A Filtered Link-Cycle Reconstruction of a Minimal Standard-Model Representation Carrier from Primitive Optical Codazzi Defects

1. Introduction and Main Results

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]. The problem considered here is more specific. A gauge-side stress tensor of non-null Rainich type is represented by a four-dimensional anisotropic Lorentzian branch, and a finite internal representation carrier is reconstructed from the filtered equivariant index data of the resolved optical link of a codimension-three defect. Finite-resolution curvature-defect mechanisms provide a separate comparison class [4].

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

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

(1)

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

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

(2)

where $\mathfrak{M}$ denotes the corresponding quotient configuration space. An exactly closed theory has $D\left[\mathrm{\Phi }\right]=0$ on physical configurations. An effective theory selects local minima of (2) within the prescribed conserved-current and topological sector.

This formulation contains familiar limiting forms. In a local representation in which the closure defect is the variational gradient of an action, $\mathrm{\Delta }\left(\mathrm{\Phi }\right)=\delta S/\delta \mathrm{\Phi }$, the condition $D\left[\mathrm{\Phi }\right]=0$ gives the Euler-Lagrange equations. If the defect is ${\mathrm{\Delta }}_{\mu \nu }=G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }-8\pi G{T}_{\mu \nu }$, the vanishing condition gives the Einstein equation. If the defect is a curvature-compatibility condition, the corresponding Bianchi-type closure is obtained. In the present paper this principle is used only in its primitive local link reduction: at fixed conserved currents and fixed topological class, the closed branch is represented by an isolated Schur-low sector, and the finite labels and phases are read from the Schur-Berry data of that sector.

Three levels are separated. First, the primitive filtered link gives a rigidity statement for the finite carrier. Second, the same primitive class has a central coefficient shadow which gives a family-response torsor and, in the unblocked case, a finite clock-shift response algebra. Third, the isolated Schur-low sector gives a finite neutral-cell benchmark after a primitive scalar unit has been fixed. The first level is the local carrier theorem. The second is algebraic and conditional on Schur-visible motion when Berry-Wilczek-Zee response is used. The third is a diagnostic of the completed spectral problem.

The Alena-type input is used below as a sufficient current-residual realization of the local source hypotheses. It is motivated by the Alena Tensor identification of [5]; in this role it supplies the residual density, the translational-current coefficient, and the vortex terms used in the branch stress response [6]. The continuum, variational, and Higgs-like branch-potential inputs are those of [7,8], and [9]. In the current-residual reading of [6], the Alena branch contains a phase-amplitude current, a rotational/vorticity response, and a spin-vorticity vortex sector. The gauge-branch reading used here fixes one additional convention. The branch tensor $Y_{\mu \nu }\left(k\right)$ is treated as the Rainich-type representative of the gauge-side stress in the closed branch sector, and the Hilbert-Belinfante comparison is used only to assign separated current and stress-response coefficient labels at frozen principal order. The residual-scalar branch used below is the closed split-conserved sector of this input: its Hilbert response separates into a translational-current part and a weighted branch-response part, the residual scalar is required to be a Codazzi multiplier, and the translational-current coefficient is then constrained by the same scalar. Section 2 records this reduction and the order-$\le 2$ Rainich-current two-jet reduction. Once the scalar singlet is separated, the non-scalar associated-graded normal source has only the $V_1$ and $V_2$ types. In the Alena-type class, these types are realized by the phase-current and vorticity/Codazzi channels. A primitive phase winding gives the thin core used below; after blow-up, its degree-one transverse trace is the simple positive class. The finite carrier theorem uses only the optical ARC branch, the Codazzi closure, the resolved link, and the filtered source data on that link. The Codazzi multiplier supplies the differential closure which complements the algebraic Rainich conditions [10]; the corresponding comparison class is the classical Rainich reconstruction problem [11].

The organizing object is a primitive filtered link cycle. On the non-degenerate Codazzi set, the two-eigenvalue splitting is optical. The corresponding null directions are geodesic and shear-free, and the defect has the standard local projective-spinor reading of an optical structure in the sense of [12]. The broader twistor comparison class is classical [13], while twistor approaches to Standard Model geometry provide a separate representation-theoretic comparison [14]. In the present construction only the projective line attached to the resolved defect is used. After real blow-up of the worldline defect $\mathrm{\Gamma }$, the boundary fiber is identified with ${\mathbb{CP}}^1$. A simple positive transverse-frame resolution gives a complex line $L_{\mathrm{\Gamma }}$ with $c_1\left(L_{\mathrm{\Gamma }}\right)=1$, hence $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$.

The primitive link therefore carries a ${\mathrm{spin}}^c$ Dirac cycle. Its equivariant index is the Borel-Weil tower; the same spaces appear as the standard monopole zero modes on $S^2$ [15] and in the Taub-NUT Dirac comparison class [16]. In Borel-Weil form [17], with representation-theoretic conventions as in [18], this index is written as

$$E_q={Ind}_{SU\left(2\right)}\left(D_{\mathrm{\Gamma },q}\right)=H^0({\mathbb{CP}}^1,L_{\mathrm{\Gamma }}^{q-1})\simeq H^0({\mathbb{CP}}^1,\mathcal{O}(q-1)),q\ge 1.$$

(3)

Here $D_{\mathrm{\Gamma },q}$ denotes the positive link Dirac operator twisted by $L_{\mathrm{\Gamma }}^{q-1}$. The usual Atiyah-Singer index framework [19] and the spinorial K-orientation background [20] are used in this local form. For regular families of defects, the corresponding refinement is the differential K-theoretic index package of [21], with the determinant-line holonomy comparison supplied by the Bismut-Freed construction [22].

The local source data make the primitive cycle filtered. For natural scalar-sector local sources of transverse order at most two, the associated-graded normal source on the link has the form

$$gr{J}_{\perp }^{\le 2}=V_0\oplus V_1\oplus V_2,{\left(gr{J}_{\perp }^{\le 2}\right)}_{ns}=V_1\oplus V_2.$$

(4)

The type $V_1$ is the phase-current channel, and $V_2$ is the Codazzi-gap channel. Thus the defect data determine the filtered equivariant link cycle

$${\mathfrak{D}}_{\mathrm{\Gamma }}^{fil}=\left({\mathbb{CP}}_{\mathrm{\Gamma }}^1,L_{\mathrm{\Gamma }},D_{\mathrm{\Gamma }};gr{J}_{\perp }^{\le 2}\right).$$

(5)

The notation records only the finite data used in the reconstruction; the analytic realization of this cycle by a gauge-fixed normal operator is treated separately in Section 6.

The visibility rule is the Toeplitz shadow of (3). On the Borel-Weil block $E_q$ one has

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

(6)

Its Toeplitz form is standard [23]; the finite-mode reading is the usual fuzzy-sphere cutoff [24], and finite-matrix brane models provide a broader comparison class [25]. It follows from (6) that the first separated visible realization of $V_1$ is $E_2$, while the first separated visible realization of $V_2$ is $E_3$. Hence a two-channel generic non-scalar filtered two-jet selects the finite support

$${\mathcal{V}}_{\mathrm{\Gamma }}=C_{\mathrm{\Gamma }}\oplus W_{\mathrm{\Gamma }},C_{\mathrm{\Gamma }}=E_3,W_{\mathrm{\Gamma }}=E_2.$$

(7)

The word “separated” refers to the two independent principal channels in (4). At the level of raw containment, $E_3$ already contains both $V_1$ and $V_2$ in its traceless endomorphisms; the carrier (7) records the first block on which each nonzero separated channel appears. The nearby alternatives and the raw-containment option are separated in Proposition 12 and Table 3. Higher multipoles do not change the compressed low support while the Schur gap remains open and the low-high coupling is subcritical. This persistence is recorded in Section 4 using the Schur estimates of Appendix B.

After (7) has been selected by the link-equivariant support rule, the equivariant action has served to fix the blocks. The compact basis group is then the unitary basis group of the selected Hermitian carrier spaces, with the unimodular top-form constraint:

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

(8)

Here and below, “basis group” means the carrier-basis group after the $SU\left(2\right)$-equivariant link action has selected the blocks. The standard exterior package on ${\mathcal{V}}_{\mathrm{\Gamma }}$ then gives the one-generation module. This is the familiar $3+2$ organization from the $SU\left(5\right)/Spin\left(10\right)$ comparison class [26]. Related Clifford-ideal realizations are discussed in [27] and [28]. In the present setting the exterior package is applied after the carrier has been reconstructed from the filtered link cycle.

A second comparison is with almost-commutative geometry. There the finite internal algebra is part of the input [29], while the spectral action provides the dynamical principle [30]. Modern refinements remain a natural comparison class [31], and Lorentzian issues remain relevant on the fermionic side [32]. Here the finite labels are attached to a quantized link of a closed four-dimensional branch problem. This places the construction closer to geometric matter models of the type considered in [33]. The fixed exterior parity used below is also compatible with differential-form realizations of fermions [34].

The same primitive class has a second finite shadow. Its integral representative gives the line $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$ used in (3). Its reduction modulo the projective-color center gives

$$c_1\left(L_{\mathrm{\Gamma }}\right)⟼{\overline{c}}_1\left(L_{\mathrm{\Gamma }}\right)\in H^2({\mathbb{CP}}^1,{\mathbb{Z}}_3)\simeq {\mathbb{Z}}_3.$$

(9)

The corresponding affine ${\mathbb{Z}}_3$ torsor carries the central family-response space. In the unblocked central response, one phase component and one adjacent-shift component generate the clock-shift algebra and hence $M_3\left(\mathbb{C}\right)$. This central response carrier is read from the same primitive class as the Borel-Weil tower, with coefficients reduced modulo the projective color center. Its interpretation as a dynamical generation multiplicity belongs to the completed spectral problem.

The analytic layer is used to represent the selected support as an isolated low sector. The relevant normal operator is of Dirac-Callias-Fredholm type; the classical Callias index mechanism [35] and its geometric formulation [36] supply the comparison class. In the regime used here, Codazzi-Callias dominance and a subcritical Schur bound isolate the low block selected by (7). For regular families, the corresponding finite-rank bundle carries the projected Berry connection [37]; its non-Abelian form is the Berry-Wilczek-Zee connection [38]. The global closed-branch problem is organized by a finite-dimensional Kuranishi obstruction map.

The finite package fixed above should be read as representation-level data. The carrier, the compact basis group, the exterior module, and the central family-response torsor are determined after the local support (7) has been selected. Yukawa operators, fermion masses, CKM and PMNS matrices, running couplings, thresholds, and the full quantum field dynamics remain closed spectral data. They enter the completed branch and are not used in the proof of the finite carrier theorem. The neutral Schur cell of Appendix D gives a finite benchmark after the primitive scalar unit has been fixed.

The main theorem concerns the first layer. The central family-response and the benchmark layers are recorded separately because they use additional algebraic or analytic data beyond the separated Toeplitz support.

 Theorem 1

(Primitive filtered carrier rigidity). Let Γ be a filtered primitive optical Codazzi defect in the sense of Definition 4. Assume that its natural scalar-sector transverse source has order at most two, that the scalar singlet has been separated, and that the principal non-scalar part of (4) is two-channel generic, so that both the $V_1$ and $V_2$ components are nonzero and separated at principal order. Then the resolved link defines the primitive filtered cycle (5). Its $SU\left(2\right)$-equivariant link index is the tower (3), and the unique minimal separated Toeplitz-visible support of its non-scalar filtered symbol is (7). After the selected blocks are read as Hermitian carrier spaces and the unimodular top-form constraint is imposed, the compact basis group is (8). The exterior algebra on ${\mathcal{V}}_{\mathrm{\Gamma }}$ gives the standard one-generation representation package. If, in addition, the gauge-fixed normal family satisfies the Codazzi-Callias gap and the subcritical Schur bound, the selected support is represented by an isolated finite low-sector bundle with Berry-Wilczek-Zee connection.

Table 1. Scope of the local reconstruction.

Layer	Output	Status
Alena gauge-branch selection	The Hilbert-Belinfante compatible current-residual sector supplies separated current and stress-response coefficient labels. After the scalar singlet is removed, these labels give the $V_1$ and $V_2$ source channels.	Sufficient local selection mechanism; recorded in Section 2.
Carrier rigidity	The primitive link, the Borel-Weil tower, and the order-$\le 2$ non-scalar source select the minimal separated support (7).	Main local theorem; proved in Section 4.
Finite representation package	The selected Hermitian carrier spaces and the unimodular top-form constraint give the carrier-basis group (8); the exterior package gives the standard one-generation representation content.	Finite representation consequence; recorded in Section 5.
Central family response	The coefficient reduction (9) gives a central ${\mathbb{Z}}_3$ family-response torsor. In the unblocked response, the phase and shift directions generate $M_3\left(\mathbb{C}\right)$.	Algebraic response datum; Berry-Wilczek-Zee response is conditional on Schur visibility.
Analytic representative	A Callias-Schur gap represents the selected support by an isolated low-sector bundle with Berry-Wilczek-Zee connection.	Conditional on the analytic gap and subcritical Schur bound.
Primitive benchmark	The neutral Schur cell gives a finite diagnostic after the primitive scalar unit has been fixed.	Appendix D; not used in the carrier theorem.
Closed spectral data	Yukawa operators, fermion masses, CKM and PMNS data, thresholds, and running are assigned to the completed branch.	Not fixed by the local filtered link cycle.

Table 1. Scope of the local reconstruction.

Layer	Output	Status
Alena gauge-branch selection	The Hilbert-Belinfante compatible current-residual sector supplies separated current and stress-response coefficient labels. After the scalar singlet is removed, these labels give the $V_1$ and $V_2$ source channels.	Sufficient local selection mechanism; recorded in Section 2.
Carrier rigidity	The primitive link, the Borel-Weil tower, and the order-$\le 2$ non-scalar source select the minimal separated support (7).	Main local theorem; proved in Section 4.
Finite representation package	The selected Hermitian carrier spaces and the unimodular top-form constraint give the carrier-basis group (8); the exterior package gives the standard one-generation representation content.	Finite representation consequence; recorded in Section 5.
Central family response	The coefficient reduction (9) gives a central ${\mathbb{Z}}_3$ family-response torsor. In the unblocked response, the phase and shift directions generate $M_3\left(\mathbb{C}\right)$.	Algebraic response datum; Berry-Wilczek-Zee response is conditional on Schur visibility.
Analytic representative	A Callias-Schur gap represents the selected support by an isolated low-sector bundle with Berry-Wilczek-Zee connection.	Conditional on the analytic gap and subcritical Schur bound.
Primitive benchmark	The neutral Schur cell gives a finite diagnostic after the primitive scalar unit has been fixed.	Appendix D; not used in the carrier theorem.
Closed spectral data	Yukawa operators, fermion masses, CKM and PMNS data, thresholds, and running are assigned to the completed branch.	Not fixed by the local filtered link cycle.

 Theorem 2

(Alena-type current-residual selection). Let the collar be a primitive non-degenerate Alena-type collar in the sense of Definition 3. Assume, in addition, that it is two-channel generic. Then the resolved regular component supplies the primitive link class $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$, and the current-built residual scalar supplies the non-scalar filtered source (4) with both $V_1$ and $V_2$ components nonzero. Hence the hypotheses of Theorem 1 are satisfied. If the collar is Schur-admissible, the selected support is represented by the low-sector bundle of Theorem 4.

 Proof. 

The primitive degree-one transverse trace gives the simple positive resolved class by Proposition 9. The residual scalar is constrained by Proposition 4, and the translational coefficient is reduced by Proposition 5. The split-conserved collar gives the principal coefficient separation by Proposition 6. The Rainich-current two-jet reduction is Proposition 3, and the Alena-type realization is Proposition 7. The two-channel genericity assumption, read through Proposition 7, gives Assumption A1. The filtered support theorem then gives (7). The final assertion is the analytic isolation statement of Theorem 4. □

 Corollary 1

(Self-describing gauge-branch carrier). Let a four-dimensional Alena-type collar be read in the closed gauge-branch sense described in Section 2. Assume that the branch is non-null Rainich, optical Codazzi, primitive, two-channel generic, and Schur-admissible, and that the frozen Hilbert-Belinfante coefficient labels are separated. Then the primitive branch reconstructs the carrier (7). Its compact basis group is (8), the exterior package on this carrier is the one-generation module of Section 5, and the primitive coefficient reduction gives the central family-response torsor (9).

 Proof. 

The gauge-branch reading and the frozen coefficient separation are recorded in Proposition 6. The worldline and projective-link step is recorded in Remark 2 together with Proposition 9. The non-scalar two-jet is (4), and the Alena-type realization is Proposition 7. The degree-one threshold comparison is Proposition 11. The separated support is then (7) by Theorem 3. The carrier-basis group (8) follows from the Hermitian carrier spaces together with (31), and Corollary 4 records the corresponding local anomaly condition. The central torsor is (9). Schur-admissibility gives the isolated representative by Theorem 4. □

 Proposition 1

(Local carrier self-closure). Under the hypotheses of Theorems 1 and 2, the finite part of the self-reconstruction loop (1) closes on the primitive carrier quotient. The reconstructed carrier is (7). If the Schur-admissibility hypotheses are also imposed, this carrier is represented by an isolated Schur-low sector. The statement is local and structural; the completed spectral data are not fixed by this proposition.

 Proof. 

The closed residual branch supplies separated principal source channels by Proposition 6. The multiplier and current reductions constrain the scalar and translational coefficients by Propositions 4 and 5. The primitive link gives the tower (3), and the order-$\le 2$ filtered source gives the non-scalar channels in (4). The support theorem gives (7). Thus reconstruction from the local closed observable data returns the same finite carrier. If the analytic hypotheses are imposed, Theorem 4 represents this support by an isolated low-sector bundle. The flavor eigenvalues, non-circulant Schur corrections, thresholds, and running belong to the completed branch. □

 Proposition 2

(Central family-response structure). Under the primitive coefficient reduction (9), the central response space is an affine ${\mathbb{Z}}_3$ torsor. If the adjacent response is unblocked, the central phase and adjacent-shift directions generate the full clock-shift algebra on this three-dimensional space. If, in addition, the low-sector family is regular, the Codazzi-Callias gap and the subcritical Schur bound hold, and these central directions are Schur-visible in the Riesz variation, the Berry-Wilczek-Zee curvature has the corresponding central non-Abelian response component. Leading circulant family operators are simultaneously diagonalizable; physical mixing requires non-circulant Schur-Berry data from the completed branch.

 Proof. 

The coefficient reduction is (9). The finite clock-shift closure is recorded in Section 5. The analytic low-sector bundle is supplied by Theorem 4. The Schur-visible curvature statement is Proposition 18. The circulant limitation is Proposition 19. □

 Remark 1

(Primitive Schur benchmark status). The neutral Schur cell is used as a finite diagnostic after the carrier (7), the compact group (8), and the primitive scalar unit have been fixed. The local carrier theorem is independent of this benchmark. Its promotion to a physical prediction requires the same scalar unit to be obtained from the completed spectral problem.

The proof of the main statements is distributed as follows. Section 2 records the Alena-type residual Lagrangian class, the gauge-branch coefficient separation, and the current-built two-jet selection. Section 3 constructs the primitive link cycle from the optical Codazzi branch and records the four-dimensional projective-link specificity. Section 4 proves the filtered Toeplitz support, the primitive degree-one threshold comparison, and the primitive low-carrier rigidity. Section 5 records the exterior package, the unimodular anomaly condition, the neutral Schur cell, and the mod-three family-response carrier. Section 6 gives the analytic representative, the Schur reduction, the Berry-Wilczek-Zee response, and the closed spectral data. Appendix A contains the component Codazzi calculation and the primitive clutching proof. Appendix B contains the normal-operator estimates and the variation formulae. Appendix C contains the finite algebraic checks. Appendix D contains the primitive Schur benchmark. Appendix E records the thin-core compactness and Kuranishi completion criteria.

Table 2. Readings and status of the primitive filtered cycle.

Reading	Output	Status
Closed-observable reading	The branch is treated as a primitive local representative of the self-reconstruction loop (1); the selected finite sector is an isolated Schur block of the closed observable algebra.	Guiding principle; only the local link reduction is used.
Rainich-current residual reading	The split-conserved residual scalar gives separated current and Codazzi traces; after the scalar singlet is removed, the order-$\le 2$ source has only the $V_1$ and $V_2$ non-scalar two-jet types.	Propositions 6, 3, and 7.
Optical Codazzi reading	The two-eigenvalue Codazzi branch gives the projective link ${\mathbb{CP}}_{\mathrm{\Gamma }}^1$ after real blow-up of the worldline defect.	Local Codazzi consequence.
Primitive topological reading	The simple positive transverse-frame class gives $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$ and the class $c_1\left(L_{\mathrm{\Gamma }}\right)$; the nonzero link class obstructs smooth filling of the normal ball.	Primitive sector and clutching.
Equivariant index reading	The twisted link Dirac operators give the Borel-Weil tower (3).	Standard link index.
Filtered source reading	The non-scalar two-jet gives the $V_1$ and $V_2$ channels of (4).	Two-channel generic stratum.
Toeplitz support reading	The visibility rule (6) gives the separated support (7).	Carrier rigidity theorem.
Finite algebraic reading	The carrier (7) gives $S\left(U\right(3)\times U(2\left)\right)$ and the one-generation exterior package.	Representation-theoretic consequence.
Coefficient reading	The reduction (9) gives the central ${\mathbb{Z}}_3$ family-response torsor.	Coefficient reduction.
Analytic and differential reading	The Callias-Schur gap gives an isolated low-sector representative, and its regular family carries the Berry-Wilczek-Zee connection.	Conditional on the analytic gap.
Global reading	The closed branch completion is organized by a Kuranishi obstruction map; flavor eigenvalues, thresholds, running, and non-circulant response data are closed spectral data.	Completed branch problem.

Table 2. Readings and status of the primitive filtered cycle.

Reading	Output	Status
Closed-observable reading	The branch is treated as a primitive local representative of the self-reconstruction loop (1); the selected finite sector is an isolated Schur block of the closed observable algebra.	Guiding principle; only the local link reduction is used.
Rainich-current residual reading	The split-conserved residual scalar gives separated current and Codazzi traces; after the scalar singlet is removed, the order-$\le 2$ source has only the $V_1$ and $V_2$ non-scalar two-jet types.	Propositions 6, 3, and 7.
Optical Codazzi reading	The two-eigenvalue Codazzi branch gives the projective link ${\mathbb{CP}}_{\mathrm{\Gamma }}^1$ after real blow-up of the worldline defect.	Local Codazzi consequence.
Primitive topological reading	The simple positive transverse-frame class gives $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$ and the class $c_1\left(L_{\mathrm{\Gamma }}\right)$; the nonzero link class obstructs smooth filling of the normal ball.	Primitive sector and clutching.
Equivariant index reading	The twisted link Dirac operators give the Borel-Weil tower (3).	Standard link index.
Filtered source reading	The non-scalar two-jet gives the $V_1$ and $V_2$ channels of (4).	Two-channel generic stratum.
Toeplitz support reading	The visibility rule (6) gives the separated support (7).	Carrier rigidity theorem.
Finite algebraic reading	The carrier (7) gives $S\left(U\right(3)\times U(2\left)\right)$ and the one-generation exterior package.	Representation-theoretic consequence.
Coefficient reading	The reduction (9) gives the central ${\mathbb{Z}}_3$ family-response torsor.	Coefficient reduction.
Analytic and differential reading	The Callias-Schur gap gives an isolated low-sector representative, and its regular family carries the Berry-Wilczek-Zee connection.	Conditional on the analytic gap.
Global reading	The closed branch completion is organized by a Kuranishi obstruction map; flavor eigenvalues, thresholds, running, and non-circulant response data are closed spectral data.	Completed branch problem.

The scope of the reconstruction is local and structural. In the terminology of (2), the paper proves the primitive finite-carrier part of a local zero-defect branch. In the Alena gauge-branch reading, the same proof chain may be read as a conditional self-description mechanism: a closed non-null gauge-stress branch supplies separated current and stress-response channels, the primitive projective link quantizes them, and the first separated Toeplitz supports give (7). The carrier (7), the group (8), and the central torsor from (9) do not determine Yukawa matrices, fermion masses, CKM or PMNS data, or the full running-coupling problem. These belong to the closed spectral problem. The neutral Schur cell discussed in Appendix D is a finite diagnostic after the carrier and the primitive finite scalar unit have been fixed.

2. Alena-Type Residual Lagrangians

The current-residual input is used as a local representative of the self-closure defect (2). A nonzero link obstruction is represented by a finite-energy residual core, while the finite carrier is read from the first non-scalar Hessian response of that core on the resolved link. The residual-scalar layer below records a sufficient local mechanism for reducing the closure defect: split k-conservation gives the principal coefficient separation, the Codazzi multiplier condition constrains the scalar, and current conservation constrains the translational coefficient. Its role is to supply, in a checkable collar class, the source and primitive-sector hypotheses of the filtered link theorem.

In the gauge-branch reading of the Alena-type sector, the branch-response tensor $Y_{\mu \nu }\left(k\right)$ is the closed-branch representative of the gauge-side stress tensor. The Hilbert variation gives the symmetric branch stress response, while the Noether-Belinfante reading fixes the current and spin-response coefficient labels in the flat representative [6]. Only the induced frozen coefficient split is used below. The finite carrier theorem does not require the full Alena dynamics.

The first two-jet step is independent of the special Alena-type normalization.

 Proposition 3

(Rainich-current two-jet reduction). Let S be a local Rainich-current residual source on an oriented codimension-three collar. Assume that its effective normal order is at most two, that the scalar singlet is separated, and that the source contains a conserved phase-current response and a vorticity/Codazzi response. Then the non-scalar associated-graded normal two-jet of S has only the $V_1$ and $V_2$ types. The $V_1$ component is the linear phase-current channel, and the $V_2$ component is the trace-free quadratic vorticity/Codazzi channel.

 Proof. 

Only the associated-graded normal two-jet is used. On the oriented normal fiber $N_{\mathrm{\Gamma }}$, the order-zero, order-one, and order-two symmetric normal terms have types $V_0$, $V_1$, and $V_0\oplus V_2$, respectively. After the scalar trace has been separated, the remaining non-scalar terms are exactly $V_1$ and $V_2$. The conserved current contributes through the first normal response, while the first anisotropic vorticity/Codazzi response contributes through the trace-free quadratic term. No type $V_{\ell }$ with $\ell \ge 3$ occurs at normal order at most two. □

The Alena-type class is a concrete realization of this two-jet.

 Definition 1

(Alena-type residual Lagrangian). A current-residual branch Lagrangian is called Alena-type on a collar if its residual part has the form

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

(10)

Here ζ 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 associated translational current is

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

(11)

The frozen amplitude block is assumed non-degenerate:

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

(12)

The Codazzi closure is imposed by a penalty-dominant term in the collar energy. The corresponding compactness and penalty estimates are recorded in Appendix E.

The residual scalar is used in a closed split-conserved sector.

 Definition 2

(Closed split-conserved residual collar). An Alena-type collar is called closed split-conserved if the product-rule Hilbert response of the residual density in (10) has a local split

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

(13)

and the two summands are separately k-conserved on the frozen collar:

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

(14)

Here ${\mathrm{\Xi }}_{\mu \nu }$ denotes the variation of the residual scalar density, while $Y_{\mu \nu }$ denotes the branch-response tensor coming from the variation of $p_{\mathrm{\Lambda }}$. The density contribution is included in $Y_{\mu \nu }$.

Put $\tau =k^{\mu \nu }{Y}_{\mu \nu }$ and $B_{\mu \nu }=Y_{\mu \nu }-{\displaystyle \frac{1}{3}}\tau k_{\mu \nu }$. The residual scalar is a Codazzi multiplier when the trace-adjusted tensor $A_{\mu \nu }=\varphi B_{\mu \nu }$ satisfies the Codazzi condition used in (21). If $C_{\alpha \mu \nu }^B={\nabla }_{\alpha }^{\left(k\right)}{B}_{\mu \nu }-{\nabla }_{\mu }^{\left(k\right)}{B}_{\alpha \nu }$ and $\theta =dlog\left|\varphi \right|$, this is equivalent to

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

(15)

 Proposition 4

(Codazzi multiplier criterion). Let X be a connected non-degenerate collar set on which $B_{\mu \nu }$ is invertible, and let ${\beta }^{\mu \nu }$ denote its inverse. If (15) holds, then θ is fixed by B:

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

(16)

Conversely, a nonzero scalar Codazzi multiplier exists on X if and only if (15) holds with $\theta ={\theta }^B$ and ${\theta }^B$ is exact. In that case ϕ is determined on X up to a nonzero constant factor.

 Proof. 

Contracting (15) with ${\beta }^{\mu \nu }$ gives (16). If a multiplier exists, then $\theta =dlog\left|\varphi \right|$, hence ${\theta }^B$ is exact. Conversely, if (15) holds with $\theta ={\theta }^B=d{f}_B$, then $\varphi =C_{\varphi }{e}^{f_B}$ gives a local Codazzi multiplier. The possible periods belong to the closed branch problem. □

 Proposition 5

(Codazzi-current reduction). Let the residual scalar in (10) be a Codazzi multiplier on a non-degenerate connected collar set. If ${\theta }^B=d{f}_B$, then

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

(17)

If, in addition, (11) holds, then

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

(18)

 Proof. 

The first identity follows from Proposition 4; substitution into (10) gives (17). The conservation law in (11) gives (18) after differentiating along U. □

 Corollary 2

(Residual-scalar freedom reduction). On a connected non-degenerate collar set on which $B_{\mu \nu }$ is invertible and ${\theta }^B$ is exact, the pair $(\varphi ,{\zeta }^2)$ is not independent branch input. The Codazzi multiplier criterion fixes ϕ up to one nonzero constant factor, and the residual-scalar identity fixes ${\zeta }^2$. If translational-current conservation is imposed, the remaining compatibility is the transport equation (18). Periods of ${\theta }^B$ and singular strata of B belong to the closed branch problem.

 Proof. 

This is the combined content of Propositions 4 and 5. □

 Proposition 6

(Hilbert-Belinfante channel separation). On a closed split-conserved residual collar, in the gauge-branch reading described above, the frozen principal source separates into the translational-current trace and the residual stress/Codazzi trace. The associated-graded coefficient channels are therefore separated before Toeplitz visibility is applied. After the scalar singlet is removed and the normal order is restricted to at most two, the current coefficient gives the $V_1$ channel and the trace-free stress/Codazzi coefficient gives the $V_2$ channel.

 Proof. 

The two equations in (14) are imposed at the frozen collar level. Their principal traces are therefore assigned to distinct coefficient labels. In the Hilbert reading the second summand in (13) is the symmetric branch-response stress. In the Belinfante reading, the current and spin-response data enter through the corresponding superpotential term. In the split-conserved frozen principal symbol, this changes the representative but not the assigned current and stress-response coefficient labels. The gauge-fixed normal operator records the same principal statement in Proposition A1. Proposition 3 then identifies the order-one current coefficient with $V_1$ and the trace-free order-two stress/Codazzi coefficient with $V_2$. □

 Definition 3

(Primitive non-degenerate Alena collar). A regular thin-core component of an Alena-type collar is called primitive and non-degenerate if:

1.

its transverse degree is one on the linking spheres, with positive orientation;

2.

the residual sector is closed split-conserved in the sense of Definition 2;

3.

the residual scalar is an admissible Codazzi multiplier in the sense of Proposition 4;

4.

the penalty-dominant limit gives a Codazzi-closed two-eigenvalue optical branch with nonzero Codazzi gap on the frozen collar.

It is called two-channel generic if the two non-scalar coefficients of the current-built normal two-jet are both nonzero. It is called Schur-admissible if, in addition, the Codazzi-Callias gap and the Schur bound hold on the parameter domain used for the analytic low-sector representative.

 Proposition 7

(Current-built Alena two-jet). Let an Alena-type residual scalar be restricted to a primitive regular collar component. Assume the closed split-conserved sector of Definition 2 and the multiplier reduction of Proposition 5. Let ${\nu }_{\mathrm{\Gamma }}^{\perp }$ be the degree-one transverse trace on the resolved normal link. Then the non-scalar part of the associated-graded normal two-jet has the form

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

(19)

The first term has type $V_1$, and the trace-free quadratic term has type $V_2$. On the locus $c_1\left(\varphi \right)c_2\left(\varphi \right)\ne 0$, Assumption A1 is satisfied.

 Proof. 

The Alena-type scalar (10) is a Rainich-current residual source of the type used in Proposition 3. The multiplier reduction (17) and the transport equation (18) constrain the phase-current coefficient. The split conservation in (14) separates this current trace from the residual Codazzi trace at frozen principal order. The primitive transverse trace gives the displayed $V_1$ and $V_2$ components. The nonzero coefficient condition gives Assumption A1. □

The degree-one condition in Definition 3 is the primitive sector used in the link construction. After real blow-up and the positive orientation convention, it gives the simple positive transverse class and hence the line $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$ by Proposition 9. Higher transverse degrees give different link quantization data and are not part of the primitive carrier theorem.

The Alena-type collar therefore supplies a sufficient source side of the reconstruction: the primitive topological class gives the Borel-Weil tower, while the universal Rainich-current reduction of Proposition 3 is realized by (19). The Toeplitz support calculation is independent of this Lagrangian realization and is carried out in Section 4.

3. From Codazzi to the Primitive Link Cycle

The local geometric input is an optical ARC branch associated with the non-null Rainich tensor identified in [5]. Only the part needed to construct the primitive link cycle is retained here. Standard gauge and spinor conventions are those of [39] and [40]; the connection is the Levi-Civita connection of the branch metric, as usual in Lorentzian geometry [41].

The section has two roles. First, the two-eigenvalue Codazzi closure is converted into the optical projective link. Second, the real blow-up of the worldline defect and the simple positive resolved frame are used to construct the unfiltered ${\mathrm{spin}}^c$ link cycle. The filtered source part is added in Section 4; the analytic normal operator and the Callias-Schur representative are treated in Appendix B.

A codimension-three worldline core is supplied here by a primitive non-degenerate Alena-type collar, with the thin-core compactness criterion recorded in Appendix E. The local reconstruction below uses only the resulting optical branch, the Codazzi closure, the real blow-up, and the primitive resolved transverse class.

3.1. Optical Codazzi Input and Projective Link

The branch metric is denoted by k, and the trace-adjusted branch tensor by A. On the non-degenerate set the algebraic Rainich type determines an orthogonal splitting

$${TM|}_{\mathcal{U}}=E\oplus F,dimE=2,dimF=2,$$

(20)

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

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

(21)

The algebraic Rainich side is classical [10]. Generic perturbations need not remain in the same algebraic stratum, as discussed in [42]. Related self-dual and conformal-metric comparison classes are those of [43] and [44].

Only the standard two-eigenvalue Codazzi mechanism is needed.

 Lemma 1

(Optical Codazzi lemma). Let $A^{♯}$ be the k-self-adjoint endomorphism defined by A, and assume that on $\mathcal{U}$ it has two distinct eigenvalues M and P with eigenbundles E and F as in (20). Then (21) is equivalent to

$$\begin{array}{cc}\hfill {\left({\nabla }_{X_1}^{\left(k\right)}{X}_2\right)}_F& ={\displaystyle \frac{k(X_1,X_2)}{M-P}}{\left({\nabla }^{\left(k\right)}M\right)}_F,X_1,X_2\in \mathrm{\Gamma }\left(E\right),\hfill \end{array}$$

(22)

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

(23)

In particular, the two null directions in E are geodesic and shear-free.

 Proof. 

The component decomposition of (21) with respect to $E\oplus F$ gives the two-eigenvalue Codazzi identities. The mixed components give (22) and (23), and the converse follows by recombining the same components. The null directions in the Lorentzian plane E are therefore tangent to geodesic congruences, and the trace-free transverse optical part vanishes. The component calculation is recorded in Appendix A. □

The underlying identities are standard in the Riemannian Codazzi setting [45]. Lorentzian Codazzi structures are discussed in [46]. Related $2+2$ optical geometries occur in [47] and [48], while the divergence-free non-null Maxwell comparison belongs to the Rainich program of [11].

Thus a non-degenerate Codazzi-closed ARC branch carries the optical structure used below: the Lorentzian principal plane gives two shear-free geodesic null directions, and the spacelike principal plane gives the corresponding screen space. No Einstein-vacuum or Petrov assumption is used in this step.

The compact-leaf obstruction supplies the worldline core. In the warped collar model recalled in Appendix E, the normalized vorticity closure has a distributional source on a compact transverse leaf. The curvature conventions used there are the standard ones of [49] and [50]. Following the core of this source along the Lorentzian plane gives the worldline defect $\mathrm{\Gamma }$.

The defect is resolved by real blow-up:

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

(24)

The boundary fiber in (24) is the link sphere of the resolved defect. The projective-spinor interpretation of the optical structure is the standard one of [12] and [51]; the broader curved-twistor comparison class is represented by [52]. Only the local projective line attached to the defect is used.

 Remark 2

(Four-dimensional projective-link specificity). For a particle-like defect, the requirement that the resolved link be the projective line is specific to four spacetime dimensions. A worldline in a Lorentzian D-manifold has link $S^{D-2}$, and $S^{D-2}\simeq S^2\simeq {\mathbb{CP}}^1$ gives $D=4$. In this dimension the non-null Rainich two-plane splitting in (20) also exhausts the tangent space.

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

$${\mathbb{R}}^4\setminus \mathbb{R}\simeq \mathbb{R}\times ({\mathbb{R}}^3\setminus \left\{0\right\}),H^1({\mathbb{R}}^4\setminus \mathbb{R};\mathbb{R})=0,H^2({\mathbb{R}}^4\setminus \mathbb{R};\mathbb{Z})\simeq \mathbb{Z}.$$

(25)

The $H^1$ statement removes the local scalar-period obstruction. The integral $H^2$ class is the topological slot occupied by the primitive link charge.

The same local topology gives the determinant-line obstruction used by the primitive sector.

 Proposition 8

(Link-class obstruction to smooth filling). Let $B^3$ be a normal ball to a candidate worldline component and let $S^2=\partial B^3$ be its linking sphere. If the ${\mathrm{spin}}^c$ determinant line on the punctured normal ball restricts to a line of Chern number n on $S^2$, then it extends as a smooth topologically trivial determinant line over $B^3$ only when $n=0$. For $n=1$, after the projective identification of the link, the restricted line is $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$.

 Proof. 

Complex line bundles on $S^2$ are classified by their first Chern class, while every complex line bundle on $B^3$ is topologically trivial. Hence a smooth extension over the filled normal ball has trivial restriction to $S^2$. A nonzero Chern number therefore obstructs smooth filling through the core. For $n=1$, the restricted line is the positive generator of $\mathrm{Pic}\left({\mathbb{CP}}^1\right)$ after the projective link has been identified. □

The optical projective-spinor reading identifies the boundary fiber in (24) with the projective spinor sphere:

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

(26)

This is the projective link used in the filtered index construction.

3.2. Blow-up, Primitive Line, and the Unfiltered Link Cycle

If the oriented transverse frame is resolved across the defect, its north-south patching on an equatorial collar is described by a clutching map $g_{NS}:S^1\to SO\left(2\right)\simeq U\left(1\right)$. In local trivializations the connection forms satisfy $A_S=A_N+g_{NS}^{-1}d{g}_{NS}$, and the curvature period is the degree of $g_{NS}$. The resolved-frame patching and the local regularity background are the standard ones of elliptic theory [53] and [54].

 Lemma 2

(Primitive clutching). Assume that, on an equatorial collar, the leading clutching of the resolved transverse frame is induced, after oriented local screen coordinates have been chosen, by an orientation-preserving invertible real linear map $A:{\mathbb{R}}^2\to {\mathbb{R}}^2$. Then the clutching degree is 1.

 Proof. 

The leading equatorial map is homotopic to $v\mapsto Av/\left|Av\right|$ on $S^1$. Since $A\in G{L}^{+}(2,\mathbb{R})$, the homotopy stays in the orientation-preserving component. This component retracts to $SO\left(2\right)$, and the induced degree is 1. □

The simple positive sector is the sector used in the reconstruction. By Lemma 2, the associated line $L_{\mathrm{\Gamma }}\to S_{\mathrm{\Gamma }}^2$ has $c_1\left(L_{\mathrm{\Gamma }}\right)=1$, hence $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$ after (26). Proposition 8 records the corresponding obstruction to smooth filling of the normal ball. Higher clutching degrees correspond to multiple local link charges and define different link quantization data. Their degree comparison is recorded in Appendix D.

The unfiltered primitive link cycle is the finite datum

$${\mathfrak{D}}_{\mathrm{\Gamma }}^0=\left({\mathbb{CP}}_{\mathrm{\Gamma }}^1,L_{\mathrm{\Gamma }},D_{\mathrm{\Gamma }}\right),L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right).$$

(27)

Here $D_{\mathrm{\Gamma }}$ denotes the ${\mathrm{spin}}^c$ link Dirac operator associated with the projective link. For $q\ge 1$, its twists by $L_{\mathrm{\Gamma }}^{q-1}$ give the index tower already written in (3). The same zero-mode spaces are the standard monopole harmonics on $S^2$ [15] and occur in the Taub-NUT Dirac comparison class [16]. The Borel-Weil identification is the standard one [17], with representation conventions as in [18].

This motivates the following definition.

 Definition 4.

A primitive optical Codazzi defect is a worldline defect Γ in a non-degenerate optical Codazzi branch such that:

1.

the branch metric and trace-adjusted tensor satisfy (20) and (21) on $M\setminus \mathrm{\Gamma }$;

2.

the real blow-up (24) has projective link (26);

3.

the resolved oriented transverse frame is simple and positive in the sense of Lemma 2.

If, in addition, a natural scalar-sector transverse source of order at most two is fixed, the defect will be called filtered primitive. The filtered source condition is introduced in Section 4.

 Proposition 9

(Primitive link cycle). Every primitive optical Codazzi defect in the sense of Definition 4 determines the unfiltered link cycle (27). Its positive twists realize the equivariant index tower (3).

 Proof. 

Lemma 1 gives the local projective-spinor structure. Real blow-up gives the link sphere in (24), and the optical projective-spinor reading identifies it with (26). The resolved transverse frame determines the equatorial clutching map. Lemma 2 gives degree 1 in the simple positive sector, so the associated line is $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$. The ${\mathrm{spin}}^c$ link Dirac operator with this line is the cycle (27), and its positive twists give (3). □

The output of this section is therefore the primitive ${\mathrm{spin}}^c$ link cycle, not yet the finite carrier. The latter requires the filtered source part of the cycle. The $SU\left(2\right)$ and spherical-tensor conventions used in the next section are the standard ones of [55] and [56].

4. The Filtered Equivariant Index and Primitive Low-Carrier Rigidity

By Proposition 9, a primitive optical Codazzi defect determines the unfiltered link cycle (27). The equivariant index of its positive twists is the Borel-Weil tower (3). The present section adds the filtered source part, proves the separated first-support rule, and records the low-carrier rigidity statement.

The link problem is local. The positive link modes are standard twisted ${\mathrm{spin}}^c$ Dirac zero modes on ${\mathbb{CP}}^1$ in the sense of [57]. The usual conic comparison class is represented by [58]; boundary compression is the standard pseudodifferential projection step of [59], and the finite functional-calculus background is that of [60]. The same zero-mode count is given by the Aharonov-Casher index formula [61].

The projective-line reading used below is local. Twistor comparison classes are discussed in [14] and [62]. Self-dual spinorial comparison classes are classical in [63] and [64]. Related chiral and pure-connection descriptions are given in [65,66,67], and [68]. These comparison structures are not used to alter the local filtered index problem.

4.1. Filtered Source Symbol

The finite labels attached to the link are treated as sector labels rather than as open local tensor indices. Closed local observables are therefore singlet-valued in the standard sense of sector reconstruction [69]. In particular, the scalar component in (4) is retained as a local singlet and is excluded from the non-singlet support problem.

A principal source on the defect link will be called a natural scalar-sector local source of transverse order at most two if, near $\mathrm{\Gamma }$, it is given by a bundle-natural differential expression in the local optical Codazzi data whose dependence on derivatives normal to $\mathrm{\Gamma }$ is of order at most two, and whose principal normal symbol has been reduced to the scalar sector after all non-normal tensor indices have been contracted or projected to closed local singlets. In defect-adapted normal coordinates $(r,\omega ,t)$, it is determined by the local germ of the branch data, transforms tensorially under changes of defect-adapted coordinates, and depends on the transverse variables only through the symmetric normal jet of order at most two. Nonlocal projections, pseudodifferential operations, and transverse derivatives of order three or higher are excluded from this class.

For such a source, only the associated-graded normal jet enters the principal link problem. The resulting filtered source part of the primitive cycle is the one recorded in (4), and the filtered link cycle is (5).

 Proposition 10

(Filtered two-jet symbol). Let Γ be a primitive optical Codazzi defect, and let the fixed source be natural scalar-sector of transverse order at most two. Then the associated-graded normal source on the link is (4). Its non-scalar part consists of the separated principal channels $V_1$ and $V_2$.

 Proof. 

Let $N_{\mathrm{\Gamma }}$ be the oriented transverse normal fiber. By the order assumption and the scalar-sector reduction, the associated-graded transverse data take values in ${Sym}^0\left(N_{\mathrm{\Gamma }}^{*}\right)\oplus {Sym}^1\left(N_{\mathrm{\Gamma }}^{*}\right)\oplus {Sym}^2\left(N_{\mathrm{\Gamma }}^{*}\right)$. After restriction to the link sphere, these are the degree 0, degree 1, and degree 2 spherical components. For an oriented rank-three transverse fiber one has the standard $SU\left(2\right)\simeq \mathrm{Spin}\left(3\right)$ identifications with $V_0$, $V_1$, and $V_0\oplus V_2$, respectively. Separating the scalar trace leaves exactly $V_1$ and $V_2$. A type $V_{\ell }$ with $\ell \ge 3$ would require a homogeneous transverse term of degree at least three. □

The principal frozen link problem is natural on the round projective line. Hence the assignment of principal coefficient channels to the principal symbol is $SU\left(2\right)$-equivariant. The analytic reduction behind this statement is recorded in Appendix B: the gauge-fixed Codazzi symbol is elliptic, the boundary Hodge symbol reduces under admissible middle-spinor data to the Clifford symbol on the projective link, and the associated-graded normal operator preserves the phase-current and Codazzi-gap coefficient channels at principal order.

For later use, a local genericity condition is fixed. A moment-resolved core means that the associated-graded transverse trace up to order two is represented by a normalized core profile on $N_{\mathrm{\Gamma }}\simeq {\mathbb{R}}^3$, with first moment $m\in {\mathbb{R}}^3$ and trace-free second moment $M_0\in {Sym}_0^2\left({\mathbb{R}}^3\right)$. In the Rainich-current reading, the first moment is the phase-current moment, while the trace-free second moment is the vorticity/Codazzi-gap moment. The scalar trace of the second moment belongs to the $V_0$ channel.

 Lemma 3

(Moment-resolved genericity). For a moment-resolved source, the assignment from $(m,M_0)$ to the non-scalar link channels $({\xi }_{V_1},{\xi }_{V_2})$ is an isomorphism of $SU\left(2\right)$ modules. Hence the locus ${\xi }_{V_1}\ne 0$ and ${\xi }_{V_2}\ne 0$ is open and dense in the model moment space.

 Proof. 

The first moment gives the degree-one link component and hence the $V_1$ channel. The trace-free second moment gives the degree-two link component and hence the $V_2$ channel. These are the standard irreducible $SU\left(2\right)\simeq \mathrm{Spin}\left(3\right)$ identifications for an oriented rank-three transverse fiber. Their direct sum gives the stated isomorphism. The excluded locus is the union of the two proper linear conditions $m=0$ and $M_0=0$. □

 Assumption A1

(Generic filtered source). The realized principal non-scalar source has nonzero projections to both $V_1$ and $V_2$.

This is the only source nondegeneracy used in the local carrier theorem. It is not part of the primitive link topology. For a general natural scalar-sector source it is a local nondegeneracy hypothesis on the realized principal symbol. For a Rainich-current residual source of order at most two, Proposition 3 gives the same two possible non-scalar types. For an Alena-type residual collar, Proposition 7 identifies the generic condition with $c_1\left(\varphi \right)c_2\left(\varphi \right)\ne 0$. In the moment-resolved reading, this is the nonvanishing of the first phase-current moment and the trace-free second vorticity/Codazzi moment, in the sense of Lemma 3.

4.2. Toeplitz Support and Carrier Rigidity

The Toeplitz visibility rule on the primitive link is (6). Its Toeplitz form on ${\mathbb{CP}}^1$ is standard [23]; the finite-mode cutoff is the fuzzy-sphere cutoff of [24], and finite-matrix brane models provide the comparison class of [25].

 Lemma 4

(Separated first-support rule). For an irreducible integer $SU\left(2\right)$ type $V_{\ell }$, the separated Toeplitz support in the primitive tower is the first Borel-Weil block $E_q$ for which $V_{\ell }$ occurs in ${End}_0\left(E_q\right)$. This block is $E_{\ell +1}$.

 Proof. 

By (6), ${End}_0\left(E_q\right)$ contains precisely the integer types $V_1,\dots ,V_{q-1}$. Hence the first occurrence of $V_{\ell }$ is at $q=\ell +1$. □

 Proposition 11

(Degree-one threshold uniqueness). Let the positive link line be replaced formally by a degree-n line $O\left(n\right)$ with $n\ge 1$. Then the first block in the corresponding positive tower on which $V_{\ell }$ is visible has index

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

(28)

Consequently, only the primitive degree-one case has separated first thresholds $V_1\mapsto E_2$ and $V_2\mapsto E_3$. For every $n\ge 2$, the $V_1$ and $V_2$ thresholds occur in the same first nontrivial block of the modified tower.

 Proof. 

For a degree-n line, the q-th positive block is $H^0({\mathbb{CP}}^1,O\left(n(q-1)\right))$. Its traceless endomorphism space contains the integer types up to $V_{n(q-1)}$. Thus the first block which contains $V_{\ell }$ is given by (28). The stated alternatives follow by substituting $\ell =1$ and $\ell =2$. □

 Theorem 3

(Primitive low-carrier rigidity). Let Γ be a filtered primitive optical Codazzi defect in the sense of Definition 4. Assume Assumption A1, and assume that the two principal channels are retained as separated coefficient labels before Toeplitz visibility is applied. Then the unique minimal separated Toeplitz-visible support of the non-scalar filtered source is the carrier (7).

 Proof. 

By Proposition 10, the non-scalar filtered source has separated components of types $V_1$ and $V_2$. By Assumption A1, both are present. Lemma 4 gives first support $E_2$ for $V_1$ and first support $E_3$ for $V_2$. Since the channels are kept separated at principal order, their first supports are recorded separately. This gives (7). Minimality follows from the same first-support rule. □

The separated condition in Theorem 3 refers to the two principal channels of the filtered source. At the level of raw containment in a single block, $E_3$ already contains both $V_1$ and $V_2$ in its traceless endomorphisms. The carrier (7) records the first block on which each nonzero separated channel appears.

The low-block rigidity used later is the same Clebsch-Gordan count. With $P_{23}$ denoting the orthogonal projection onto $E_2\oplus E_3$ inside the positive Borel-Weil tower, no integer $V_{\ell }$ with $\ell \ge 3$ occurs in ${End}_0(E_2\oplus E_3)$. Indeed, ${End}_0\left(E_2\right)=V_1$ and ${End}_0\left(E_3\right)=V_1\oplus V_2$, while the mixed blocks $Hom(E_2,E_3)$ and $Hom(E_3,E_2)$ carry only half-integer types by the Clebsch-Gordan rule [18]. Hence any principal integer multipole $T^{(\ell )}$ with $\ell \ge 3$ satisfies $P_{23}{T}^{(\ell )}{P}_{23}=0$.

Since $E_q$ carries spin $(q-1)/2$, the first possible low-high occurrences are also fixed by the same count. For $\ell \ge 3$, the type $V_{\ell }$ occurs in $Hom(E_3,E_r)$ first for $r=2\ell -1$, and in $Hom(E_2,E_r)$ first for $r=2\ell$. Thus $V_3$ has no low-high channel through $E_4$; its first occurrences are $E_3\leftrightarrow E_5$ and $E_2\leftrightarrow E_6$.

Thus the order-$\le 2$ hypothesis is a minimal low-sector hypothesis. Higher transverse moments are not excluded by the framework. They define higher multipole channels and enter the closed high-sector problem. Their effect on the low carrier is controlled by the Schur gap and by the low-high coupling.

 Proposition 12

(Separated-support alternatives). In the primitive degree-one tower, the separated first-support rule used in Theorem 3 has the alternatives displayed in Table 3. In particular, if both $V_1$ and $V_2$ are present, the primitive separated carrier is (7). The raw single-block containment in $E_3$ does not satisfy the separated first-support rule for the $V_1$ channel.

 Proof. 

Lemma 4 gives first support $E_2$ for $V_1$ and first support $E_3$ for $V_2$. If only one of the two channels is present, only its first support is selected. If both are present and the channels are kept separated at principal order, their first supports are recorded separately, giving (7). The single block $E_3$ contains both channels as raw endomorphism types, but it is not the first support of the $V_1$ channel. The case $\ell \ge 3$ follows from the low-block rigidity count above. Higher-degree link sectors change the primitive tower data and are therefore different link quantization data. □

Table 3. Primitive separated-support alternatives and local negative controls.

Input sector	$V_1$ channel	$V_2$ channel	Selected separated support
$V_1\ne 0$, $V_2\ne 0$	first on $E_2$	first on $E_3$	$E_3\oplus E_2$
$V_1\ne 0$, $V_2=0$	first on $E_2$	absent	$E_2$
$V_1=0$, $V_2\ne 0$	absent	first on $E_3$	$E_3$
$V_1=0$, $V_2=0$	absent	absent	no non-scalar low carrier
principal $V_{\ell }$, $\ell \ge 3$	absent at principal order	absent at principal order	high-sector data
degree $n>1$ link	altered tower	altered tower	non-primitive link sector

Table 3. Primitive separated-support alternatives and local negative controls.

Input sector	$V_1$ channel	$V_2$ channel	Selected separated support
$V_1\ne 0$, $V_2\ne 0$	first on $E_2$	first on $E_3$	$E_3\oplus E_2$
$V_1\ne 0$, $V_2=0$	first on $E_2$	absent	$E_2$
$V_1=0$, $V_2\ne 0$	absent	first on $E_3$	$E_3$
$V_1=0$, $V_2=0$	absent	absent	no non-scalar low carrier
principal $V_{\ell }$, $\ell \ge 3$	absent at principal order	absent at principal order	high-sector data
degree $n>1$ link	altered tower	altered tower	non-primitive link sector

The table records local negative controls for the support selection. Degeneration of either principal channel changes the selected low support, while a higher-degree link changes the tower itself.

 Corollary 3

(Uniqueness of the primitive separated carrier). In the primitive degree-one tower, among natural scalar-sector transverse sources of order at most two with nonzero separated $V_1$ and $V_2$ principal channels, the carrier (7) is the unique minimal separated Toeplitz-visible support. The raw single-block realization inside $E_3$ is not minimal for the $V_1$ channel in the separated-support sense. Higher integer multipoles and higher-degree link sectors belong to different, respectively high-sector or non-primitive, data.

 Proof. 

This is the two-channel case of Proposition 12. The first supports of the two separated channels are $E_2$ and $E_3$ by Lemma 4. The low-block rigidity count excludes principal integer $V_{\ell }$, $\ell \ge 3$, from the compressed support (7), while higher-degree links change the primitive tower data. □

4.3. Support Refinement and Analytic Separation

The theorem above is representation-theoretic. A positive support cost gives the corresponding effective selection statement. For one separated channel, let P be a finite visible support and let

$${\mathcal{E}}_a(\mathcal{B},P)=\langle \mathcal{B},{\mathcal{A}}_{a,P}\mathcal{B}\rangle -2\mathrm{Re}\langle \mathcal{B},{\mathcal{S}}_a\rangle +{\lambda }_a\mathrm{Tr}P,a=1,2,$$

(29)

where ${\mathcal{A}}_{a,P}>0$ on the active channel and ${\lambda }_a>0$. In the collar model of Appendix E, ${\lambda }_a$ may be taken proportional to the Codazzi gap.

 Proposition 13

(Positive support refinement). Assume that (29) is block diagonal at principal order in the separated $V_1$ and $V_2$ channels. If a visible enlargement of a support carries no principal source component, then it is not selected by the minimum of (29). The conclusion is stable under lower-order mixed terms whose relative bound is smaller than the support gap.

 Proof. 

For fixed P, the Euler equation is ${\mathcal{A}}_{a,P}\mathcal{B}={\mathcal{S}}_a$. On a passive summand the source projection vanishes, hence the minimizing response on that summand is zero. The source term is unchanged by adding the passive summand, while the last term in (29) increases by ${\lambda }_a\mathrm{Tr}{P}_{\mathrm{pass}}$. In the Schur reduction of Appendix B, a passive high summand with gap $g_H$ and low-high coupling $\mathcal{K}$ contributes at most ${\parallel \mathcal{K}\parallel }^2/g_H$ in operator norm. The passive summand remains excluded when this gain is smaller than the support penalty. A relatively bounded lower-order perturbation changes the minimum continuously, so a strict support gap persists. □

 Proposition 14

(Persistence under higher multipoles). Let $P_{23}$ be the projection onto the carrier (7). Allow a higher-order perturbation of the source with integer multipoles $V_{\ell }$, $\ell \ge 3$, and assume that the primitive tower data are unchanged. If the high-sector inverse in the Schur complement exists on the low spectral window, and if the lower-order mixed block has relative bound below the support gap, then the support selected by Theorem 3 is unchanged.

 Proof. 

The principal low-low compression of every integer $V_{\ell }$, $\ell \ge 3$, to $E_2\oplus E_3$ vanishes by the Clebsch-Gordan count above. Thus higher multipoles can affect the selected low support only through low-high coupling and the corresponding Schur correction. Proposition A3 gives the effective low-sector reduction when the high-sector inverse exists. Proposition A2 gives stability when the lower-order mixed gain is smaller than the strict support gap. Proposition 13 then excludes passive visible enlargements without principal source component. □

The Codazzi gap has two local uses. It is the scale of the two-eigenvalue Codazzi splitting, and it is the support barrier in (29). In the locked spectral sector it also supplies the zeroth-order input for the Callias estimate in Appendix B. The limit in which the gap closes lies on the boundary of the admissible optical phase; the two-eigenvalue splitting, the projective-link reading, and the Callias locking estimate degenerate together.

The content of Theorem 3 is local. The link cycle supplies the Borel-Weil tower, the filtered source fixes the non-scalar $SU\left(2\right)$ content, and Toeplitz visibility identifies the first separated support of each nonzero channel. Scale setting, Yukawa data, mixing, running couplings, higher clusters, and family refinement are not part of this support theorem. They enter through the finite package and the analytic family response described in Section 5 and Section 6.

5. The Finite Standard-Model Package

The support theorem of Section 4 has fixed the finite carrier (7). The structures recorded in this section are read after that support has been selected. The representation-theoretic comparison with the usual $SU\left(5\right)$/Spin$\left(10\right)$ organization is standard [26]. Related Clifford-ideal realizations are discussed in [27] and [28]; the octonionic ladder-operator comparison is represented by [70]. Differential-form realizations of fermions are reviewed in [34].

Set

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

(30)

Then $dimC=3$ and $dimW=2$ by (7). Each selected Borel-Weil block carries its standard $SU\left(2\right)$-invariant $L^2$ Hermitian structure induced by the link geometry on ${\mathbb{CP}}^1$. The $SU\left(2\right)$-equivariant link action has already been used in the separated-support selection. At the finite carrier level the admissible basis changes are the unitary changes of the selected Hermitian carrier spaces. Thus, before unimodularity, the carrier-basis group is $U\left(C\right)\times U\left(W\right)$. The unimodular reduction is imposed by fixing the carrier top form on ${\mathrm{\Lambda }}^{5}V$; the admissible basis changes satisfy

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

(31)

This gives the compact group (8) in the carrier-basis sense. The link-equivariant commutant of the irreducible Borel-Weil blocks remains the Schur commutant. Equivalently, the induced global form is $S(U\left(3\right)\times U\left(2\right))\simeq (SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y)/{\mathbb{Z}}_6$. No finite internal algebra is inserted at this stage; the finite labels are those of the support (7).

5.1. Exterior Representation Package

The one-generation module is taken to be

$${\mathcal{F}}_{\mathrm{\Gamma }}^{\mathrm{even}}:={\mathrm{\Lambda }}^{\mathrm{even}}V.$$

(32)

Its dimension is $1+\left({\displaystyle \genfrac{}{}{0pt}{}{5}{2}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{5}{4}}\right)=16$. The hypercharge convention used below is the degree-counting normalization on ${\mathrm{\Lambda }}^{•}V$ compatible with (31). If $N_C$ and $N_W$ denote the C- and W-degree operators, then

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

(33)

The electric charge is $Q=T_3+Y$, with $T_3$ acting in the usual way on the $SU\left(2\right)$ factor.

 Corollary 4

(Unimodular anomaly condition). For the carrier (30), let $Y_{a,b}=a{N}_C+b{N}_W$. The infinitesimal form of the top-form constraint (31) is $3a+2b=0$. The local $SU{\left(3\right)}^{2}U\left(1\right)$, $SU{\left(2\right)}^{2}U\left(1\right)$, gravitational-$U\left(1\right)$, and cubic $U\left(1\right)$ anomaly sums of ${\mathrm{\Lambda }}^{\mathrm{even}}V$ are all proportional to this same factor. Hence (33) is the normalized degree-counting representative of the unimodular anomaly-free class.

 Proof. 

The general anomaly factors are recorded in Appendix C. Each contains the factor $3a+2b$. The normalization $a=-1/3$, $b=1/2$ gives (33). □

The identifications in Table 4 use the top-form trivialization determined by (31). In particular, ${\mathrm{\Lambda }}^{2}C\otimes {\mathrm{\Lambda }}^{2}W$ is identified with $C^{*}$, and ${\mathrm{\Lambda }}^{3}C\otimes W$ is identified with $W^{*}$. With the convention (33), the standard anomaly cancellations hold on (32). The linear gravitational-$U{\left(1\right)}_Y$ sum, the cubic $U{\left(1\right)}_Y$ anomaly, and the mixed $SU{\left(3\right)}^{2}U{\left(1\right)}_Y$ and $SU{\left(2\right)}^{2}U{\left(1\right)}_Y$ anomalies cancel in the usual $10\oplus \overline{5}\oplus 1$ pattern [26]. The explicit bookkeeping is placed in Appendix C. Global inflow and cobordism refinements of the resolved worldline defect are not part of this finite representation check.

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

$${\mathcal{F}}_{\mathrm{\Gamma }}^{\mathrm{odd}}:={\mathrm{\Lambda }}^{\mathrm{odd}}V.$$

(34)

Every element of $V\oplus V^{*}$ acts by Clifford multiplication on ${\mathrm{\Lambda }}^{•}V$, and hence defines an odd map from (32) to (34). The restriction to $W\oplus W^{*}$ is the weak odd map used in the local one-Higgs bookkeeping. At the local interface, the corresponding odd Dirac-type term has the form $\overline{\mathrm{\Psi }}(i{\gamma }^{\mu }{\nabla }_{\mu }^{\mathrm{eff}}-m_C{\mathrm{\Gamma }}_C-y_{\mathrm{\Phi }}c(\mathrm{\Phi }+{\mathrm{\Phi }}^{†}))\mathrm{\Psi }$, with $\mathrm{\Phi }\in W$. Thus the principal weak insertion is a one-Higgs channel on the reconstructed finite module.

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

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

(35)

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

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

(36)

This condition is part of the closed branch operator and does not change the finite representation content selected by the Borel-Weil support theorem. Fermionic statistics belong to the quantization of the corresponding branch spinor field.

The neutral local channels are then fixed by Table 4. The active lepton doublet sits in ${\mathrm{\Lambda }}^{3}C\otimes W\simeq W^{*}$, while the neutral singlet sits in ${\mathrm{\Lambda }}^{0}V$. With (33), the one-Higgs Dirac channel formed by the lepton doublet, the weak factor, and the singlet is neutral. The complete local one-Higgs selection is Proposition A6. The active Majorana bilinear has its first neutral active class in the usual two-weak-factor form. 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 [71], while baryon-violating contacts belong to the usual proton-decay effective comparison class [72]. The detailed selection-rule table is recorded in Appendix C.

5.2. Central Coefficient Shadow

The primitive class also has a coefficient shadow. Its integral representative has already been used to obtain the line $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$ and the index tower (3). Its reduction modulo the projective color center was recorded in (9). Since closed local color observables are singlet-valued, the global color form is naturally projective on the color block. Principal-bundle classification at this level is standard [73], and characteristic or secondary classes give the corresponding comparison language [74]. The same global form is detected by line operators [75].

Let ${\mathfrak{L}}_{\mathrm{\Gamma }}$ be the affine torsor of central projective-color sectors associated with (9). The corresponding finite central space is

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

(37)

After an auxiliary origin has been chosen, (37) is identified with the regular representation of ${\mathbb{Z}}_3$. With $\omega =exp(2\pi i/3)$, the clock and shift generators may be written as

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

(38)

The phase direction is fixed by the central grading. The adjacent direction is the nearest-neighbour transport on the torsor. A finite Dirichlet response on the torsor has the form

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

(39)

Its quadratic operator is

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

(40)

The central response is called unblocked when ${\nu }_{\mathrm{\Gamma }}>0$.

 Proposition 15

(Finite Heisenberg central response). In the unblocked central response, the central phase and one adjacent shift generate $M_3\left(\mathbb{C}\right)$ on (37). The traceless anti-Hermitian Lie closure is $\mathfrak{su}\left(3\right)$.

 Proof. 

The monomials $Z^m{S}^n$, $m,n\in {\mathbb{Z}}_3$, form the finite Heisenberg basis. The matrix units are obtained by Fourier inversion:

$$E_{rs}={\displaystyle \frac{1}{3}}{\displaystyle \sum _{a=0}^2}{\omega }^{-ar}{Z}^a{S}^{s-r},r,s\in {\mathbb{Z}}_3.$$

(41)

Hence the generated associative algebra is $M_3\left(\mathbb{C}\right)$. Its traceless anti-Hermitian part is $\mathfrak{su}\left(3\right)$. □

In the unblocked central response, the Hermitian generated space is ${Herm}_3$, and the repeated-spectrum locus is a proper algebraic subset. A generic Hermitian central response therefore has three simple eigenvalues.

At the carrier level, the same primitive sector which fixes the positive link line also fixes the three-dimensional central family-response space. Thus the central torsor fixes the first algebraic support of family response. Its three points label the central carrier sectors. Their physical generational splitting requires a sectoral operator on (37), and this operator is a closed-branch Schur/Berry-Wilczek-Zee datum. The low-cluster use of (40), the Schur-visible corrections, and the Berry-Wilczek-Zee response are treated in Section 6. Flavor eigenvalues, CKM or PMNS data, threshold conversion, and running remain part of the closed spectral problem.

5.3. Neutral Schur Cell

The neutral Schur cell is another finite reading of the same carrier. It is independent of the proof of the support theorem. After (7), (31), and (33) have been fixed, the weak angular directions, the determinant direction, and the radial closure determine a basis-independent reduced cell. The covariant-derivative expansion background is represented by [76]; precision electroweak matching belongs to the comparison class of [77].

The closed neutral cell is spanned by the two charged weak angular directions, the determinant direction, and the radial closure. The weak angular directions contribute two copies of the weak block, while the determinant direction contributes one copy of the color block. Thus

$$D_{\mathrm{cell}}=2dimW+dimC=7.$$

(42)

This number is fixed by the ranks in (7).

 Proposition 16

(Closed-cell basis independence). After the carrier (7) and the top-form constraint (31) have been fixed, the closed neutral Schur cell is basis-independent. Its weak, determinant, and radial coordinates are determined by ranks, Frobenius traces, and the unimodular scalar direction.

 Proof. 

The admissible basis changes act by $U\left(C\right)\times U\left(W\right)$ with the determinant restriction (31). The weak root directions, determinant direction, and radial scalar direction are defined by the block decomposition (30) and by the fixed top form. Ranks, traces, and Frobenius norms are invariant under the allowed unitary changes. □

The leading closed-cell weights are

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

(43)

Here $A_0$ is the weak angular entry, $B_0$ is the determinant entry, and $C_0$ is the radial entry in the minimal closed cell. With this normalization, the determinant entry is $dimCdimW/D_{\mathrm{cell}}^2$.

The first angular/radial Schur correction is recorded by one local scalar unit u:

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

(44)

In the first angular/radial Schur step, the determinant entry is unchanged. The weak angular generators are traceless, so their first Schur correction has no projection to the determinant direction. The radial channel absorbs one scalar Schur unit, and the weak entry is fixed by the unimodular scalar balance between the color and weak blocks. The corresponding scale-free neutral angle is

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

(45)

The primitive normalization of u, the numerical benchmark, and the degree comparison are recorded in Appendix D. The carrier theorem and the central response construction are independent of this benchmark.

 Corollary 5

(Scheme independence of the primitive carrier). In a fixed primitive degree-one filtered link cycle, with the analytic contour gap kept open, the finite carrier (7), the compact basis group (8), the exterior package (32), and the central torsor (37) are independent of renormalization or matching scheme. Scheme-dependent data enter through the completed spectral problem: sectoral eigenvalues, non-circulant Schur corrections, threshold conversion, and running.

 Proof. 

The primitive class fixes $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$ and hence the tower (3). The visibility thresholds in (6) are representation-theoretic. Thus the support (7) is fixed before spectral matching is performed. The compact basis group and the exterior package are algebraic constructions from this carrier, while the central torsor is the coefficient shadow of the same primitive class. A change of renormalization or matching convention may change the sectoral operators on the completed branch, but it does not change the primitive class, the tower, or the visibility thresholds. If the normal-family representative is used, the Riesz projection remains in the same finite support as long as the analytic contour gap remains open. □

6. Analytic Representative and Global Response

The preceding sections identify the filtered link cycle, its separated Toeplitz support, and the finite package attached to that support. The analytic role is narrower. A gauge-fixed normal family is used to represent the selected support (7) as an isolated low sector and to define its differential response over regular families of completed branches.

The relevant operator is of Dirac-Callias type. The comparison class is the Callias index mechanism [35] and its geometric Fredholm formulation [36]. The boundary Dirac reduction, the elliptic normal symbol, and the Schur estimates are recorded in Appendix B.

Primitive protection and analytic isolation.

The primitive line $L_{\mathrm{\Gamma }}\simeq \mathcal{O}\left(1\right)$ fixes the local ${\mathrm{spin}}^c$ link class and hence the index class of the positive link cycle in the usual Atiyah-Singer sense [19]. In the primitive holomorphic link model this same datum gives the Borel-Weil tower (3), and the filtered visibility thresholds are fixed by (6). The analytic low sector is the separated spectral representative of these filtered data. The projection (47) is defined after the Codazzi-Callias gap and the Schur bound have supplied a separated contour. Hence the link index tower, the visibility rule, and the separated support are fixed by the primitive filtered cycle, while the Riesz projection persists over a branch family as long as the analytic contour gap remains open. The mod-three reduction (9) gives the central torsor used for the family carrier; spectral isolation of (48) belongs to the Callias-Schur hypotheses.

6.1. Callias-Schur Isolation

Let B be a regular parameter space of completed primitive branches, and let $N_{+}\left(b\right)$ be the positive gauge-fixed normal operator over $b\in B$. The low block is the finite support selected in (7). With respect to the low-high decomposition, the normal family is written as

$$N_{+}\left(b\right)=\left(\begin{array}{cc}{N}_{LL}\left(b\right)& K{\left(b\right)}^{*}\\ K\left(b\right)& N_{HH}\left(b\right)\end{array}\right).$$

(46)

The Codazzi-Callias gap gives a uniform separation of the high block from the low cluster, and the Schur bound requires the off-diagonal term $K\left(b\right)$ to be subcritical relative to that separation. The model collar in Appendix E supplies a sufficient non-empty regime for this condition.

Let $\mathrm{\Omega }\subset \mathbb{C}$ be a contour enclosing the selected low cluster and no other spectrum of $N_{+}\left(b\right)$. The corresponding spectral projection is

$$P_b={\displaystyle \frac{1}{2\pi i}}{\oint }_{\mathrm{\Omega }}{(z-N_{+}\left(b\right))}^{-1}dz.$$

(47)

The rank of $P_b$ is constant as long as the contour remains separated from the high spectrum.

 Theorem 4

(Analytic representative). Assume that the positive gauge-fixed normal family satisfies the Codazzi-Callias gap and the subcritical Schur bound on a regular parameter domain B. Then the support (7) is represented by an isolated finite low-sector bundle

$${\mathcal{E}}_{\mathrm{low}}=\mathrm{ran}P⟶B.$$

(48)

The projection (47) is stable under lower-order perturbations whose relative bound is below the Schur margin.

 Proof. 

The Callias term gives the high-sector coercive estimate after gauge fixing. The Schur complement of (46) remains invertible on the high complement when the off-diagonal block is subcritical. Hence the selected finite cluster is separated by a positive contour gap. The Riesz projection (47) is therefore well-defined and varies smoothly with b. The perturbative stability follows from the standard resolvent estimate for separated spectral clusters. The detailed normal-operator estimate is given in Appendix B. □

 Corollary 6

(Gap as the isolated-sector condition). In the analytic realization of the primitive filtered cycle, the Codazzi-Callias gap and the subcritical Schur bound are the conditions which make the selected support (7) a stable isolated sector. While the gap remains open, the Riesz projection (47) has constant rank and the projected Berry-Wilczek-Zee connection is defined on (48). If the contour gap closes, the isolated-sector interpretation is lost.

 Proof. 

Theorem 4 gives the isolated finite low-sector bundle from the Codazzi-Callias gap and the subcritical Schur bound. The constancy of the rank follows from the stability of the Riesz projection under perturbations preserving the contour gap. The connection (49) and curvature (50) are defined from this projection. If the contour meets the high spectrum or the low cluster is no longer separated, the Riesz projection is no longer defined by (47). □

 Corollary 7

(Callias-Schur scale). In a Schur-admissible primitive branch, the Codazzi-Callias gap defines the local spectral scale which protects the isolated low-sector representative. This scale controls the persistence of (47) and the validity of the low-sector Schur reduction. It is an analytic isolation scale; its identification with a physical mass scale requires an additional completed-branch operator.

 Proof. 

The high-sector coercive term and the subcritical Schur bound give the contour gap used in Theorem 4. The same gap bounds the high-sector inverse in the Schur complement and controls the resolvent estimate for (47). Thus it is the local scale of spectral isolation. A particle mass, a threshold, or a singlet effective mass is obtained only after a sectoral operator has been specified on the completed branch. □

Theorem 4 does not change the support computed in Theorem 3. It supplies the analytic realization of that support by a finite-rank spectral bundle. If the Codazzi gap closes, the two-eigenvalue optical splitting, the projective link reading, and the Callias separation degenerate together.

6.2. Family Response and Berry-Wilczek-Zee Connection

The bundle (48) carries the projected connection. If ${\nabla }^0$ denotes the trivial connection in a local Hilbert trivialization, then

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

(49)

In a local orthonormal frame $\mathrm{\Psi }$ of ${\mathcal{E}}_{\mathrm{low}}$, the connection form is ${\mathrm{\Psi }}^{†}d\mathrm{\Psi }$. The corresponding curvature is

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

(50)

This is the Berry connection in the non-degenerate case [37] and the Berry-Wilczek-Zee connection in the degenerate case [38]. In the families-index language, its determinant-line trace belongs to the Bismut-Freed holonomy comparison class [22]; the corresponding differential K-theoretic index comparison is represented by [21].

Three finite family data are kept separate. The coefficient reduction (9) gives the algebraic central torsor. A sectoral low-cluster operator on (37) gives closed spectral splitting. Berry-Wilczek-Zee curvature requires Schur-visible motion of the isolated projection. The following statements record these layers.

 Lemma 5

(Central cubic selection). Let z be a local adjacent central-response coordinate of degree one for the central ${\mathbb{Z}}_3$ torsor. Then the first scalar phase-sensitive monomials in a ${\mathbb{Z}}_3$-invariant closed-branch functional are $z^3$ and ${\overline{z}}^3$. The quadratic invariant ${\left|z\right|}^2$ is radial. Equivalently, in the degree-one component of an equivariant Kuranishi obstruction map, after the linear kernel term has been removed, the first phase-sensitive term has type ${\overline{z}}^2$.

 Proof. 

Under the generator of ${\mathbb{Z}}_3$, z transforms as $z\mapsto \omega z$, where ${\omega }^3=1$. Hence $z^p{\overline{z}}^q$ has central degree $p-q$ modulo 3. Scalar monomials have degree zero, so $p-q\equiv 0(mod3)$. Up to degree two, the only scalar monomial is ${\left|z\right|}^2$, which is radial. The first scalar monomials depending on the central phase are $z^3$ and ${\overline{z}}^3$. For a degree-one equivariant obstruction component the condition is instead $p-q\equiv 1(mod3)$. After the linear kernel term has been removed in the Kuranishi reduction, the first phase-sensitive term is ${\overline{z}}^2$. □

 Proposition 17

(Closed central-cycle coefficient). Let $H_0,H_1,H_2$ be the three fibers of the central ${\mathbb{Z}}_3$ torsor, and let $V_{01}:H_0\to H_1$, $V_{12}:H_1\to H_2$, and $V_{20}:H_2\to H_0$ be adjacent central response maps. Then $C_3={\mathrm{Tr}}_{H_0}\left(V_{20}{V}_{12}{V}_{01}\right)$ is invariant under independent unitary changes of frames in the three fibers. If $C_3\ne 0$, its phase is a closed-cycle datum of the torsor.

 Proof. 

Under frame changes $U_i$ on $H_i$, the maps transform as $V_{ij}\mapsto U_j{V}_{ij}{U}_i^{-1}$. Hence $V_{20}{V}_{12}{V}_{01}$ transforms by conjugation with $U_0$. The trace is invariant under conjugation. Since the total central degree of three adjacent shifts is zero modulo three, this is the first phase-sensitive oriented adjacent closed cycle on the torsor. □

The proposition is algebraic. In a completed branch, a cubic central Kuranishi coefficient, if generated by Schur elimination of adjacent central detector blocks, is represented by the corresponding closed Schur product with high-sector resolvents. Its nonvanishing and normalization are closed spectral data.

The Schur detector is the low-high part of the variation of $N_{+}\left(b\right)$. For $X\in T_{b}B$, set

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

(51)

Taking the $(1-P_b)$-$P_b$ block of the derivative of (47) gives the inverse Sylvester map determined by the separated low and high spectra. Hence a nonzero component of (51) cannot be removed by passing to ${\partial }_X{P}_b$ as long as the cluster remains isolated.

 Proposition 18

(Schur-visible central response). Assume that the central torsor response of SubSection 5.2 is unblocked and Schur-visible on an open subset of B. If the phase direction and an adjacent-shift direction contribute nontrivially to (50), then the traceless Berry-Wilczek-Zee holonomy algebra on the corresponding central family bundle contains the $\mathfrak{su}\left(3\right)$ generated by the clock-shift algebra.

 Proof. 

Schur visibility places the adjacent central component in (51). The inverse Sylvester map transfers this component to the motion of $P_b$. Together with a phase component, its commutator contributes to the low-low curvature block in (50). The finite central algebra generated by the phase and adjacent shift is $M_3\left(\mathbb{C}\right)$ by (41). The traceless anti-Hermitian closure is therefore $\mathfrak{su}\left(3\right)$. □

 Theorem 5

(Central family-response theorem). In a primitive degree-one filtered link cycle, the coefficient reduction (9) defines the central ${\mathbb{Z}}_3$ family-response torsor. If the central response is unblocked, the phase and adjacent-shift directions generate $M_3\left(\mathbb{C}\right)$ on (37). If, in addition, the completed branch family satisfies the Codazzi-Callias gap and the subcritical Schur bound, and these directions are Schur-visible in the variation of (47), then the Berry-Wilczek-Zee curvature has the corresponding central non-Abelian response component.

 Proof. 

The torsor is the coefficient shadow (9). The unblocked finite algebraic closure is Proposition 15. The Schur-visible curvature statement is Proposition 18. The analytic bundle on which the curvature is defined is supplied by Theorem 4. □

This theorem is a family-response statement. The algebraic carrier is fixed by the primitive coefficient reduction; the nonzero curvature contribution is an analytic property of the completed branch. In a finite-dimensional boundary trace space, the blocked condition for the adjacent central conductance is closed. The unblocked, Schur-visible condition is therefore stable on the complement of this closed locus whenever the relevant detector coordinate is defined.

A sectoral family operator obtained from the closed low-cluster response may have a leading circulant central part

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

(52)

Its eigenvalues are

$$y_{f,k}=A_f+2\left|{\beta }_f\right|cos\left({\vartheta }_f+{\displaystyle \frac{2\pi k}{3}}\right),k\in {\mathbb{Z}}_3,$$

(53)

where ${\vartheta }_f=arg{\beta }_f$ up to the convention for S. The corresponding scale-free shape parameter, when $\mathrm{Tr}{Y}_f^{\left(0\right)}\ne 0$, is

$$s_f={\displaystyle \frac{\mathrm{Tr}{\left(Y_f^{\left(0\right)}-\frac{1}{3}\mathrm{Tr}\left(Y_f^{\left(0\right)}\right)\mathbf{1}\right)}^2}{\mathrm{Tr}{\left(\frac{1}{3}\mathrm{Tr}\left(Y_f^{\left(0\right)}\right)\mathbf{1}\right)}^2}}={\displaystyle \frac{2|{\beta }_f{|}^2}{A_f^2}}.$$

(54)

 Proposition 19

(Circulant leading shape and non-circulant flavor data). Let two sectoral leading family responses be of the form (52). Then they are simultaneously diagonalized by the finite Fourier transform on the central torsor, and therefore give no physical mixing data. If one sector has simple leading spectrum, any mixing correction must have a component outside the circulant algebra $\mathbb{C}\left[S\right]$. A CP-sensitive invariant requires, in addition, a non-removable complex component in such a non-circulant correction.

 Proof. 

The leading responses lie in the commutative algebra $\mathbb{C}\left[S\right]$. Since $S^3=1$, this algebra is diagonalized by the finite Fourier transform, so any two leading responses are simultaneously diagonalized. Hence their relative diagonalizing matrix is trivial up to permutations and phases. If one leading spectrum is simple, its commutant is $\mathbb{C}\left[S\right]$, so a correction which changes the relative eigenspaces must have a component outside this algebra. A correction which can be made real by sectoral rephasings may give nontrivial eigenspaces but gives no CP-odd invariant. A CP-sensitive invariant therefore requires a non-removable complex component in the same non-circulant Schur/Berry-Wilczek-Zee correction. □

6.3. Self-Describing Branch Synthesis

The preceding sections give the local self-describing branch chain. In a closed Alena gauge-branch collar, the Hilbert-Belinfante coefficient split gives the current and stress-response labels of Proposition 6. The projective-link step is fixed by Remark 2 and Proposition 9. The primitive degree-one threshold comparison is Proposition 11. The separated support theorem then gives the carrier (7), and the finite package gives (8), the exterior module, and the coefficient shadow (9).

This chain is local. It may fail in three visible ways: the branch may leave the non-null optical Rainich-Codazzi stratum; one of the two principal source channels may vanish or cease to be separated; or the Schur/Callias gap may close, allowing high-sector data to enter the low cluster. In the primitive gap-open sector, additional low chiral carriers are therefore not produced by the order-$\le 2$ link mechanism. Such carriers require a change of source order, link degree, topological sector, or analytic gap.

The central torsor fixes only the first algebraic family-response support. By Proposition 19, leading circulant family operators do not give physical mixing. The flavor eigenvalues, non-circulant Schur corrections, CP-sensitive invariants, thresholds, and running remain closed spectral data of the completed branch.

6.4. Kuranishi Completion and Closed Spectral Data

The local collar data are completed by a nonlinear Fredholm problem. Let $\mathcal{F}$ be the gauge-fixed branch map on the weighted edge spaces used in Appendix E, and let $L=D{\mathcal{F}}_{u_0}$ be its linearization at an edge-regular reference branch $u_0$. After the gauge directions have been removed, L is Fredholm. Choose closed complements

$$X=kerL\oplus X^{\prime },Y=\mathrm{ran}L\oplus \mathrm{coker}L.$$

(55)

The nonlinear Fredholm reduction gives a finite-dimensional obstruction map

$$\kappa :kerL\supset U⟶\mathrm{coker}L,$$

(56)

defined near the origin. Nearby finite-action branches in this chart are exactly the points of ${\kappa }^{-1}\left(0\right)$.

 Proposition 20

(Primitive completion criterion). Let a primitive current-residual collar datum be represented in a Kuranishi chart of the gauge-fixed branch problem. If its kernel parameter lies in ${\kappa }^{-1}\left(0\right)$, the collar extends to a nearby completed branch. If $\mathrm{coker}L=0$, the obstruction map is identically zero in the chart.

 Proof. 

Writing a nearby configuration as $u_0+a+w$, with $a\in kerL$ and $w\in X^{\prime }$, the projected equation to $\mathrm{ran}L$ is solved uniquely for $w=\mathrm{\Psi }\left(a\right)$ by the inverse function theorem. Substitution into the cokernel component gives (56). Thus the full equation is equivalent to $\kappa \left(a\right)=0$. If the cokernel vanishes, there is no remaining obstruction. □

When the completed branch is optical ARC, the realized source is natural of transverse order at most two, and the principal non-scalar moments satisfy Assumption A1, the proof chain closes. The primitive collar supplies the link cycle of Section 3; Lemma 3 identifies the nonzero $V_1,V_2$ channels in the moment-resolved case; Theorem 3 gives the separated support; Section 5 gives the exterior package and the central coefficient shadow; and Theorem 4 supplies the isolated low-sector representative. This is the global completion route used here: primitive current-residual collar, Kuranishi extension, filtered link cycle, carrier, central torsor, and analytic low-sector bundle.

The remaining spectral data are attached to the completed branch. Flavor eigenvalues, the ordering of the sectoral low modes, non-circulant Schur corrections, CP-sensitive invariants, threshold conversion, and running-coupling matching are not fixed by the local filtered link cycle. They belong to the closed spectral problem determined by the Kuranishi branch and its finite response operators.

7. Conclusions and Discussion

The reconstruction has been stated as a primitive local instance of the self-reconstruction loop (1). The full principle is used here only through its local finite-sector consequence. After the conserved-current and topological data have been fixed, the resolved optical link and the filtered source select an isolated Schur-low carrier. In the terminology of (2), this gives the primitive finite-carrier part of a local closed-observable branch.

The first outcome is the carrier rigidity theorem. A primitive optical Codazzi defect gives the projective link, the simple positive line, and the Borel-Weil tower. For a natural scalar-sector source of order at most two, after the scalar singlet has been separated, the non-scalar principal symbol has the $V_1$ and $V_2$ channels. Under two-channel genericity and separated principal labels, the separated first-support rule gives the carrier (7). The nearby alternatives are recorded in Table 3. Higher integer multipoles belong to the high-sector problem; they preserve the low carrier while the Schur gap remains open and the low-high coupling is subcritical. Thus the central finite-carrier step is representation-theoretic and local.

The carrier (7) gives the carrier-basis group (8) after the selected Hermitian carrier spaces and the unimodular top-form constraint have been fixed. The exterior package (32), with hypercharge fixed by (33), gives the standard one-generation representation content displayed in Table 4. The finite algebraic checks are contained in Appendix C. This representation package is attached after the carrier has been selected; it is not an independent input to the support theorem.

The same primitive class has a central coefficient shadow. The reduction (9) gives an affine ${\mathbb{Z}}_3$ family-response torsor and the finite central space (37). In the unblocked response, the clock and adjacent-shift directions generate the full matrix algebra on this space by Proposition 15. This is the algebraic family carrier. Its interpretation as physical generation splitting requires a sectoral operator on the completed branch.

The analytic layer supplies the isolated spectral representative. Under the Codazzi-Callias gap and the subcritical Schur bound, the Riesz projection (47) defines the low-sector bundle (48). The projected connection and curvature are (49) and (50). If the central phase and adjacent-shift directions are Schur-visible, Theorem 5 gives the corresponding non-Abelian central response component. Leading circulant family operators are simultaneously diagonalizable by Proposition 19; mixing and CP-sensitive invariants therefore require non-circulant Schur-Berry data from the completed branch.

The Callias-Schur gap also gives the local isolation scale of the selected sector. It controls the high-sector inverse in the Schur complement and the persistence of the Riesz projection. Its identification with a particle mass, a threshold, or a singlet effective mass requires an additional sectoral operator. In particular, the singlet Majorana scale, if represented by the same locked Callias block, is a closed spectral datum rather than a consequence of the carrier theorem alone.

The neutral Schur cell gives a separate finite benchmark. Once the carrier, top-form constraint, hypercharge convention, and primitive finite scalar unit have been fixed, the reduced cell gives the diagnostic values recorded in Appendix D. The benchmark is independent of the proof of carrier rigidity. Its promotion to a physical prediction requires the same scalar unit to be obtained from the completed spectral problem, for example from the completed normal operator, a regularized determinant, or an equivalent closed spectral mechanism.

The construction therefore separates three mechanisms. The primitive link mechanism fixes the finite carrier through the Borel-Weil tower and the filtered Toeplitz visibility rule. The central response mechanism attaches the mod-three family torsor and, when unblocked, the clock-shift algebra. The completed Schur-Berry mechanism carries the remaining spectral data: family eigenvalues, non-circulant corrections, CP-sensitive invariants, threshold conversion, and running-coupling matching. Only the first mechanism enters the proof of the minimal separated carrier. The second is algebraic at the torsor level and analytic when Berry-Wilczek-Zee curvature is used. The third is part of the completed branch problem.

The Alena-type collar is used as a sufficient source realization of the primitive filtered hypotheses. The carrier theorem itself uses the optical Codazzi branch, the primitive resolved link, and the filtered order-$\le 2$ source data. An independent realization of the same hypotheses by another current-residual or geometric defect model would lead to the same local carrier calculation. Conversely, degeneration of the two-channel source, loss of the primitive degree-one link, or closure of the Schur gap changes the conclusion according to the alternatives recorded in Section 4.

Several extensions are left in a form suitable for direct tests. First, primitive collars beyond the Alena-type realization should be classified by the same source hypotheses and by the two-channel genericity condition. Second, the non-circulant part of the Schur-Berry response should be computed from the Kuranishi branch, since this is the first place where physical mixing and CP-sensitive data can enter. Third, the primitive scalar unit used in Appendix D should be derived or falsified inside the completed spectral problem. These three problems respectively test the robustness of the carrier, the family-response geometry, and the neutral-cell benchmark.

In this form, the result is a local finite-carrier theorem with a conditional analytic representative and a finite diagnostic layer. The carrier (7), the group (8), the exterior package (32), and the central torsor (37) are fixed by the primitive filtered cycle and the stated genericity assumptions. Yukawa matrices, fermion masses, CKM and PMNS data, thresholds, and running remain closed spectral data of the completed branch.

Appendix D.1. Primitive Chern-Weil Normalization

Let $d{\mu }_{\mathrm{\Gamma }}$ be the normalized link measure on ${\mathbb{CP}}^1$, and let $F_{\mathrm{\Gamma }}$ be the curvature of a unitary connection on $L_{\mathrm{\Gamma }}$. Since the primitive sector has $c_1\left(L_{\mathrm{\Gamma }}\right)=1$,

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

(A34)

 Lemma A3

(Primitive Chern-Weil action). In the primitive class, the normalized Chern-Weil action is minimized by the constant-curvature representative and equals

$$S_{\mathrm{prim}}=\underset{c_1\left(L_{\mathrm{\Gamma }}\right)=1}{inf}{\int }_{{\mathbb{CP}}^1}{\left|F_{\mathrm{\Gamma }}\right|}^{2}d{\mu }_{\mathrm{\Gamma }}=4{\pi }^2.$$

(A35)

 Proof. 

The curvature representative of a unitary connection on the degree-one line has fixed integral by (A34). With the normalized link measure, the $L^2$ norm is minimized by the constant representative in this class. Substitution of the degree-one period gives (A35). □

 Definition A2

(Primitive finite scalar unit). The primitive finite scalar unit is defined as the inverse primitive Chern-Weil action per state of the reconstructed even exterior module.

 Lemma A4

(Value of the primitive scalar unit). With Definition A2,

$$u_{\mathrm{prim}}={\displaystyle \frac{1}{dim{\mathcal{F}}_{\mathrm{\Gamma }}^{\mathrm{even}}{S}_{\mathrm{prim}}}}={\displaystyle \frac{1}{16S_{\mathrm{prim}}}}={\displaystyle \frac{1}{64{\pi }^2}}.$$

(A36)

 Proof. 

The even exterior module has dimension 16 by (32). The result follows from (A35). □

The value (A36) is a finite normalization of the primitive Schur cell. Its use as a physical scalar unit requires that the same unit be obtained from the completed normal operator, a regularized determinant, or an equivalent closed spectral mechanism.

Appendix D.2. Neutral-Cell Diagnostic

In the primitive benchmark, the local Schur unit in (44) is set equal to (A36). This gives

$${sin}^2{\theta }_{\mathrm{link}}=0.223184071\dots .$$

(A37)

This is the value of (45) after the primitive finite normalization has been chosen. In the on-shell electroweak convention, the comparison quantity is

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

(A38)

Using the values quoted in [92], one obtains

$${sin}^2{\theta }_W^{\mathrm{on}-\mathrm{shell}}=0.223071223\dots ,\left|{sin}^2{\theta }_{\mathrm{link}}-{sin}^2{\theta }_W^{\mathrm{on}-\mathrm{shell}}\right|=1.13\times {10}^{-4}.$$

(A39)

The comparison in (A39) is with the on-shell quantity extracted from $M_W$ and $M_Z$. The scheme-dependent effective leptonic mixing angle is a different observable.

For a diagnostic mass reading at an external electroweak scale, put $v={\left(\sqrt{2}{G}_F\right)}^{-1/2}$, with $G_F$ taken from [92]. The reduced neutral cell gives

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

(A40)

With $v=246.2196719\mathrm{GeV}$ this gives

$$m_W^{\mathrm{link}}=80.3707\mathrm{GeV},m_Z^{\mathrm{link}}=91.1882\mathrm{GeV},m_h^{\mathrm{link}}=125.2399\mathrm{GeV}.$$

(A41)

 Proposition A7

(Neutral-cell benchmark). Equations (A37)-(A41) give a reduced Schur-cell diagnostic after the primitive scalar unit (A36) has been chosen and the external Fermi scale has been inserted. The local carrier theorem is independent of this diagnostic.

 Proof. 

The carrier theorem uses the primitive filtered link cycle, the Borel-Weil tower, and the separated Toeplitz support. The neutral-cell entries used in (A40) are read only after (7), (31), and (33) have been fixed. The external scale enters through $G_F$. Hence the diagnostic does not enter the proof of Theorem 3. □

Appendix D.3. Sensitivity to the Scalar Unit

Let the local Schur unit be perturbed by $u=u_{\mathrm{prim}}(1+\epsilon )$. Then (44) gives

$$A\left(\epsilon \right)=A_0-{\displaystyle \frac{3}{2}}{u}_{\mathrm{prim}}(1+\epsilon ),B\left(\epsilon \right)=B_0,C\left(\epsilon \right)=C_0+u_{\mathrm{prim}}(1+\epsilon ).$$

(A42)

The corresponding scale-free angle is

$${sin}^2{\theta }_{\mathrm{link}}\left(\epsilon \right)={\displaystyle \frac{B\left(\epsilon \right)}{A\left(\epsilon \right)+B\left(\epsilon \right)}}.$$

(A43)

 Proposition A8

(Scalar-unit sensitivity). The benchmark values in (A37) and (A41) are sensitive only through the scalar unit u in (44). A promotion of the benchmark to a physical prediction requires an independent derivation of $u_{\mathrm{prim}}$ from the completed spectral problem.

 Proof. 

The dependence on u enters the neutral cell through (44). Substitution gives (A42) and (A43). The link carrier, the exterior package, and the central torsor do not depend on u. □

Appendix D.4. Primitive Collar Scale Diagnostic

The primitive collar normalization gives a second reading of the same reduced Schur block. With the spin-vorticity normalization $g=1/2$ fixed in [6], and with (43) and (44), the neutral scalar amplitude is

$$v_H^{\mathrm{prim}}={\displaystyle \frac{M_{\mathrm{Pl}}}{g}}exp\left(-S_{\mathrm{prim}}-{\displaystyle \frac{1}{144{\pi }^2}}\right)\sqrt{{\displaystyle \frac{C_0}{C}}}.$$

(A44)

With the electroweak convention $v_{\mathrm{EW}}^{\mathrm{prim}}=\sqrt{2}{v}_H^{\mathrm{prim}}$, this becomes

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

(A45)

Using the unreduced Planck mass and the constants quoted in [92], one obtains

$$v_{\mathrm{EW}}^{\mathrm{prim}}=246.2205\mathrm{GeV}.$$

(A46)

For comparison, the Fermi-constant value used in (A40) is $246.2197\mathrm{GeV}$ for the same central constants. Substitution of (A45) into (A40) gives

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

(A47)

These numbers use the same finite Schur block as (A41), with the external Fermi scale replaced by the primitive collar scale (A45). This is a collar-scale diagnostic. Its use as an independent physical scale requires the collar normalization to be derived inside the completed branch.

Appendix D.5. Degree Comparison

For a positive line of degree n, the same Chern-Weil normalization gives

$$S_n=n^2{S}_{\mathrm{prim}}.$$

(A48)

The positive modes become

$$E_q^{\left(n\right)}=H^0({\mathbb{CP}}^1,\mathcal{O}\left(n(q-1)\right)),dim{E}_q^{\left(n\right)}=n(q-1)+1.$$

(A49)

The visibility threshold for $V_{\ell }$ is then

$$q=1+⌈{\displaystyle \frac{\ell }{n}}⌉.$$

(A50)

The comparison (A48)-(A50) is used only as a rigidity diagnostic. The carrier theorem uses the primitive degree-one sector fixed by Proposition 9.