The mirror complement is an analytic sector. In a locked Fredholm realization it is separated by a gap condition of the form \[\Pi_{\rm mir}K_{\rm lock}\Pi_{\rm mir} \geq m_\Gamma\Pi_{\rm mir}, \qquad m_\Gamma>0 . \label{eq:sm-mirror-gap}\tag{36}\] This condition belongs to the closed branch operator and does not change the finite representation content selected in Theorem 2. 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 \(\Lambda^3C\otimes W\simeq W^*\), while the neutral singlet sits in \(\Lambda^0V\). With 30 , 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 [76]. Baryon-violating contact classes belong to the usual proton-decay comparison range [77]; 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 19 and the Borel-Weil tower 20 . After the split determinant global form 28 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 35 . The reduction used in the central reading is \[c_1(L_\Gamma) \longmapsto [c_1(L_\Gamma)]_3 \in H^2(\mathbb{CP}^1,\mathbb{Z}_3) \simeq \mathbb{Z}_3 . \label{eq:sm-mod-three-shadow}\tag{37}\] 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 [78]; recent Clifford-family variants provide another comparison [79].
 
Let \(\mathfrak L_\Gamma\) be the affine \(\mathbb{Z}_3\) torsor determined by 37 . The corresponding central response space is \[\mathcal{H}_{\rm cen} = \ell^2(\mathfrak L_\Gamma). \label{eq:sm-central-torsor-space}\tag{38}\] After an auxiliary origin has been chosen, 38 is identified with the regular representation of \(\mathbb{Z}_3\). With \(\omega=\exp(2\pi i/3)\), the clock and shift generators are \[Z e_a = \omega^a e_a, \qquad S e_a = e_{a+1}, \qquad a\in\mathbb{Z}_3, \qquad ZS=\omega SZ . \label{eq:sm-central-clock-shift}\tag{39}\] The clock records the central grading. The adjacent shift records nearest-neighbour transport on the torsor. A finite Dirichlet response is \[\mathcal{E}_{\rm cen}(\psi) = \nu_\Gamma \sum_{a\in\mathbb{Z}_3} |\psi_{a+1}-e^{i\theta_\Gamma}\psi_a|^2, \qquad \nu_\Gamma\geq0 , \label{eq:sm-central-dirichlet-response}\tag{40}\] with \(\theta_\Gamma\) the discrete holonomy. Thus 40 is the quadratic form of the magnetic graph Laplacian on the affine three-cycle. Its quadratic operator is \[Y_{\rm cen} = 2\nu_\Gamma{\boldsymbol{1}} - \nu_\Gamma e^{-i\theta_\Gamma}S - \nu_\Gamma e^{i\theta_\Gamma}S^\dagger . \label{eq:sm-central-response-operator}\tag{41}\] The central response is called unblocked when \(\nu_\Gamma>0\). In the Fourier-character basis, the eigenvalues of 41 are \[y_k = 2\nu_\Gamma \left[ 1-\cos\left( \theta_\Gamma+\frac{2\pi k}{3} \right) \right], \qquad k\in\mathbb{Z}_3 . \label{eq:sm-central-eigenvalues}\tag{42}\] Away from the finite degeneracy locus of 42 , 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 \(\operatorname{End}(\mathcal{H}_{\rm cen})\simeq M_3(\mathbb{C})\). The corresponding traceless anti-Hermitian closure is \(\mathfrak{su}(3)\). This is an algebraic statement about the central response space 38 ; 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 \[C_3 = \operatorname{Tr}_{H_0} \left( V_{02}V_{21}V_{10} \right), \label{eq:sm-central-closed-cubic}\tag{43}\] 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 40 , \[C_3 = -\nu_\Gamma^3 e^{-3i\theta_\Gamma}. \label{eq:sm-central-cubic-value}\tag{44}\] The phase of 44 is a closed-cycle datum of the torsor. Under independent unitary changes of the three torsor fibers, the adjacent maps in 43 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.
 
If the primitive locked kernel over the central torsor has rank \(r\), then the family-response multiplicity is \[N_{\rm fam} = 3r . \label{eq:sm-family-multiplicity}\tag{45}\] The torsor fixes the factor \(3\). The rank \(r\) and the sectoral eigenvalue splitting are completed-branch data. The simple primitive locked-kernel case has \(r=1\) and gives three families. In this reading, the three family labels are the finite magnetic-cycle labels of 40 , while the physical splitting is supplied only by the completed sectoral operator. Leading family operators supported only in the circulant algebra \(\mathbb{C}[S]\) are simultaneously diagonalized by the finite Fourier transform on 38 . Hence physical mixing requires a completed-branch contribution outside \(\mathbb{C}[S]\). 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 23 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. 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 \[\mathcal{N}_{\rm cl} = \begin{pmatrix} 0 & L_{\Phi_0}^{*}\\ L_{\Phi_0} & 0 \end{pmatrix} + \Psi_{\rm Callias} + \Psi_{\rm ext} + \Psi_{\rm cen} + \Psi_{\rm hol}. \label{eq:spectral-closed-normal-representative}\tag{46}\] 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 46 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 19 .
 
Let \(B\) be a local parameter domain in the completed branch problem, and let \(N_+(b)\) be the positive gauge-fixed normal family obtained from 46 . 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. For \(b\in B\), the Riesz projection onto the selected low window is \[P_b = \frac{1}{2\pi i} \int_{\gamma} (z-N_+(b))^{-1}\,dz , \label{eq:spectral-riesz-projection}\tag{47}\] where \(\gamma\) encloses the low cluster and no other spectrum. The finite-rank bundle \(\mathcal{E}_{\rm low}\to B\) is defined by \(\mathcal{E}_{{\rm low},b}=P_b\mathcal{H}\).

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 23 .

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 \[\nabla^{\rm BWZ} = P\nabla^0P . \label{eq:spectral-bwz-connection}\tag{48}\] Its curvature is \[F^{\rm BWZ} = P(dP)\wedge(dP)P . \label{eq:spectral-bwz-curvature}\tag{49}\] 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 [80].
 
On the full selected carrier, 48 is read as the projected gauge connection of the moving low-sector bundle. In a local orthonormal frame \(e_i(b)\) of \(\mathcal{E}_{\rm low}\), its connection matrix is \[(A^P_\mu)_{ij} = \langle e_i,\partial_\mu e_j\rangle . \label{eq:spectral-projected-gauge-potential}\tag{50}\] Under a local change of low-sector frame \(u(b)\), the usual gauge transformation law is obtained: \[A^P_\mu \longmapsto u^{-1}A^P_\mu u+u^{-1}\partial_\mu u . \label{eq:spectral-projected-gauge-transformation}\tag{51}\] After the determinant reduction 26 , the structure algebra is \[\mathfrak g_{\rm low} = \mathfrak s(\mathfrak u(C)\oplus\mathfrak u(W)) \simeq \mathfrak s(\mathfrak u(3)\oplus\mathfrak u(2)), \label{eq:spectral-low-gauge-algebra}\tag{52}\] 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 \[p_\Lambda^P = \frac{1}{4} \sum_a \epsilon_a g_a^{-2} \operatorname{tr}_a \left( F^{P,a}_{\mu\nu}F^{P,a\,\mu\nu} \right), \label{eq:spectral-projected-curvature-invariant}\tag{53}\] where \(a\) runs over the simple and central factors of 52 . 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] and [32].
 
The projected curvature stress of 53 is the bosonic comparison object used in the Alena branch reading. Let \(\Upsilon^P_{\mu\nu}\) denote the gauge-side Hilbert stress obtained from 53 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 \(Y^{\rm br}_{\mu\nu}[k]\).
 
A projected gauge sector is called Alena-compatible on a connected patch when \(\Upsilon^P_{\mu\nu}\) is symmetric, traceless, and of non-null Rainich type, \[\Upsilon^{P\,\mu}{}_{\mu}=0, \qquad \Upsilon^{P\,\mu}{}_{\alpha}\Upsilon^{P\,\alpha}{}_{\nu} = \rho^2\delta^\mu{}_\nu, \qquad \rho>0, \label{eq:spectral-rainich-stress-condition}\tag{54}\] the associated \(2+2\) splitting is non-degenerate, and the finite-anisotropy ratio \(r=u/p_\Lambda^P\) satisfies \(|r|<1\), where \(u=\Upsilon^P_{UU}\) 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 53 .
 
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 \[p_\Lambda^P=p_0K^2, \qquad Y^{\rm br}_{\mu\nu}[k]=-\Upsilon^P_{\mu\nu}, \label{eq:spectral-alena-rainich-identification}\tag{55}\] where \(K=\eta^{\mu\nu}k_{\mu\nu}\) and \[Y^{\rm br}_{\mu\nu}[k] = p_0K^2 \left( \frac{4}{K}k_{\mu\nu}-\eta_{\mu\nu} \right). \label{eq:spectral-branch-response-tensor}\tag{56}\] Indeed, in an adapted \(\eta\)-orthonormal tetrad \((U,N,W,S)\), the non-null Rainich stress has canonical components \(\Upsilon^P_{UU}=u\), \(\Upsilon^P_{NN}=-u\), and \(\Upsilon^P_{WW}=\Upsilon^P_{SS}=u\). With \(\tanh\chi=r\), define \[k_{\mu\nu} = U_\mu U_\nu - N_\mu N_\nu - e^{2\chi} \left( W_\mu W_\nu+S_\mu S_\nu \right). \label{eq:spectral-reconstructed-branch-metric}\tag{57}\] Then \(K=2+2e^{2\chi}=4/(1-r)\), and the normalization above gives \(p_0K^2=p_\Lambda^P\). Since \(4/K=1-r\) and \(e^{2\chi}=(1+r)/(1-r)\), substitution in 56 gives \(Y^{\rm br}_{UU}=-u\), \(Y^{\rm br}_{NN}=u\), and \(Y^{\rm br}_{WW}=Y^{\rm br}_{SS}=-u\). Thus 55 holds tensorially. The sign in \(Y^{\rm br}=-\Upsilon^P\) 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 4. 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 \(X\in T_bB\), put \[V_X = (1-P_b)(\partial_XN_+(b))P_b . \label{eq:spectral-schur-detector}\tag{58}\] 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 49 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 \(\mathbb{C}[S]\). Let \(\Pi_{\rm circ}\) denote the Hilbert-Schmidt orthogonal projection from \(M_3(\mathbb{C})\) onto \(\mathbb{C}[S]\). Since the Weyl monomials \(Z^mS^n\) 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 \[F^{\rm cen}_{XY} = \Delta^{-2} \left( \bar a b Z^\dagger S - \bar b a S^\dagger Z \right), \label{eq:spectral-weyl-curvature}\tag{59}\] and \[\Pi_{\rm circ}(F^{\rm cen}_{XY})=0, \qquad \|F^{\rm cen}_{XY}\|_{\rm HS}^2 = 6|ab|^2\Delta^{-4}. \label{eq:spectral-weyl-curvature-norm}\tag{60}\] 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 \[Y_f^{(0)} = A_f{\boldsymbol{1}} + \beta_f S + \overline{\beta}_fS^\dagger . \label{eq:spectral-leading-circulant-family-operator}\tag{61}\] They are simultaneously diagonalized by the finite Fourier transform on 38 . Hence a family sector whose completed response remains inside \(\mathbb{C}[S]\) has no physical mixing from this central layer. If one leading sector has simple spectrum, a mixing correction must have a component outside \(\mathbb{C}[S]\). A CP-sensitive invariant also requires a non-removable complex component in the same non-circulant Schur-Berry correction.
 

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 \[Y_f = \Pi_f K_{\rm eff}\Pi_f, \qquad K_{\rm eff} = K_{LL}-K_{LH}K_{HH}^{-1}K_{HL}+\cdots , \label{eq:spectral-sectoral-schur-operator}\tag{62}\] 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 62 gives the structural form of the sectoral operator; its numerical eigenvalues are not fixed by the carrier theorem.
 
The local weak bridge is the odd Clifford insertion from \(W\oplus W^*\) between 29 and 34 . It supplies the one-Higgs Dirac channels on the finite module. The Majorana and contact classes described after 35 are therefore read as sectoral closed-branch operators of the form 62 . 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, \(\delta F^P_{\mu\nu}=D_\mu\delta A^P_\nu-D_\nu\delta A^P_\mu\), one obtains \[\delta \int p_\Lambda^P\,d{\rm vol}_\eta = - \int \sum_a \epsilon_a g_a^{-2} \operatorname{tr}_a \left( D_\mu F^{P,a\,\mu\nu}\delta A^P_\nu \right) d{\rm vol}_\eta + {\rm boundary}. \label{eq:spectral-yang-mills-variation}\tag{63}\] 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 29 , the completed low-sector covariant derivative contains the branch spin connection and the projected gauge connection: \[D_\mu^{\rm low} = \partial_\mu+\omega_\mu^{\rm branch}+A^P_\mu . \label{eq:spectral-low-dirac-covariant-derivative}\tag{64}\] 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 46 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 2.
 
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 6. The entries \(A\), \(B\), and \(C\) are completed-branch finite coefficients, while \(C_0\), \(u_{\rm prim}\), and \(S_{\rm prim}\) 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 \(\dim C=3\) and \(\dim W=2\), \[A_0=\frac{\dim C}{\dim C+2\dim W}=\frac{3}{7}, \qquad B_0=A_0\frac{\dim W}{\dim C+2\dim W}=\frac{6}{49}, \qquad C_0=A_0\frac{\dim C}{\dim C+\dim W}=\frac{9}{35}. \label{eq:spectral-neutral-cell-leading-weights}\tag{65}\] The constant-curvature representative in the primitive class 19 gives \(S_{\rm prim}=4\pi^2\). Since \(\dim\Lambda^{\rm even}V=16\), the scalar unit used in the first neutral Schur step is \(u_{\rm prim}=1/(16S_{\rm prim})=1/(64\pi^2)\).
 
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 \[\delta S_{\rm sc} = u_{\rm prim}P_C-\frac{3}{2}u_{\rm prim}P_W . \label{eq:spectral-scalar-schur-step}\tag{66}\] Indeed, \(\mathop{\mathrm{Tr}}_{C\oplus W}\delta S_{\rm sc}=0\) gives \(3u_{\rm prim}+2\delta A=0\). Thus \[A=A_0-\frac{3}{2}u_{\rm prim}, \qquad B=B_0, \qquad C=C_0+u_{\rm prim}. \label{eq:spectral-neutral-cell-corrected}\tag{67}\] The coefficient \(-3/2\) is the dimension ratio \(-\dim C/\dim W\). 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 \[\sin^2\theta_{\rm link} = \frac{B}{A+B}. \label{eq:spectral-link-angle}\tag{68}\] In the primitive neutral-cell reading used here, 68 gives \[\sin^2\theta_{\rm link} = 0.223184\ldots . \label{eq:spectral-link-angle-value}\tag{69}\] The comparison value extracted from the on-shell masses is \[\sin^2\theta_W^{\rm on-shell} = 1-\frac{M_W^2}{M_Z^2}. \label{eq:spectral-onshell-angle}\tag{70}\] Using the values quoted in [37], the difference between 69 and 70 is \(1.13\times10^{-4}\). The effective leptonic mixing angle is a scheme-dependent observable and is not identified with 70 .
 
The same finite Schur block can be read with a primitive collar scale. With the spin-vorticity normalization \(g=1/2\) fixed in [39], the zero-remainder relative determinant scale is \[v_{\rm EW}^{\rm prim} = 2\sqrt2\,M_{\rm Pl} \exp\left( -S_{\rm prim} - \frac{1}{144\pi^2} \right) \sqrt{\frac{C_0}{C_0+u_{\rm prim}}}. \label{eq:spectral-primitive-ew-scale}\tag{71}\] A finite scalar determinant remainder may be recorded by \(v_{\rm EW}=e^{-\delta_{\rm sc}}v_{\rm EW}^{\rm prim}\). The radial determinant ratio is a normalization of the scalar direction, not an additional neutral-cell coefficient. The promotion of 71 to a prediction requires an independent completed-branch derivation of the finite determinant target.
 
With the same central constants as above, 71 gives \(v_{\rm EW}^{\rm prim}=246.2205\,{\rm GeV}\). The Fermi value \(v_F=(\sqrt2G_F)^{-1/2}\), with \(G_F\) taken from [37], is \(246.2196719\,{\rm GeV}\). The zero-remainder primitive value differs from \(v_F\) by \(8.3\times10^{-4}\,{\rm GeV}\), equivalently by \(3.4\) ppm or by \(0.34\) quoted \(G_F\) standard deviations. Equivalently, the Fermi comparison corresponds to \(\delta_{\rm sc}\simeq3.4\times10^{-6}\). The standard-deviation count records the size of the required scalar determinant remainder.
 
At the zero-remainder primitive scale, the neutral cell gives the diagnostic mass reading \[m_\gamma^{\rm prim}=0, \qquad m_W^{\rm prim}=\frac{v_{\rm EW}^{\rm prim}}{2}\sqrt A, \qquad m_Z^{\rm prim}=\frac{v_{\rm EW}^{\rm prim}}{2}\sqrt{A+B}, \qquad m_h^{\rm prim}=v_{\rm EW}^{\rm prim}\sqrt C . \label{eq:spectral-primitive-mass-reading}\tag{72}\] Substitution into 72 gives \[m_\gamma^{\rm prim}=0, \qquad m_W^{\rm prim}=80.3710\,{\rm GeV}, \qquad m_Z^{\rm prim}=91.1885\,{\rm GeV}, \qquad m_h^{\rm prim}=125.2403\,{\rm GeV}. \label{eq:spectral-primitive-mass-values}\tag{73}\] Relative to the current PDG averages [37], these values differ by \(0.14\), \(0.25\), and \(0.37\) quoted standard deviations, respectively. The comparison is a zero-remainder diagnostic of the primitive neutral cell.
 
The Higgs mode is the radial reading of the same gap. If the primitive neutral cell is locked under the radial scale variation, \(\partial_{\log v}A=\partial_{\log v}B=\partial_{\log v}C=0\), the \(hWW\) and \(hZZ\) couplings are obtained by differentiating 72 with respect to \(v\): \[g_{hWW}=\frac{\partial m_W^2}{\partial v}=\frac{2m_W^2}{v}, \qquad g_{hZZ}=\frac{\partial m_Z^2}{\partial v}=\frac{2m_Z^2}{v}. \label{eq:spectral-higgs-gap-derivative-couplings}\tag{74}\] Thus the Standard-Model normalization of the gauge-boson Higgs couplings is recovered as locked gap scaling. A radial leakage operator \(L_{\rm rad}=\partial_{\log v}S_{\rm eff}\) gives \[\kappa_W-1=\frac{1}{2}\frac{\ell_W}{A}, \qquad \kappa_Z-1=\frac{1}{2}\frac{\ell_W+\ell_B}{A+B}. \label{eq:spectral-higgs-leakage-kappas}\tag{75}\] For a trace-neutral scalar leakage with \(\ell_B=0\) one obtains the correlation \[\kappa_Z-1 = \frac{A}{A+B}(\kappa_W-1). \label{eq:spectral-higgs-leakage-correlation}\tag{76}\] This is a finite neutral-cell test. A nonzero deviation from 76 measures radial determinant leakage.
 
The fermionic reading is compared with the spin-vorticity vortex relation of [39], \[y_f=\cosh\varphi_f-1, \qquad m_f=\frac{v}{\sqrt2}y_f . \label{eq:spectral-vortex-yukawa-reading}\tag{77}\] If the vortex angle is locked, \(\partial_{\log v}\varphi_f=0\), then \[g_{hff} = \frac{\partial m_f}{\partial v} = \frac{m_f}{v}. \label{eq:spectral-higgs-fermion-gap-coupling}\tag{78}\] More generally, \[\kappa_f-1 = \coth\left(\frac{\varphi_f}{2}\right) \partial_{\log v}\varphi_f . \label{eq:spectral-vortex-angle-leakage}\tag{79}\] The finite test is therefore \(Q_f=(\kappa_f-1)\tanh(\varphi_f/2)\). 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.
 
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 \[\Delta_{i\to F} = m_i-\sum_{a\in F}m_a . \label{eq:spectral-decay-gap-excess}\tag{80}\] A channel is open only when 80 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 \[\Gamma(h\to f\overline{f}) = N_c\frac{m_hm_f^2}{8\pi v^2} \left( 1-\frac{4m_f^2}{m_h^2} \right)^{3/2}. \label{eq:spectral-higgs-fermion-decay-width}\tag{81}\] Using running quark masses at the Higgs scale for the quark channels, as in the standard Higgs-decay treatment [81], the primitive gap reading gives the hierarchy recorded in Table 7. 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 73 , since \(m_h-2m_W=-35.50\,{\rm GeV}\) and \(m_h-2m_Z=-57.14\,{\rm GeV}\). The channels \(WW^*\) and \(ZZ^*\) 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 7, the summed Higgs width is approximately \(4.0\,{\rm MeV}\), 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 73 , one has \(\Delta_{t\to bW}\simeq88\,{\rm GeV}\). With the standard charged-current normalization this gives \(\Gamma_t\simeq1.35\,{\rm GeV}\) 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 3 is not a proton-decay channel.
 
For the family sector, write the Schur-completed non-circulant Weyl coefficient as \[\Xi_f = \sum_r \overline{a}_{f,r}b_{f,r}\Delta_{f,r}^{-2}. \label{eq:spectral-sectoral-schur-weyl-coefficient}\tag{82}\] If the non-circulant correction is represented at first order by \(\lambda_f Z+\overline{\lambda}_fZ^\dagger\), with \(\lambda_f\) proportional to 82 , and if the leading circulant eigenvalues are \(y_{f,k}\), then the first change of the sectoral eigenbasis has adjacent entries \[(\Theta_f)_{k+1,k} = \frac{\lambda_f}{y_{f,k}-y_{f,k+1}} . \label{eq:spectral-sectoral-basis-variation}\tag{83}\] For two sectors \(f,g\), the corresponding relative mixing is, to first order, \[V_{fg} = {\boldsymbol{1}}+\Theta_g-\Theta_f+O(\Theta^2). \label{eq:spectral-sectoral-relative-mixing}\tag{84}\] 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 83 .
 
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 one-parameter completed-branch matching of the determinant shadow is recorded by \[\lambda_{\rm F}(\kappa) = \frac{\kappa B}{A+\kappa B}. \label{eq:spectral-central-matching-lambda}\tag{85}\] The value \(\kappa=1\) corresponds to a common normalization of the neutral and central shadows of the primitive class. A value \(\kappa\neq1\) is read as a finite threshold or Schur-Berry matching coefficient on the completed branch. The Schur-length diagnostic is \[s_{12}^{q} = \lambda_{\rm F}, \qquad s_{23}^{q} = \sqrt{\frac{\dim W}{\dim C}}\lambda_{\rm F}^2, \qquad s_{13}^{q} = \frac{1}{\dim C}\lambda_{\rm F}^3 . \label{eq:spectral-ckm-angle-diagnostic}\tag{86}\] The phase-sensitive invariant is compared with the closed central cubic 43 . In the same normalization, \[J_q^{\rm diag} = \frac{1}{4}\lambda_{\rm F}^6 . \label{eq:spectral-jarlskog-diagnostic}\tag{87}\] The invariant \(J\) is the usual rephasing invariant of the CKM matrix [36].
 
Matching 85 to the PDG value \(\lambda_{\rm CKM}=0.22501\) quoted in [37] gives \[\kappa_{\rm CKM} = 1.0105566095\ldots . \label{eq:spectral-kappa-ckm}\tag{88}\] Substitution into 86 and 87 gives \[s_{12}^{q}=0.2250100000\ldots, \qquad s_{23}^{q}=0.0413388137\ldots, \qquad s_{13}^{q}=0.0037973813\ldots, \qquad J_q^{\rm diag} = 3.2445\times10^{-5}. \label{eq:spectral-ckm-values}\tag{89}\]

Using the standard CKM convention and 87 , one obtains \(\delta_q^{\rm diag}=1.2332\,{\rm rad}\), or \(70.66^\circ\). The corresponding moduli are \[|V_{\rm CKM}^{\rm diag}| = \begin{pmatrix} 0.974349 & 0.225008 & 0.003797\\ 0.224868 & 0.973512 & 0.041339\\ 0.008798 & 0.040570 & 0.999138 \end{pmatrix}. \label{eq:spectral-ckm-matrix-diagnostic}\tag{90}\] The associated unitarity-triangle angles are \[\alpha_{\rm diag}=85.99^\circ, \qquad \beta_{\rm diag}=23.39^\circ, \qquad \gamma_{\rm diag}=70.62^\circ . \label{eq:spectral-unitarity-triangle-diagnostic}\tag{91}\] The coefficient 88 is matched in this diagnostic. Its derivation would require the completed Schur-Berry matching operator, which also fixes the non-circulant sectoral Yukawa 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 \(e^{i(\phi_{k+1}-\phi_k)}q_k\). Hence the cyclic product is rephasing-invariant, and the first loop-type CP-odd scalar is \[J^{\rm loop}_{fg} = \Im(q_0q_1q_2). \label{eq:spectral-cp-loop-invariant}\tag{92}\] In the one-gap minimal central model, 92 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 45 . 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. The primitive optical Codazzi defect supplies the link and the primitive line. The scalar-sector two-jet supplies the two non-scalar source channels. Separated central implementability supplies the minimal carrier support. The finite determinant count supplies the local determinant kernel. The Callias-Schur assumptions supply the analytic low-sector representative.

The status of the ingredients is summarized in Table 9. This separation is useful because the carrier theorem, the determinant package, the family torsor, and the numerical branch diagnostics enter at different levels of the construction.

The finite local reconstruction has the following form. A primitive optical Codazzi defect defines the projective link and the primitive line. The scalar-sector two-jet defines the non-scalar source types. The Borel-Weil tower, Toeplitz visibility, and separated central implementability select the rank-five block \(E_3\oplus E_2\). The finite determinant-obstruction kernel gives the split determinant basis group; the even exterior package then gives the one-generation module. 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, and the remaining finite variables are constrained by the Kuranishi equation of the completed branch.
 
The quantities collected in Section 5.3 are therefore diagnostic outputs of the completed spectral layer. The neutral-cell angle, the mass readings, the primitive electroweak scale, the Higgs-coupling and decay readings, the CKM matching coefficient, and the loop-type CP invariant are finite tests of a completed branch. Their numerical use 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 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 as the requirement that the Schur-compressed operators, determinant data, central response, and projected connection 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 45 , the scalar determinant remainder, the high-sector Schur sums, the sectoral Yukawa eigenvalues, the non-circulant family corrections, 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 [82]. Standard principal-bundle and connection conventions are those of [83].
 
The Lorentzian input belongs to the Rainich-Codazzi side rather than to a higher-dimensional metric ansatz. The background Lorentzian conventions are those of [84]. The Codazzi part uses the two-eigenvalue closure discussed in Section 3.1; the Riemannian comparison class is represented by [85]. The algebraic non-null stress reading follows the Rainich line [5], with perturbative restrictions of Rainich algebraic types as in [47]. The aligned type-D and shear-free null-congruence comparisons are those of [6] and [54].
 
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] and [55]; curved-twistor geometry is represented by [56]. The self-dual and pure-connection comparison classes include the Plebanski and Urbantke descriptions [48], [50]. Modern pure-connection formulations are represented by [86] and [87], while spinorial higher-spin comparisons are represented by [88]. Twistor-space Standard-Model comparisons are those of [10] and [89].
 
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 \(SU(2)\) Clebsch-Gordan count in the conventions of [12]. Spherical analysis on homogeneous bundles gives the broader harmonic-analysis language [90]. 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 [59] and the Atiyah-Bott-Shapiro spinorial orientation [60] provide the stable reference. Standard spin geometry is represented by [91]. The self-dual Dirac comparison of [92] and the Clifford-spinor conventions of [93] give related spinorial settings. The Aharonov-Casher formula gives the two-dimensional zero-mode comparison [94], 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 46 is treated as a boundary problem. The elliptic boundary estimates of [95] and [96], together with the non-homogeneous boundary theory of [97], give the standard analytic setting. Seeley calculus for singular integrals and boundary problems is represented by [98]; the conic degeneration comparison is represented by [99]. The functional-calculus background of [100] 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 [101]. Pseudodifferential Callias theory is represented by [102]; recent Callias index formulations are represented by [103]. APS and Callias boundary comparisons are given by [104] and [69]. The Cauchy-data and boundary-Dirac comparison is represented by [105] and [106].
 
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 [57] and [58]. The PDE conventions are compatible with [107].
 
The finite Standard-Model package is close to the usual grand-unified exterior-algebra organization. The \(SU(5)\) and \(\operatorname{Spin}(10)\) 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 [65], [66]. Recent algebraic unification and symmetry-breaking comparisons include [108], [63], and [64].
 
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 [109]. Lorentzian refinements are represented by [23]. No-doubling and internal finite-space questions are discussed in [110] and [111]. Further noncommutative and internal-space comparisons are given by [112], [113], [114], and [115].
 
The finite labels are treated here as sector labels of the reconstructed carrier. This is compatible with the superselection viewpoint of [116]. 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 [70], [71]. The \(S_3\) family models of [78] and the Clifford-family variants of [79] 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 [73]. Defect-localized zero modes are represented by the Jackiw-Rebbi mechanism [74]. Domain-wall fermions give the lattice comparison [75]. Differential-form fermion models are represented by [67]. The seesaw comparison is the usual one [76], and the baryon-violating contact comparison is the standard proton-decay one [77].
 
The quantitative part of Section 5.3 is a completed-branch diagnostic. The Alena-Rainich reconstruction 55 , the bosonic checks of Table 4, the weak-bridge channels of Table 3, the decay readings of Table 7, and the family criterion of Table 5 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 [81]. The CKM comparison uses the usual quark mixing structure of [33], [34], and [35]. The CP-sensitive invariant is compared with [36], and the numerical reference values are those of [37].
 
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 its separated central implementability condition. 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. 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. The split group \(S(U(3)\times U(2))\) 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 40 . The weak \(\mathbb{Z}_2\) shadow is used only through the exterior parity locking 35 . The equality \(N_{\rm fam}=3\) 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.
 
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 19 . The completed determinant norm has to fix the finite determinant target entering 71 , or the corresponding scalar remainder has to be retained. The bridge-resolvent matrix elements entering the decay readings of Table 7 have to be obtained from the same completed low-sector operator. The sectoral Schur-Weyl coefficients in 82 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 92 has to be compared with the Schur-completed version of 43 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 74 , the leakage relation 76 , and the decay hierarchy in Table 7 test the same Callias-Schur gap reading. Similarly, once 85 is matched to \(s_{12}^q\), the remaining quantities in 86 and 87 are fixed within the one-parameter central matching. 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.

Statements↩︎

 
Author has no relevant financial or non-financial interests to disclose.
 
Author did not receive support from any organization for the submitted work.
 
All data, symbolic computations, numerical evaluations, and plotting routines used in this article are contained in the accompanying supplementary materials, where applicable.
 
During the preparation of this manuscript, the author used generative AI tools for language editing, formatting, consistency checks, and organization of selected passages. These tools were not used to generate research data, perform the scientific analysis, or draw the conclusions. All mathematical statements, citations, and scientific claims were reviewed and verified by the author, who takes full responsibility for the final manuscript.

References↩︎