Minimal Borel-Weil Support from Equivariant Normal Operators on Optical Codazzi Defects: Application to the 3+2 Exterior-Algebra Multiplet Package

1. Introduction and Statement of 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 question 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 sector is recovered from the quantized optical link of a defect. The basic data are the branch geometry, the Codazzi closure, the compact link, and the isolated low spectral sector.

The Alena Tensor identification (16) motivates the ARC input class used below [4]. In this role it supplies the residual density, the translational-current coefficient, and the vortex terms used in the branch stress response [5]. The continuum, variational, and Higgs-like branch-potential inputs are those of [6,7,8]. After these branch data have been fixed, the local normal-operator and Borel-Weil reconstruction depends only on the optical ARC branch, the Codazzi closure, the blown-up link, the spinorial boundary data, and the isolated low spectral sector. The restrictions used below are separated in Table 1. They define the geometric and analytic class; generic source conditions and global spectral matching are kept separate from the support-selection statement. The Codazzi multiplier supplies the differential closure which complements the algebraic Rainich conditions [9]; the corresponding comparison class is the classical Rainich reconstruction problem [10].

The support theorem is local on the edge collar of a transverse-frame resolved optical ARC defect. It applies to edge-regular finite-action completions whose collar data lie in the admissible class. The local warped model of Appendix A supplies the model collar data for the normal operator, while the closed branch problem fixes spectral normalizations, fermion masses, mixing matrices, threshold-sensitive quantities, and higher clusters. These global data are not used in the principal Toeplitz support selection.

The local branch is built from a tetrad $(U,N,W,S)$. The Rainich plane is $span(U,N)$, and the transverse two-plane is $span(W,S)$. The branch metric leaves the Rainich plane unscaled and rescales the transverse plane. The same anisotropy is read from the algebraic gauge-to-branch relation and from the normalized vorticity-flux closure. The trace-adjusted Codazzi tensor gives the Levi-Civita divergence identity for the weighted branch stress, so the flat-reference stress response is represented by the branch geometry. These statements are made precise in Section 2.

On the non-degenerate Codazzi set, the two-eigenvalue splitting is optical. The Rainich principal null directions are geodesic and shear-free, and the local branch determines a projective-spinor reading of the link in the standard sense of spinor and twistor geometry [11]. Related optical and projective-spinor structures are part of the standard twistor comparison class [12]; twistor approaches to Standard Model geometry provide a separate representation-theoretic comparison [13]. In the present setting only the local projective line attached to the optical defect is used. A compact-leaf obstruction gives a branch worldline $\Gamma$. Its link is a two-sphere, and the optical structure identifies it with $S_{\Gamma }^2\simeq {\mathbb{CP}}^1$. The codimension-three topology of a worldline defect is used in the local collar: the scalar period obstruction is absent in the model ${\mathbb{R}}^4\setminus \mathbb{R}\simeq S^2$, while the integral link charge occupies the corresponding $H^2$ slot.

The defect is treated on the real blow-up $[M;\Gamma ]$. The link sphere is then the boundary fiber $S(N\Gamma )$, and the link problem is an edge boundary problem. The gauge-fixed Codazzi complex has an elliptic principal symbol after the divergence gauge is included. Its boundary operator is of Hodge type on $S_{\Gamma }^2$ before the optical spinorial projection is taken. The admissible optical middle-spinor data identify the projected principal symbol with Clifford multiplication. The projected boundary operator is therefore of Dirac type, and its difference from the transverse-frame spin^c Dirac operator is of zeroth order. This edge operator is the source of the equivariant normal operator used below.

The normal operator is the frozen gauge-fixed Codazzi-Calderon operator on the blown-up collar. It records the radial normal part, the boundary Hodge-Dirac part, and the frozen zeroth-order response. After middle-spinor projection and compression to a positive spin^c block, it is a Dirac-type link operator with a finite zeroth-order block response. Its principal associated-graded part is taken with respect to the normal-jet filtration. In the split-conserved collar this associated-graded operator preserves the coefficient-channel decomposition into the phase-current channel and the Codazzi-gap channel. Consequently, the leading support Hessian has no coefficient map from the Codazzi-gap channel to the phase-current channel. Lower-order mixed terms are kept in the closed-branch spectral problem and are controlled by the finite Toeplitz stability estimate.

A transverse-frame resolved defect supplies the integral spin^c determinant line on $S_{\Gamma }^2$. The positive link zero modes are the standard monopole zero modes on $S^2$ [14]; the same realization appears in the Taub-NUT Dirac setting [15]. Equivalently, this is the ${\mathbb{CP}}^1$ case of Borel-Weil [16], with the representation-theoretic conventions of [17]. Thus

$$E_q=ker{D}_q^{+}\simeq H^0({\mathbb{CP}}^1,\mathcal{O}(q-1)).$$

(1)

Here $D_q$ denotes the transverse-frame spin^c Dirac operator on the projective link. The zeroth-order term in (73) is treated as part of the frozen block response; under the $SU\left(2\right)$-equivariant compression in Lemma 2, it is scalar on each $E_q$. Formula (1) is the finite link quantization used throughout the paper.

The normal second jet of a regular optical defect has a natural filtration by normal order. Its principal associated-graded part contains a scalar component, a degree-one dipole, and a traceless degree-two quadrupole. The scalar component is a closed local branch datum. The non-scalar components are the phase-current source family of highest order one and the Codazzi-gap source family of highest order two. The support comparison is made after passing to the associated graded. The standard ${\mathbb{CP}}^1$ Toeplitz comparison is used in [18]. In finite-mode form this is the fuzzy-sphere cutoff of [19]; brane and fuzzy-sphere models give the broader finite-matrix comparison class [20].

For the rank-one projective link, the visibility rule is rigid. Since $E_q\simeq {Sym}^{q-1}{\mathbb{C}}^2$, the traceless endomorphism space contains precisely the spherical types $V_1,\dots ,V_{q-1}$. The phase-current dipole first appears on $E_2$, while the Codazzi-gap quadrupole first appears on $E_3$. The channel separation supplied by the principal normal operator keeps the phase-current and Codazzi-gap source families distinct before Toeplitz visibility is applied. The minimal active support is

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

(2)

By (1), this is the Borel-Weil support $H^0({\mathbb{CP}}^1,\mathcal{O}\left(2\right))\oplus H^0({\mathbb{CP}}^1,\mathcal{O}\left(1\right))$. A positive support functional is used only to exclude passive visible enlargement after the active low source has been fixed.

After ${\mathcal{V}}_{\Gamma }$ has been reconstructed, changes of orthonormal zero-mode basis give $U\left(C_{\Gamma }\right)\times U\left(W_{\Gamma }\right)$. Fixing the total top form reduces this group to $S\left(U\right(3)\times U(2\left)\right)$. As a representation-theoretic corollary, the standard $3+2$ exterior-algebra package familiar from $SU\left(5\right)$/Spin$\left(10\right)$ constructions [21] and from Clifford-ideal descriptions [22,23] may then be applied to the reconstructed module. In the present construction the exterior package is not an input to the support theorem; it is applied after the $3+2$ module has been obtained from the quantized defect link.

The comparison with almost-commutative geometry is structural. In the noncommutative-geometric Standard Model the finite internal algebra is part of the input [24], and the spectral action supplies the dynamical principle [25]. Modern spectral-geometric refinements give the corresponding comparison class [26]; Lorentzian refinements and no-doubling issues are relevant for the fermionic sector [27]. The finite labels considered here are boundary and link labels of a closed four-dimensional branch problem. This places the construction close to geometric matter models in which topology and geometry recover particle labels without introducing them as local tensor indices [28].

The fermionic sector is obtained by diagonal locking of branch chirality with total exterior parity. This finite grading is compatible with the branch Dirac term, the weak Clifford-odd map, and charge conjugation. Fixed exterior parity is also compatible with differential-form descriptions of fermions [29]. When the chiral Fredholm gap condition holds, the mirror exterior-parity sector is separated from the isolated low cluster. The positive operator $K_F=Q_F^{†}{Q}_F$ contains the branch Dirac term, the weak Clifford-odd map, and the holonomy-Codazzi response terms. The exterior-algebra identity fixes the representation structure, while numerical fermion masses are spectral data of the closed branch operator.

The Codazzi gap ${\Delta }_C$ has two roles in the local argument. It is the scale of the two-eigenvalue splitting and the scale entering the support barrier. In the locked spectral sector the middle-spinor projection of the frozen Codazzi response has a leading Clifford-odd zeroth-order part proportional to ${\Delta }_C$, up to a controlled lower-order remainder. Under the corresponding Callias estimate the warped collar model gives a lower bound for $K_F$. Together with the stability estimate of Proposition 17, this gives only the persistence of the isolated locked cluster. The numerical Yukawa matrices and mixing data remain global spectral data.

The same normal operator gives the finite Hessian used for the bosonic comparison. After the low-high splitting of ${\mathcal{N}}_{\Gamma }^{†}{\mathcal{N}}_{\Gamma }$, the effective low Hessian is its Schur complement. The determinant entry associated with the unimodular $3+2$ split is fixed by the dimensions of $E_3$ and $E_2$ and by the top-form constraint; this gives the protected closed-cell value $B_0=6/49$. The scale-independent reading of the finite Schur block is the on-shell neutral link angle. The remaining scalar Schur unit and the radial and angular entries require the frozen heavy-block normalization and are treated separately from the local support theorem.

The family refinement is global. Since closed local color observables are singlet-valued, the closed color holonomy is naturally projective. The projective color link has a central-sector torsor classified by $H^2(S_{\Gamma }^2,{\mathbb{Z}}_3)\simeq {\mathbb{Z}}_3$, and its canonical finite quantization gives the regular three-dimensional central sector. Boundary traces of the linearized Codazzi response may act as central-degree intertwiners. When the adjacent components are nonzero, the central phase and the shift generate $M_3\left(\mathbb{C}\right)$. Flavor relations require additional spectral input; in the sector considered below, the relevant restricted source is the lowest central clock-shift sector.

The paper is organized as follows. Section 2 constructs the optical ARC branch, the compact-leaf defect, the blown-up edge geometry, and the equivariant normal operator. Section 3 gives the spin^c link quantization, the normal second-jet source, and the minimal Borel-Weil support (2). Section 4 reconstructs the unimodular zero-mode basis group, the exterior multiplet package, the projected zero-mode connection, and the locked low operator. Section 5 gives the bosonic Schur Hessian, the protected determinant entry, projective color sectors, central family response, and global finite-action completion. Section 6 summarizes the structural output and the remaining global spectral inputs. The explicit collar model and the bosonic Schur computation are collected in the appendices.

2. Optical ARC Branches and Codazzi Defects

The ARC branch used below is the local closure mechanism which produces the optical link. It is fixed by a non-null Rainich stress type, a Codazzi-density closure, a regular translational-current coefficient, and the compact-leaf obstruction of the resulting optical branch. For the local reconstruction it may be read as the mathematical input class. The same data also fix the edge boundary problem from which the normal operator of SubSection 2.3 is obtained. The stress-response terms used for the boundary Hessian and the explicit warped collar model are collected in Appendix A.

The construction is local. A flat Lorentzian metric ${\eta }_{\mu \nu }$ is used as the reference metric for the gauge-side description, while $k_{\mu \nu }$ denotes the anisotropic branch metric. The flow field $U^{\mu }$ is normalized in both descriptions, ${\eta }_{\mu \nu }{U}^{\mu }{U}^{\nu }=1$ and $k_{\mu \nu }{U}^{\mu }{U}^{\nu }=1$. Gauge-side contractions are made with ${\eta }_{\mu \nu }$ unless the branch metric is written explicitly. Standard gauge and spinor conventions are as in [30,31]; the branch connection is the Levi-Civita connection of $k_{\mu \nu }$ [32].

2.1. ARC Branch and Two-Eigenvalue Codazzi Geometry

The branch is built from a local orthonormal tetrad $(U,N,W,S)$. The vector U is timelike, N is selected by the normalized vorticity flux, and $(W,S)$ spans the transverse two-plane. With tetrad covectors lowered by ${\eta }_{\mu \nu }$,

$${\eta }_{\mu \nu }=U_{\mu }{U}_{\nu }-N_{\mu }{N}_{\nu }-W_{\mu }{W}_{\nu }-S_{\mu }{S}_{\nu }.$$

(3)

The anisotropic branch metric is

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

(4)

so the Rainich plane $span(U,N)$ is unchanged and the transverse plane is rescaled by $\chi$. Its $\eta$-trace and density are

$$k={\eta }^{\mu \nu }{k}_{\mu \nu }=2+2e^{2\chi },$$

(5)

$$p_{\Lambda }=p_0{k}^2.$$

(6)

The construction is restricted to the non-degenerate Lorentzian region of k.

The vorticity tensor is ${\omega }_{\mu \nu }={{\Delta }_{\mu }}^{\alpha }{{\Delta }_{\nu }}^{\beta }{\nabla }_{[\alpha }^{\left(k\right)}{U}_{\beta ]}$, where ${{\Delta }^{\mu }}_{\nu }={{\delta }^{\mu }}_{\nu }-U^{\mu }{U}_{\nu }$. The vorticity flux vector is

$$D_{\omega }^{\mu }=a_{\nu }{\omega }^{\nu \mu }+{\displaystyle \frac{c^2}{2}}{{\Delta }^{\mu }}_{\nu }{\nabla }_{\lambda }^{\left(k\right)}{\omega }^{\lambda \nu },a^{\mu }=U^{\alpha }{\nabla }_{\alpha }^{\left(k\right)}{U}^{\mu }.$$

(7)

The normalized flux branch is the branch for which $q^{\mu }=p_{\Lambda }{N}^{\mu }$. If $D_{\omega }^{\mu }\ne 0$, then

$$N^{\mu }=-{\displaystyle \frac{D_{\omega }^{\mu }}{|D_{\omega }|}},\left|D_{\omega }\right|=\sqrt{-D_{\omega \alpha }{D}_{\omega }^{\alpha }},\alpha ={\displaystyle \frac{1}{|D_{\omega }|}}.$$

(8)

The remaining transverse frame may be fixed locally from ${{\omega }^{\mu }}_{\nu }{N}^{\nu }$. Degenerate points of this normalization are excluded.

The branch tensor is

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

(9)

In the tetrad above its nonzero diagonal components are

$$Y_{UU}=4p_0(1-e^{4\chi }),Y_{NN}=-Y_{UU},Y_{WW}=Y_{SS}=Y_{UU}.$$

(10)

Equivalently,

$${\displaystyle \frac{Y_{UU}}{p_{\Lambda }}}=-tanh\chi .$$

(11)

The mixed tensor ${Y^{\mu }}_{\nu }$ has non-null Rainich type $(y,y,-y,-y)$:

$${\eta }^{\mu \nu }{Y}_{\mu \nu }=0,{Y^{\mu }}_{\alpha }{Y^{\alpha }}_{\nu }={\displaystyle \frac{1}{4}}{Y}_{\alpha \beta }{Y}^{\alpha \beta }{{\delta }^{\mu }}_{\nu }.$$

(12)

This is the algebraic type used in the Rainich reconstruction problem [9]. Infinitesimal branch variations are constrained by preservation of this stratum:

$$\begin{array}{cc}\hfill \delta \left({\eta }^{\mu \nu }{Y}_{\mu \nu }\right)& =0,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \delta \left({Y^{\mu }}_{\alpha }{Y^{\alpha }}_{\nu }-{\displaystyle \frac{1}{4}}{Y}_{\alpha \beta }{Y}^{\alpha \beta }{{\delta }^{\mu }}_{\nu }\right)& =0.\hfill \end{array}$$

(14)

Equivalently, if ${\mathcal{R}}_{\eta }$ is the non-null Rainich stratum and $Y=\rho J$ with $J^2=1$ and $trJ=0$, then admissible tangent vectors have the form $h=aJ+\rho K$, where $K_{\eta }^{†}=K$, $trK=tr\left(JK\right)=0$, and $JK+KJ=0$. The integrating curve is

$$J\left(t\right)=e^{t\mathcal{B}}J{e}^{-t\mathcal{B}},\rho \left(t\right)=\rho +ta,$$

(15)

with $\mathcal{B}=-JK/2$. Generic stress perturbations leave this stratum [33]. Self-dual and two-form descriptions of this tensor class occur in chiral gauge-gravity formulations [34]; Urbantke reconstruction gives the conformal-geometric comparison [35].

The convention used below is

$$Y_{\mu \nu }=-{{\rm Y}}_{\mu \nu }.$$

(16)

Thus the gauge-side tensor is represented only in a tetrad in which

$${{\rm Y}}_{NN}=-{{\rm Y}}_{UU},{{\rm Y}}_{WW}={{\rm Y}}_{SS}={{\rm Y}}_{UU},$$

(17)

and the anisotropy is fixed by

$$tanh\chi ={\displaystyle \frac{{{\rm Y}}_{UU}}{p_{\Lambda }}}.$$

(18)

Equivalently,

$$e^{4\chi }=1+{\displaystyle \frac{{{\rm Y}}_{UU}}{4p_0}}.$$

(19)

The real branch condition is

$$1+{\displaystyle \frac{{{\rm Y}}_{UU}}{4p_0}}>0.$$

(20)

This completes the algebraic gauge-to-branch map.

Put

$$R_{\omega }={\displaystyle \frac{{\omega }^2}{|D_{\omega }|}},{\omega }^2={\omega }^{\mu \nu }{\omega }_{\mu \nu }.$$

(21)

The translational-current coefficient is regularized by a phase-amplitude pair

$${\Psi }_{\zeta }={\rho }_{\zeta }{e}^{i{f}_{\zeta }},{\mu }_{\zeta }=e^{-\vartheta \left({\rho }_{\zeta }\right)},{\mu }_{\zeta }>0.$$

(22)

The vacuum normalization is ${\mu }_{\zeta }=1$ on the unexcited link background. The scalar branch factor is

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

(23)

The coefficient $\zeta$ is the translational-current coefficient of the flat-reference Alena Tensor description [5]. The scalar $\varphi$ is the branch residual used below as the Codazzi multiplier.

The translational current is

$$J_{\mathrm{tr}}^{\mu }=p_{\Lambda }{\zeta }^2{U}^{\mu }.$$

(24)

Its conservation is imposed by the phase term

$$S_{\mathrm{cur}}={\int }_X{f}_{\zeta }{\nabla }_{\mu }^{\left(k\right)}\left(p_{\Lambda }{\zeta }^2{U}^{\mu }\right)d{\mathrm{vol}}_k.$$

(25)

Variation with respect to $f_{\zeta }$ gives

$${\nabla }_{\mu }^{\left(k\right)}{J}_{\mathrm{tr}}^{\mu }=0.$$

(26)

The amplitude sector is used through its frozen-link stability condition,

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

(27)

The corresponding branch action is

$$S_{\mathrm{ARC}}^{\left(\zeta \right)}=\pm {\int }_X\varphi p_{\Lambda }d{\mathrm{vol}}_{\eta }+S_{\mathrm{cur}}+S_{{\rho }_{\zeta }}.$$

(28)

Here $S_{{\rho }_{\zeta }}$ denotes the local stiffness functional given in Appendix A.

The closed branch sector is the sector in which the material and residual parts are separately k-conserved:

$${\nabla }_{\mu }^{\left(k\right)}{T}_{\mathrm{matter}}^{\mu \nu }=0,$$

(29)

$${\nabla }^{\left(k\right)\mu }\left(\varphi Y_{\mu \nu }\right)=0.$$

(30)

The first identity is the material Alena balance in the same branch geometry [5]. At a current-supported regular link, the frozen principal boundary problem of (29) is natural on $S_{\Gamma }^2$. Its principal part is therefore $SU\left(2\right)$-equivariant, and the inequivalent low types are separated by the spherical-harmonic decomposition. For tensor traces the corresponding notation is that of tensor spherical harmonics [36,37].

The same branch fixes the material value of $\chi$. In the canonical vorticity plane, under the diagonal shear assumptions stated in Appendix A, the reduced anisotropic closure gives

$$tanh\chi ={\mu }_{\zeta }{\displaystyle \frac{2\varpi {\sigma }_{NS}+{\varpi }^2}{2|D_{\omega }|}}.$$

(31)

Together with (18), this gives the gauge-matter-vorticity compatibility condition

$${\displaystyle \frac{{{\rm Y}}_{UU}}{p_{\Lambda }}}={\mu }_{\zeta }{\displaystyle \frac{2\varpi {\sigma }_{NS}+{\varpi }^2}{2|D_{\omega }|}}.$$

(32)

Thus the same anisotropy is read from the gauge stress and from the normalized shear-vorticity branch.

Closure of the residual force-density term requires differential data in addition to the algebraic Rainich conditions. Put $\tau =k^{\mu \nu }{Y}_{\mu \nu }$ and

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

(33)

The residual scalar $\varphi \ne 0$ is a Codazzi multiplier if $A_{\mu \nu }=\varphi B_{\mu \nu }$ satisfies (40). If $C_{\alpha \mu \nu }^B={\nabla }_{\alpha }^{\left(k\right)}{B}_{\mu \nu }-{\nabla }_{\mu }^{\left(k\right)}{B}_{\alpha \nu }$ and $\theta =\mathrm{d}log\left|\varphi \right|$, this is the scalar multiplier equation

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

(34)

On a non-degenerate open set the scalar is fixed by the branch data.

Proposition 1 

(Codazzi multiplier criterion). Let X be a connected non-degenerate open set on which $B_{\mu \nu }$ is non-degenerate, and let ${\beta }^{\mu \nu }$ be its inverse. If (34) holds, then θ is fixed by B:

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

(35)

Conversely, define ${\theta }_B$ by (35). A nonzero scalar Codazzi multiplier exists on X if and only if (34) holds with $\theta ={\theta }_B$ and ${\theta }_B$ is exact. Equivalently, $\mathrm{d}{\theta }_B=0$ and all periods of ${\theta }_B$ vanish. In that case ϕ is determined on X up to multiplication by a nonzero constant.

Proof. 

Contracting (34) with ${\beta }^{\mu \nu }$ gives (35). If a multiplier exists, then $\theta =\mathrm{d}log\left|\varphi \right|$, so ${\theta }_B$ is exact. Conversely, if (34) holds with $\theta ={\theta }_B$ and ${\theta }_B=\mathrm{d}f$, then $\varphi =e^f$ gives a local Codazzi multiplier, with an arbitrary nonzero constant factor. The global condition is the vanishing of $\left[{\theta }_B\right]\in H_{\mathrm{dR}}^1\left(X\right)$. □

The translational-current coefficient is then reduced by the same scalar.

Proposition 2 

(Codazzi-current reduction of the translational coefficient). Let the residual scalar (23) be a Codazzi multiplier on a non-degenerate connected open set. If ${\theta }_B=\mathrm{d}{f}_B$, then

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

(36)

for a nonzero constant C. If, in addition, the translational current (24) is conserved, then

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

(37)

Proof. 

The first statement is the scalar part of Proposition 1 applied to (23). The conservation law (26) gives (37) after substituting (36). □

For the normalized branch, the trace-adjusted tensor is

$$A_{\mu \nu }=\varphi p_{\Lambda }\left({\displaystyle \frac{1}{3}}\left(\overline{k}-{\displaystyle \frac{4}{k}}\right)k_{\mu \nu }-{\eta }_{\mu \nu }\right),\overline{k}:=k^{\mu \nu }{\eta }_{\mu \nu }=2+2e^{-2\chi }.$$

(38)

It is chosen so that

$$\varphi Y_{\mu \nu }=A_{\mu \nu }-\mathcal{A}{k}_{\mu \nu },\mathcal{A}=A_{\alpha \beta }{k}^{\alpha \beta }.$$

(39)

Here k is the $\eta$-trace from (5), while $\overline{k}$ is the k-trace of ${\eta }_{\mu \nu }$.

Proposition 3 

(Residual divergence closure). If

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

(40)

then (30) holds.

Proof. 

Contracting (40) and using metric compatibility gives ${\nabla }^{\left(k\right)\mu }{A}_{\mu \nu }={\partial }_{\nu }\mathcal{A}$. Together with (39), this gives

$${\nabla }^{\left(k\right)\mu }\left(\varphi Y_{\mu \nu }\right)=0.$$

(41)

□

If $C_{\alpha \mu \nu }^A={\nabla }_{\alpha }^{\left(k\right)}{A}_{\mu \nu }-{\nabla }_{\mu }^{\left(k\right)}{A}_{\alpha \nu }$, then

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

(42)

Thus (41) is the contracted closure, while (40) is the full branch-integrability condition.

In the k-orthonormal frame $E_0=U$, $E_1=N$, $E_2=e^{-\chi }W$, $E_3=e^{-\chi }S$, the endomorphism ${A^a}_b$ has type $(M,M,P,P)$, with

$$M=\varphi p_{\Lambda }{\displaystyle \frac{tanh\chi (tanh\chi -3)}{3(1+tanh\chi )}},P=\varphi p_{\Lambda }{\displaystyle \frac{tanh\chi (tanh\chi +3)}{3(1+tanh\chi )}}.$$

(43)

Only the standard two-eigenvalue Codazzi splitting is used. The Riemannian background may be compared with [38]; Codazzi spacetime structures are treated in [39].

Theorem 1 

(Rainich-Codazzi optical decomposition). Let $A^{♯}$ be the k-self-adjoint endomorphism determined by $A_{\mu \nu }$. On an open set $\mathcal{U}$ on which the two eigenvalues in (43) are distinct, put

$$T\mathcal{U}=E\oplus F,E=span(U,N),F=span(W,S).$$

(44)

Then (40) holds if and only if

$$\begin{array}{cccc}\hfill d_{E}M& =0,\hfill & \hfill d_{F}P& =0,\hfill \end{array}$$

(45)

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

(46)

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

(47)

In particular, for

$$\ell =U+N,r=U-N,$$

(48)

one has

$${\nabla }_{\ell }^{\left(k\right)}\ell \Vert \ell ,{\nabla }_r^{\left(k\right)}r\Vert r,$$

(49)

and

$${\sigma }_{\ell }=0,{\sigma }_r=0.$$

(50)

Proof. 

The component decomposition of (40) with respect to (44) gives the standard two-eigenvalue Codazzi identities. The pure $EEE$ and $FFF$ components give (45), and the mixed components give (46) and (47); the converse is obtained by recombining the same components. For ℓ and r in (48), (46) removes the F-component of the corresponding accelerations. Metric compatibility leaves the E-component orthogonal to the same null vector, hence proportional to it. The trace-free transverse optical parts vanish by (47). □

The $EEE$ and $FFF$ parts of Theorem 1 give, after absorbing the numerical factor in (43) into the constants, the two first integrals

$$\varphi p_{\Lambda }{\displaystyle \frac{tanh\chi (tanh\chi -3)}{1+tanh\chi }}=C_{UN},E_0{C}_{UN}=E_1{C}_{UN}=0,$$

(51)

$$\varphi p_{\Lambda }{\displaystyle \frac{tanh\chi (tanh\chi +3)}{1+tanh\chi }}=C_{WS},E_2{C}_{WS}=E_3{C}_{WS}=0.$$

(52)

Using (23), the first integral may also be written as

$${\zeta }^2=1-{\mu }_{\zeta }{R}_{\omega }-{\displaystyle \frac{C_{UN}(1+tanh\chi )}{p_{\Lambda }tanh\chi (tanh\chi -3)}}.$$

(53)

The Codazzi gap is denoted

$${\Delta }_C=M-P.$$

(54)

It sets the local scale of the two-eigenvalue splitting and the scale used in the link support barrier.

Theorem 1 separates the full Codazzi closure from the optical input. The divergence closure (41) depends only on the contracted defect (42), while the link construction uses (49) and (50). The resulting structure is a Goldberg-Sachs type optical structure at the Codazzi level. The same $2+2$ geometry appears in type $2+2$ conformal Killing tensors [40] and in umbilical space-time structures with two shear-free geodesic null congruences [41]. The divergence-free non-null Maxwell-Minkowski case gives the Rainich comparison class [10]; Codazzi spacetime structures give another comparison class [39]. This $2+2$ splitting is the coefficient geometry used by the normal-operator reduction below.

Definition 1 

(Optical ARC branch). A non-null Alena-Rainich branch tensor is called optical on an open set if:

(i)

it satisfies the Rainich algebraic type (12) and determines a non-degenerate orthogonal splitting $T\mathcal{U}=E\oplus F$, where E is Lorentzian, F is spacelike, and the associated trace-adjusted endomorphism has two distinct eigenvalues;

(ii)

there exists a nonzero scalar ϕ such that the branch divergence closure (41) holds;

(iii)

the two principal planes are umbilic: for some $H_E\in \Gamma \left(F\right)$ and $H_F\in \Gamma \left(E\right)$,

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

(55)

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

(56)

A Codazzi-closed ARC branch is the calibrated case of this definition, with $H_E$ and $H_F$ fixed by (46) and (47).

Definition 1 supplies the local projective spinor sphere attached to the branch, in the standard sense of spinor and twistor geometry [11,42]. In the link construction below this sphere is the Riemann sphere $S_{\Gamma }^2\simeq {\mathbb{CP}}^1$. Twistor unification models use related projective-spinor geometry at the level of the full twistor space [13]. Curved twistor constructions give the broader comparison class [43].

The local sector is non-empty. In the warped realization of Appendix A, the Codazzi equations reduce to two first integrals, and a smooth hyperbolic representative is obtained. Put $s=tanh\chi$. The warped-product identities use the curvature convention of [44,45]. The normalized vorticity closure in that model gives

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

(57)

On a compact transverse leaf $\Sigma$, integration of (57) gives

$$0={\int }_{\Sigma }{\Delta }_{\gamma }\alpha d{A}_{\gamma }=\epsilon {\displaystyle \frac{2a{\left(s\right)}^2{D}_o^2}{c^{2}s}}{\mathrm{Area}}_{\gamma }(\Sigma ).$$

(58)

For $D_o\ne 0$ and $0<s<1$, this is incompatible with smooth global $\alpha$. A distributional closure has the form

$${\Delta }_{\gamma }\alpha =\epsilon {\displaystyle \frac{2a{\left(s\right)}^2{D}_o^2}{c^{2}s}}+\sum _j{q}_j{\delta }_{p_j},\sum _j{q}_j=-\epsilon {\displaystyle \frac{2a{\left(s\right)}^2{D}_o^2}{c^{2}s}}{\mathrm{Area}}_{\gamma }(\Sigma ).$$

(59)

Thus compact transverse leaves supply the boundary-source mechanism. A defect core followed along a branch trajectory in the E-plane gives a branch worldline $\Gamma$; a small two-sphere linking it is the optical link sphere. Compatible local ARC branches patch on $X=M\setminus \Gamma$ by the ordinary tensorial cocycle condition on k, Y, A, $\varphi$, $\chi$, the translational phase data, and the $2+2$ splitting.

The distributional source in (59) is a real boundary source. The integral spin^c link used in Section 3 is obtained when the defect resolution includes patching of the oriented transverse frame and the regular phase-current sector. Standard elliptic patching and removable-singularity theory provide the comparison class [46]. The local ARC sector remains governed by the standard elliptic regularity background [47].

2.2. The Defect, Blow-Up, and the Edge Boundary Problem

Let $X_e=[M;\Gamma ]$ denote the real blow-up of the branch worldline. In a normal collar of $\Gamma$ one uses local variables $(t,y)$, with t along $\Gamma$ and $y\in {\mathbb{R}}^3$ normal to $\Gamma$. Writing $y=r\omega$, $\omega \in S^2$, gives the blown-up collar with boundary fiber $S_{\Gamma }^2$:

$$\partial X_e=S(N\Gamma ),S{(N\Gamma )}_t\simeq S_{\Gamma }^2.$$

(60)

Thus the link sphere is the boundary fiber of the resolved defect. The optical structure of Definition 1 identifies this fiber with the projective spinor sphere ${\mathbb{CP}}^1$.

The codimension-three topology removes the scalar period obstruction in the local single-defect model. Indeed,

$${\mathbb{R}}^4\setminus \mathbb{R}\simeq \mathbb{R}\times ({\mathbb{R}}^3\setminus \left\{0\right\})\simeq S^2.$$

(61)

Consequently,

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

(62)

The first identity makes the global scalar multiplier condition in Proposition 1 automatic whenever $\mathrm{d}{\theta }_B=0$ in this local topology. The second identity is the topological slot occupied by the integral link class used in the spin^c quantization. For a nontrivial ambient topology the same statement is applied to the admissible collar class; global $H^1$ periods are then part of the completion problem.

The linearized Codazzi part has a simple principal symbol. For a symmetric variation $h_{\mu \nu }$ of the trace-adjusted tensor, put

$$C{\left(h\right)}_{\alpha \mu \nu }={\nabla }_{\alpha }{h}_{\mu \nu }-{\nabla }_{\mu }{h}_{\alpha \nu }.$$

(63)

At a covector $\xi \ne 0$,

$${\sigma }_{\xi }C{\left(h\right)}_{\alpha \mu \nu }={\xi }_{\alpha }{h}_{\mu \nu }-{\xi }_{\mu }{h}_{\alpha \nu }.$$

(64)

If $\xi =e^1$ in an orthonormal coframe, ${\sigma }_{\xi }C\left(h\right)=0$ gives $h_{i\nu }=0$ for $i>1$. With $h_{\mu \nu }=h_{\nu \mu }$, the remaining component is $h_{11}$. Hence

$$ker{\sigma }_{\xi }C=\{a\xi \otimes \xi \}.$$

(65)

The pure Codazzi equation is therefore overdetermined with a longitudinal symbol kernel. This kernel is removed by the divergence gauge $\delta h_{\nu }={\nabla }^{\mu }{h}_{\mu \nu }$. Its principal symbol is ${\sigma }_{\xi }\delta {\left(h\right)}_{\nu }={\xi }^{\mu }{h}_{\mu \nu }$.

Lemma 1 

(Gauge-fixed Codazzi symbol). For $\xi \ne 0$,

$${\displaystyle \frac{1}{2}}|{\sigma }_{\xi }{C\left(h\right)|}^2+|{\sigma }_{\xi }{\delta \left(h\right)|}^2={\left|\xi \right|}^2{\left|h\right|}^2.$$

(66)

In particular, $C\oplus \delta$ has injective principal symbol.

Proof. 

By homogeneity choose $\xi =e^1$. Then ${\displaystyle \frac{1}{2}}|{\sigma }_{\xi }{C\left(h\right)|}^2={\sum }_{i>1,\nu }{\left|h_{i\nu }\right|}^2$, while $|{\sigma }_{\xi }{\delta \left(h\right)|}^2={\sum }_{\nu }{\left|h_{1\nu }\right|}^2$. The sum is ${\left|h\right|}^2$. □

The corresponding formal adjoint has the usual Hodge form. If ${\beta }_{\alpha \mu \nu }=-{\beta }_{\mu \alpha \nu }$, then, up to the lower-order connection terms,

$${\left(C^{*}\beta \right)}_{\mu \nu }=-2{Sym}_{\mu \nu }{\nabla }^{\alpha }{\beta }_{\alpha \mu \nu },{\left({\delta }^{*}u\right)}_{\mu \nu }=-{Sym}_{\mu \nu }{\nabla }_{\mu }{u}_{\nu }.$$

(67)

Thus the self-adjoint first-order block built from $C\oplus \delta$ has square with principal symbol ${\left|\xi \right|}^{2}I$. This is the local elliptic core of the edge-Calderon reduction.

Near the blown-up boundary, the normal derivatives have the form

$${\partial }_{y_i}={\omega }_i{\partial }_r+{\displaystyle \frac{1}{r}}{\Omega }_i,{\Omega }_i\in \Gamma \left(T{S}^2\right).$$

(68)

Consequently any first-order normal operator obtained from the gauge-fixed Codazzi block has the form

$$D=A_r{\partial }_r+{\displaystyle \frac{1}{r}}{D}_{S^2}+A_t{\partial }_t+A_0,$$

(69)

or, equivalently,

$$rD=A_r\left(r{\partial }_r\right)+D_{S^2}+r{A}_t{\partial }_t+r{A}_0.$$

(70)

Before the optical spinorial projection is applied, the boundary operator associated with $C\oplus \delta$ is the Hodge-type operator on $S_{\Gamma }^2$, with the remaining tensor indices as coefficients. Its principal symbol is already of Clifford type. Let $B_{\mathrm{Hodge}}$ denote the principal boundary operator of the gauge-fixed Codazzi block. For $\xi \in T^{*}{S}_{\Gamma }^2$,

$${\sigma }_1\left(B_{\mathrm{Hodge}}\right)\left(\xi \right)=\epsilon \left(\xi \right)-\iota \left({\xi }^{♯}\right),$$

(71)

where $\epsilon$ is exterior multiplication and $\iota$ is contraction with the metric dual vector. Under the optical middle-spinor identification, (71) is the Clifford multiplication symbol on the projective link:

$${\sigma }_1\left(P_{\mathrm{mid}}^{\mathrm{spin}}{B}_{\mathrm{Hodge}}{P}_{\mathrm{mid}}^{\mathrm{spin}}\right)\left(\xi \right)=c_{S^2}\left(\xi \right).$$

(72)

The spin^c determinant line changes the connection and the global domain, but not the principal symbol. The frozen form of this projected edge operator is the boundary part of the normal operator introduced in SubSection 2.3.

Definition 2 

(Admissible optical middle-spinor data). Optical middle-spinor boundary data on an edge-regular ARC collar are called admissible if the projection $P_{\mathrm{mid}}^{\mathrm{spin}}$ is smooth on $S(N\Gamma )$, is compatible with the transverse-frame spin^c determinant line, and identifies the projected Hodge symbol (71) with the Clifford symbol (72). The frozen link data are called $SU\left(2\right)$-equivariant when the link metric, the projection $P_{\mathrm{mid}}^{\mathrm{spin}}$, and the determinant-line connection are $SU\left(2\right)$-equivariant.

Hypothesis 1 

(Spinorial boundary data). The edge-regular optical ARC collar carries admissible optical middle-spinor boundary data. When scalar compression on a Borel-Weil block is used, the corresponding frozen link data are assumed to be $SU\left(2\right)$-equivariant.

The spinorial assumption is part of the edge boundary input. It selects edge-regular collars for which the Hodge-type boundary symbol can be read in the spinorial link variables. In the standard $SU\left(2\right)$-invariant ${\mathbb{CP}}^1$ collar with an integral transverse-frame determinant line, this condition is realized by the usual spin^c link data [48]. The support theorem uses only the induced Dirac-type principal symbol and the scalar nature of the equivariant zeroth-order compression on each Borel-Weil block. The global compatibility of such data with a completed ARC branch is included in the finite-action completion problem; the boundary comparison is the standard one for Dirac-type operators [49].

Put $B_{\mathrm{mid}}=P_{\mathrm{mid}}^{\mathrm{spin}}{B}_{\mathrm{Hodge}}{P}_{\mathrm{mid}}^{\mathrm{spin}}$.

Lemma 2 

(Dirac-type boundary normal form). Under Assumption 1, the projected boundary operator has the form

$$B_{\mathrm{mid}}=D_{S^2,q}^{{\mathrm{spin}}^{\mathrm{c}}}+R_0,$$

(73)

where $R_0$ is a zeroth-order endomorphism. If the frozen link data are $SU\left(2\right)$-equivariant and $P_q$ denotes the Borel-Weil projection onto $E_q$, then $P_q{R}_0{P}_q$ is scalar on $E_q$.

Proof. 

By (72), $B_{\mathrm{mid}}$ and $D_{S^2,q}^{{\mathrm{spin}}^{\mathrm{c}}}$ have the same Clifford principal symbol. Hence their difference has vanishing principal symbol and is a zeroth-order endomorphism. This gives (73). This is the standard Dirac-type normal form [48]; the boundary comparison is the usual one for Dirac-type boundary operators [49].

If the frozen link data are $SU\left(2\right)$-equivariant, both first-order operators in (73) are $SU\left(2\right)$-intertwiners. Hence $R_0$ is an $SU\left(2\right)$-intertwiner. Since $E_q$ is the irreducible Borel-Weil module in (1), Schur’s lemma gives the scalar compression. □

Thus the principal symbol of the projected boundary operator is fixed by (72), while Lemma 2 separates the spin^c Dirac part from the zeroth-order remainder. The next section freezes the edge operator on the collar, passes to the principal associated graded normal operator, and records the coefficient-channel separation used in the finite Borel-Weil support problem.

2.3. The Equivariant Normal Operator of the Collar

The edge form obtained in Section 2 supplies the local operator which is used for support selection. Only its frozen collar class is needed at this stage. The gauge-fixed Codazzi block has injective principal symbol by Lemma 1; after blow-up its first-order form is recorded in (70). The optical middle-spinor projection then gives the Dirac-type boundary normal form of Lemma 2.

The normal operator is the frozen edge operator on the boundary fiber $S_{\Gamma }^2$. Its role is to separate the coefficient channels before finite Toeplitz visibility is applied. The lower-order mixed terms which are not present in the associated graded normal operator are kept as part of the closed-branch spectral problem.

Let ${\mathcal{E}}_{\Gamma }$ denote the frozen boundary trace bundle of the gauge-fixed Codazzi-Calderon block on the collar of $\Gamma$. It contains the tensor trace variables, the divergence-gauge variables, and the coefficient labels carried by the phase-current and Codazzi-gap sources. The normal operator is defined by freezing the coefficients of the rescaled edge operator (70) at $r=0$ and retaining the zeroth-order frozen response:

$${\mathcal{N}}_{\Gamma }=A_r\left(r{\partial }_r\right)+B_{\Gamma }+{\mathcal{V}}_{\Gamma }^{\left(0\right)}.$$

(74)

Here $B_{\Gamma }$ denotes the boundary operator induced by the gauge-fixed Codazzi block, and ${\mathcal{V}}_{\Gamma }^{\left(0\right)}$ denotes the frozen zeroth-order response. The tangential and non-frozen remainder terms in (70) are lower order in the collar filtration.

After admissible middle-spinor projection, Lemma 2 gives

$$P_{mid}^{spin}{\mathcal{N}}_{\Gamma }{P}_{mid}^{spin}=A_r\left(r{\partial }_r\right)+D_{S^2,q}^{spi{n}^c}+R_{0,q}+{\mathcal{V}}_{\Gamma ,q}^{\left(0\right)}.$$

(75)

The term $R_{0,q}$ is scalar on $E_q$ under the $SU\left(2\right)$-equivariant compression of Lemma 2. Thus it shifts the frozen block response but does not change the Toeplitz visibility of non-scalar source types.

The normal jet filtration of the boundary source is denoted by

$$J_{\perp }^0\subset J_{\perp }^1\subset J_{\perp }^2.$$

(76)

The support problem uses the associated graded of this filtration. The scalar graded piece is treated as a closed local singlet. The non-scalar coefficient channels are

$$H_{bdry}^{prin}=H_W\oplus H_C.$$

(77)

Here $H_W$ is generated by the phase-current trace, and $H_C$ is generated by the Codazzi-gap traces. The decomposition in (77) is a coefficient-channel decomposition. It is not a decomposition only by spherical degree, since the Codazzi-gap family also contains a degree-one companion.

Let $P_q$ denote the Borel-Weil projection onto $E_q$ once the spin^c link block has been fixed. The finite normal compression is

$${\mathcal{N}}_{\Gamma ,q}^{fin}=P_q{P}_{mid}^{spin}{\mathcal{N}}_{\Gamma }{P}_{mid}^{spin}{P}_q.$$

(78)

The support Hessian used below is the quadratic form obtained from the principal associated graded normal operator:

$$H_{support}^{prin}={\left(gr{\mathcal{N}}_{\Gamma }\right)}^{*}gr{\mathcal{N}}_{\Gamma }.$$

(79)

The split-conserved collar separates the material-current trace from the residual Codazzi trace at frozen principal order. The first trace is governed by (29); the second is governed by (30). This separation is inherited by the associated graded normal operator.

Proposition 4 

(Principal coefficient superselection). In the frozen split-conserved ARC collar, the principal associated graded normal operator preserves the coefficient-channel decomposition (77). Equivalently,

$${\Pi }_{W}gr{\mathcal{N}}_{\Gamma }{\Pi }_C=0.$$

(80)

Consequently the principal support Hessian has no Codazzi-to-current coefficient block:

$${\Pi }_W{H}_{support}^{prin}{\Pi }_C=0.$$

(81)

Proof. 

The principal symbol of the gauge-fixed Codazzi block is the tensorial symbol of $C\oplus \delta$ from Lemma 1. It acts on the tensor and form indices of the boundary trace. It does not change the coefficient label carried by a phase-current or Codazzi-gap source. The boundary principal symbol is the Hodge-Clifford symbol (71), and after middle-spinor projection it is the Clifford symbol (72). This symbol is again diagonal on the coefficient labels.

The associated graded is taken before finite Toeplitz visibility is applied. At this level the phase-current trace is the generator of $H_W$, while the Codazzi-gap traces are the generators of $H_C$. Since the collar is split-conserved at frozen principal order, no coefficient map from the residual Codazzi trace to the material-current trace is present in the principal symbol. This gives (80). Equation (81) follows from (79). □

With respect to (77), the principal associated-graded normal operator has the block form

$$gr{\mathcal{N}}_{\Gamma }=\left(\begin{array}{cc}{\mathcal{N}}_W& 0\\ 0& {\mathcal{N}}_C\end{array}\right).$$

(82)

The off-channel terms are therefore collar-lower-order or frozen closed-branch terms in the support comparison.

The remaining mixed terms split into a collar-lower-order part and a possible frozen closed-branch off-channel part. On a fixed finite set of Toeplitz candidate supports these terms are finite-dimensional after compression.

Proposition 5 

(Lower-order mixed stability). Let $H_{\rho }$ be the support Hessian on an edge-regular collar of radius ρ, and let $H_{support}^{prin}$ be (79). Put

$$K_{\rho }^{mix}={\Pi }_W\left(H_{\rho }-H_{support}^{prin}\right){\Pi }_C.$$

(83)

For every finite set of candidate supports there is a constant $C_{\Gamma }$ such that

$$∥K_{\rho }^{mix}-K_0^{mix}∥\le C_{\Gamma }\rho .$$

(84)

In particular, if

$$∥K_0^{mix}∥+C_{\Gamma }\rho <{\epsilon }_{*},$$

(85)

where ${\epsilon }_{*}$ is the stability threshold in Proposition 11, then the support selected by the principal associated graded comparison is unchanged.

Proof. 

In the rescaled edge form (70), the non-frozen terms not retained in (74) carry a factor r or the difference between a $C^1$ coefficient and its frozen boundary value. After compression to a fixed finite family of candidate supports, the corresponding finite operator has the form ${\mathcal{N}}_{\rho }={\mathcal{N}}_0+\rho {\mathcal{R}}_{\rho }$, with $\parallel {\mathcal{R}}_{\rho }\parallel$ uniformly bounded for $0<r<\rho$. Hence $H_{\rho }-H_0$ is the sum of terms linear in $\rho$ and a quadratic $O\left({\rho }^2\right)$ term. The mixed block therefore satisfies (84). A frozen off-channel closed-branch response is finite-dimensional after Toeplitz compression and is recorded as $K_0^{mix}$. The last statement is Proposition 11. □

Thus the support comparison used in Section 3 is a principal associated graded comparison. The coefficient-channel block diagonal form is fixed by Proposition 4; lower-order mixed pieces are allowed only within the finite stability range of Proposition 5.

3. Borel-Weil Link Quantization and Minimal Support

The compact-leaf obstruction in Section 2 supplies the place where finite link data enter the closed branch problem. In the sector used below the obstruction is resolved by patching the oriented transverse frame and by the regular phase-current sector of (24). By (60), the link is the boundary fiber of the blown-up defect. Assumption 1 supplies the spin^c boundary data on $S_{\Gamma }^2\simeq {\mathbb{CP}}^1$, and Lemma 2 gives the corresponding Dirac-type boundary operator.

The finite link spaces used below are the positive spin^c Dirac blocks recorded in (1). The zeroth-order term in (73) belongs to the frozen block response and does not change the Toeplitz support under the scalar compression of Lemma 2. The principal low source is read from the normal second jet. Its non-scalar associated-graded part consists of the phase-current dipole and the trace-free Codazzi-gap quadrupole. Proposition 4 separates these coefficient channels at principal associated-graded order. The Toeplitz visibility cutoff then fixes the first Borel-Weil blocks on which the two source families can be seen.

3.1. Spin^c Link Modes and Toeplitz Visibility

On the non-degenerate set ${\Delta }_C\ne 0$, mixed frame rotations between the Rainich and transverse planes are massive relative to the Codazzi eigenvalue gap (54). With the k-orthonormal transverse frame $E_2=e^{-\chi }W$ and $E_3=e^{-\chi }S$, the transverse connection on $F=span(W,S)$ is denoted

$$A_{WS}=-k(E_2,{\nabla }^{\left(k\right)}{E}_3).$$

(86)

The convention is chosen so that local connection forms transform by $A\mapsto A+\mathrm{d}\beta$ under an oriented frame rotation by $\beta$. Its curvature is

$$F_{WS}=\mathrm{d}{A}_{WS}.$$

(87)

A compact-leaf defect is called transverse-frame resolved when the resolution on $X=M\setminus \Gamma$ includes the patching of the oriented transverse frame around the link sphere.

Proposition 6 

(Transverse-frame integrality). If a compact-leaf defect is transverse-frame resolved, then the transverse link curvature has integral period:

$${\displaystyle \frac{1}{2\pi }}{\int }_{S_{\Gamma }^2}{F}_{WS}\in \mathbb{Z}.$$

(88)

Consequently the transverse-frame resolution supplies an integral spin^c determinant line on the link.

Proof. 

Let $U_N$ and $U_S$ be two frame patches on $S_{\Gamma }^2$. On $U_N\cap U_S$ the oriented k-orthonormal transverse frames differ by a map to $SO\left(2\right)$, written as $g_{NS}=e^{i\beta }$. With the convention in (86), the local connection forms satisfy $A_S=A_N+\mathrm{d}\beta$. By Stokes’ theorem on the two hemispheres,

$${\int }_{S_{\Gamma }^2}{F}_{WS}=-{\int }_{S^1}\mathrm{d}\beta .$$

(89)

The transition function $e^{i\beta }$ is single-valued on the equator, so the last integral is an integer multiple of $2\pi$. This gives (88). □

The topological class in (88) is the integral class allowed by (62). General real distributional solutions of (59) may lie outside this integral sector. After vacuum selection the residual transverse holonomy is read as the electromagnetic connection, while the full electroweak group appears below as the stabilizer of the reconstructed two-dimensional block.

For an optical ARC branch carrying a transverse-frame resolved compact-leaf defect, the link sphere carries the projective-spinor reading supplied by Definition 1. The finite zero-mode spaces are the positive spin^c link modes in (1). Changing the small sphere or the transverse frame changes the construction by the corresponding spin^c bundle isomorphism.

The self-dual three-dimensional comparison uses the standard bilinear structure on self-dual forms [50] and its spinorial descriptions [51]. Chiral gravitational variables are used only for the branch geometry [52,53]. Pure-connection and self-dual formulations give the corresponding comparison class [54,55].

Finite non-singlet labels are treated as link or boundary data. A closed local branch observable is invariant under changes of zero-mode basis. Hence an open vector in a nontrivial irreducible finite multiplicity space cannot define a closed local branch tensor. For color this leaves local singlet closures, such as ${\overline{q}}_A{q}^A$ and ${ϵ}_{ABC}{q}^A{q}^B{q}^C$, while open color indices occur only as link, boundary, or sector labels. This is the local branch form of the separation between closed observables and finite sector labels in Doplicher-Roberts reconstruction [56].

Let $\Gamma$ be a branch worldline produced by (58) and (59). The optical branch structure identifies the small link sphere with the projective spinor sphere of the branch; in the Codazzi sector this structure is supplied by Theorem 1. Thus the finite link module is assigned to the Borel-Weil realization of the branch twistor line. This is the local analogue of using projective twistor geometry for internal symmetry in twistor unification [13,57].

Near the branch worldline, after separating the tangential direction along $\Gamma$, the normal spinorial boundary problem has the spin^c form on a collar or conic neighbourhood [48,58]. Let $D_q$ be the resulting spin^c Dirac operator on the link sphere, with determinant degree fixed by (88). With the chosen orientation, the positive link modes are the standard monopole zero modes on the projective sphere [14]. Equivalently, they are the ${\mathbb{CP}}^1$ Borel-Weil spaces [16], as in (1). The Aharonov-Casher count gives the same index reading [59]. Hence ${dim}_{\mathbb{C}}{E}_q=q$, with the standard $SL\left(2\right)$ convention used below [17]. The same zero-mode realization appears in the Taub-NUT Dirac setting [15].

The finite link sector is therefore the spin^c Berezin-Toeplitz quantization of the projective optical link. The integer q fixes the quantum block, while the low boundary source is read through the finite Toeplitz image on that block. Let ${\mathcal{H}}_{\ell }$ denote the degree-ℓ scalar spherical harmonics on $S_{\Gamma }^2$. The Toeplitz operator on $E_q$ is $T_q\left(f\right)=P_q{M}_f{P}_q$, where $P_q$ is the orthogonal projection to $E_q$. Standard ${\mathbb{CP}}^1$ Toeplitz quantization gives the finite-mode comparison class [18]. The same cutoff has the fuzzy-sphere form [19], with brane/fuzzy-sphere models providing the broader finite-matrix setting [20].

Lemma 3 

(Toeplitz visibility cutoff). The traceless endomorphism space of $E_q$ decomposes as

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

(90)

Equivalently,

$$T_q\left({\mathcal{H}}_{\ell }\right)=\left\{\begin{array}{cc}{V}_{\ell },\hfill & 0\le \ell \le q-1,\hfill \\ 0,\hfill & \ell \ge q,\hfill \end{array}\right.$$

(91)

after projection to the irreducible $SU\left(2\right)$ types.

Proof. 

Equation (1) identifies $E_q$ with the spin $(q-1)/2$ irreducible $SU\left(2\right)$ module. The Clebsch-Gordan decomposition of $E_q\otimes E_q^{*}$ gives the summands $V_{\ell }$ for $0\le \ell \le q-1$. Standard ${\mathbb{CP}}^1$ Toeplitz quantization gives a nonzero equivariant map on each visible summand and no finite-mode image for $\ell \ge q$. □

The visible non-scalar Toeplitz content of $E_q$ is read from (90). Hence $E_2$ is the first dipole-only block and $E_3$ is the first block with a degree-two response.

Let $L_a$ be the $SU\left(2\right)$ generators on $E_q$. The fuzzy adjoint Laplacian is

$${\Delta }_{q}X={\displaystyle \sum _{a=1}^3}[L_a,[L_a,X]].$$

(92)

The Toeplitz link energy on an active response $B\in {End}_0\left(E_q\right)$ is

$${\mathcal{E}}_q\left(B\right)={\kappa }_{q}Tr\left(B^{†}{\Delta }_{q}B\right),{\kappa }_q>0.$$

(93)

On the summand $V_{\ell }$ this energy is proportional to $\ell (\ell +1)$.

3.2. Second-Jet Sources

The regular phase current and the Codazzi gap give the two non-scalar low sources. The phase source is first order on the link, while the gap source contains the normal Hessian of $log|{\Delta }_C|$. Put

$$\begin{array}{cc}\hfill S_W& ={\Pi }_{\ell =1}^{\mathrm{offdiag}}\left(\mathrm{d}{f}_{\zeta }+A_{WS}\right),\hfill \end{array}$$

(94)

$$\begin{array}{cc}\hfill S_C^{\left(1\right)}& ={\Pi }_{\ell =1}\left(d_{\perp }log\left|{\Delta }_C\right|\right),\hfill \end{array}$$

(95)

$$\begin{array}{cc}\hfill S_C^{\left(2\right)}& ={\Pi }_{\ell =2}{\left({\nabla }_{\perp }^{2}log\left|{\Delta }_C\right|\right)}_0.\hfill \end{array}$$

(96)

Here the projections are taken after the frozen boundary-trace map from link one-forms and normal tensors to the corresponding Toeplitz source space. The symbol ${\Pi }_{\ell =1}^{\mathrm{offdiag}}$ denotes the off-diagonal part of the degree-one projected response on the dipole block, $d_{\perp }$ and ${\nabla }_{\perp }$ are normal derivatives to $\Gamma$, and the subscript 0 denotes the trace-free part. The color source is $S_C=S_C^{\left(1\right)}+S_C^{\left(2\right)}$.

The projected link traces of the linearized branch response are denoted by

$$b_C^{\left(2\right)}={\Pi }_{\ell =2}{T}_{\Gamma }\left(\delta A\right),b_W^{\mathrm{offdiag}}={\Pi }_{\mathrm{offdiag}}{\Pi }_{\ell =1}{T}_{\Gamma }\left(\delta A\right).$$

(97)

The weak dipole is tied to the phase-current link one-form, while the color quadrupole is tied to the normal Hessian of the Codazzi gap.

The normal two-jet gives the low harmonic content directly. In a normal frame, a representative boundary field has expansion $F\left(y\right)=F_0+F_i{y}_i+{\displaystyle \frac{1}{2}}{F}_{ij}{y}_i{y}_j+{O\left(\right|y|}^3)$. On the link $y=rn$, the term $F_i{n}_i$ is of type $V_1$, while $F_{ij}^0{n}_i{n}_j$, with $F_{ij}^0$ trace-free, is of type $V_2$. The scalar part belongs to $V_0$. Equivalently, for the scalar normal model the indicial equation is

$$\rho (\rho +1)=\ell (\ell +1),$$

(98)

so the regular roots $\rho =\ell$ give the low trace orders $\ell =0,1,2$ for the normal two-jet.

The projected Calderon trace in these three low components is stable under small smooth perturbations of the frozen collar. The corresponding singular-integral and functional-calculus background is the standard one [60,61].

Proposition 7 

(Low Codazzi link traces). The link components $\ell =0,1,2$ survive the principal linearized Codazzi equation on a punctured normal neighbourhood of Γ. In the annulus model of Appendix A, the boundary trace map has nonzero image in these three components. This low-mode non-vanishing is stable under sufficiently small smooth perturbations of the normal branch geometry. Consequently, the central, degree-one, and degree-two low responses used in (97) are realized by boundary traces of linearized branch responses whenever the corresponding projected low trace is nonzero.

Proof. 

The principal annulus trace calculation separates the spherical components of the frozen normal problem. The regular indicial roots contain the families $\ell =0,1,2$, and the projected trace has nonzero image in those components. Stability follows from the finite-dimensional stability of the projected Calderon trace under small smooth perturbations. The assignment to the phase-current and Codazzi-gap channels is made by (94)-(96). □

The same low-mode language is compatible with the anisotropy variation. If $\chi ={\chi }_0+\epsilon f$, $\zeta ={\zeta }_0+\epsilon z$, and ${\mu }_{\zeta }{R}_{\omega }=B_0+\epsilon b$, with $T_0=tanh{\chi }_0$ and ${\varphi }_0=1-{\zeta }_0^2-B_0$, then the linearized first-integral difference gives

$${\displaystyle \frac{c_{WS}}{A_{WS,0}}}-{\displaystyle \frac{c_{UN}}{A_{UN,0}}}={\varphi }_0{\displaystyle \frac{6(1-T_0^2)}{9-T_0^2}}f.$$

(99)

Here $c_{UN}$ and $c_{WS}$ are the variations of the first-integral constants, while $A_{UN,0}$ and $A_{WS,0}$ denote the unperturbed scalar factors in (51) and (52). In the real non-degenerate branch, the scalar multiplier in (99) is nonzero. Thus, at the frozen-coefficient level on a small link sphere, the linearized anisotropy component f is not removed by the Codazzi first-integral difference. Its axisymmetric low components are the Legendre components of the spherical modes used in the annulus trace calculation.

A projected low source on the quantized optical link is called second-jet if, in a normal collar of $\Gamma$, the principal boundary source of the normal spinorial Codazzi problem is determined, modulo higher link-energy terms, by the normal two-jet of the trace-adjusted branch tensor, the regular translational current (24), and the normal Codazzi gap (54). It is called non-degenerate if the central projected trace is nonzero, the $V_1$ phase-current projection is nonzero, and the $V_2$ Codazzi-gap projection is nonzero. The color-generating locus is the subset on which $S_C^{\left(1\right)}$ and $S_C^{\left(2\right)}$ generate the traceless algebra on the minimal quadrupole-capable block after Toeplitz projection.

Proposition 8 

(Second-jet link-source reduction). For a non-degenerate second-jet link source in the split-conserved matter-Codazzi sector (29)-(30), the principal non-scalar low source has exactly the degree-one and degree-two harmonic types. At the frozen-coefficient link level these are the irreducible $SU\left(2\right)$ types $V_1$ and $V_2$. The corresponding primitive source families are the phase-current first-order family and the Codazzi-gap second-order family.

Proof. 

Restriction of the normal two-jet to $S_{\Gamma }^2$ gives the types $V_0$, $V_1$, and $V_2$ described above. The trace part of the quadratic term belongs to $V_0$. Higher harmonic degrees occur only in higher normal jets. The non-degenerate condition gives the nonzero $V_1$ and $V_2$ projections. The frozen link operator is $SU\left(2\right)$-equivariant, and irreducible spherical types of different degree are inequivalent. This gives the stated primitive source families. The residual part is closed by (30). □

Corollary 1 

(Generic active second-jet stratum). Let $\mathcal{B}$ be a Banach space of admissible linearized boundary data for which the projected trace maps to the central component, to the degree-one phase-current component, and to the degree-two Codazzi-gap component are continuous linear maps and are not identically zero. Then the subset ${\mathcal{B}}^{nd}\subset \mathcal{B}$ on which all three projected traces are nonzero is open and dense. If the degree-one color companion is not symmetry-forbidden, the color-generating locus is open and dense inside ${\mathcal{B}}^{nd}$.

Proof. 

The zero set of each nonzero continuous projected trace map is a closed proper linear subspace of $\mathcal{B}$. Its complement is open and dense. The first statement follows by finite intersection. The failure of the degree-one phase channel to contain the required off-diagonal component is a proper linear condition. The non-generating color pairs are cut out by the corresponding commutator minors in the finite-dimensional space ${End}_0\left(E_3\right)$, hence form a proper closed exceptional set whenever one generating pair exists. □

Lemma 4 

(Non-empty color-generating locus). In the minimal quadrupole-capable block $E_3$, the complexified color-generating locus of Corollary 1 is non-empty.

Proof. 

Identify $E_3$ with the spin-one basis of ${Sym}^2{\mathbb{C}}^2$, and let $e_{ij}$ be the corresponding matrix units. In the Clebsch-Gordan decomposition ${End}_0\left(E_3\right)=V_1\oplus V_2$, one may take $X=e_{12}+e_{23}$ in $V_1$ and $Q=e_{21}-e_{32}$ in $V_2$. Their first commutator gives $[X,Q]=diag(1,-2,1)$. Commuting this diagonal element with X and Q separates the adjacent matrix units. Further commutators give $e_{13}$ and $e_{31}$, and the remaining diagonal traceless elements follow from adjacent commutators. Hence the complex Lie closure is ${\mathfrak{sl}}_3\left(\mathbb{C}\right)$. Equivalently, the compact real form has complexified closure ${\mathfrak{sl}}_3\left(\mathbb{C}\right)$. □

3.3. Second-Jet Support Energy and the Minimal Module

A Borel-Weil link quantization of a second-jet source is called filtered when the normal jet filtration $J_{\perp }^0\subset J_{\perp }^1\subset J_{\perp }^2$ is passed to its associated graded before finite Toeplitz visibility is applied. The scalar graded piece is treated as a closed local singlet. The primitive non-scalar source families are the phase-current family and the Codazzi-gap family specified by (94)-(96). Each primitive family is assigned to the initial Borel-Weil block $E_q$ which sees its highest nonzero normal order.

Proposition 9 

(Filtered second-jet assignment). For a non-degenerate second-jet link source, filtered link quantization assigns the phase-current source family to $E_2$ and the Codazzi-gap source family to $E_3$.

Proof. 

Proposition 8 identifies the principal non-scalar associated-graded types with $V_1$ and $V_2$. By Lemma 3, the initial Borel-Weil block which sees $V_{\ell }$ is $E_{\ell +1}$. The phase-current family has highest nonzero normal order one and is assigned to $E_2$. The Codazzi-gap family has highest nonzero normal order two and is assigned to $E_3$; its degree-one companion is visible in the same block by (90). The scalar piece has no open finite index. This gives the stated filtered assignment. □

The filtered second-jet construction fixes the primitive source families. The support tension is used only for the finite support comparison against passive enlargement of the zero-mode blocks. The separation used below is a separation of boundary source families, not only a separation by harmonic degree: the Codazzi-gap family includes its degree-one companion together with its degree-two principal component.

The support-selection Hessian is the principal associated-graded Hessian of the frozen split-conserved collar. Thus the normal-order filtration is applied before finite Toeplitz visibility is used. Proposition 4 gives the coefficient-channel separation of this principal Hessian. The phase-current source (94) belongs to $H_W$, while the Codazzi-gap traces (95) and (96) belong to $H_C$. In the associated-graded support problem these primitive source families are retained before Toeplitz projection. Mixed terms which are lower order in this filtration are part of the closed-branch spectral problem and are kept within the stability range of Proposition 5.

Let $P_W$ project to a dipole support $E_{q_W}$ and let $P_C$ project to a quadrupole-capable support $E_{q_C}$. The two support tensions are scaled by the Codazzi gap:

$${\lambda }_W={\lambda }_{W,0}|{\Delta }_C{|}_{\Gamma },{\lambda }_C={\lambda }_{C,0}{\left|{\Delta }_C\right|}_{\Gamma },{\lambda }_{W,0},{\lambda }_{C,0}>0.$$

(100)

The associated-graded support energy is

$$\begin{array}{cc}\hfill {\mathcal{E}}_{\Gamma }& ={\kappa }_W{\mu }_{\zeta ,\Gamma }Tr\left(B_W^{†}{\Delta }_{P_W}{B}_W\right)+{\kappa }_{C}Tr\left(B_C^{†}{\Delta }_{P_C}{B}_C\right)+{\lambda }_{W}Tr{P}_W+{\lambda }_{C}Tr{P}_C\hfill \\ \hfill & -2\mathrm{Re}\langle B_W,S_W\rangle -2\mathrm{Re}\langle B_C,S_C\rangle .\hfill \end{array}$$

(101)

Here ${\Delta }_{P_W}$ and ${\Delta }_{P_C}$ are the fuzzy adjoint Laplacians on the selected supports, while $B_W$ and $B_C$ are the active low-mode responses.

Proposition 10 

(Gap-current Euler equations). For fixed $P_W$ and $P_C$, the stationary active responses of (101) satisfy

$$\begin{array}{cc}\hfill {\kappa }_W{\mu }_{\zeta ,\Gamma }{\Delta }_{P_W}{B}_W& =S_W,\hfill \end{array}$$

(102)

$$\begin{array}{cc}\hfill {\kappa }_C{\Delta }_{P_C}{B}_C& =S_C.\hfill \end{array}$$

(103)

Consequently all unforced visible components vanish in the ground state.

Proof. 

Variation of (101) with respect to $B_W^{†}$ and $B_C^{†}$ gives (102) and (103). On every visible summand with zero source, the corresponding equation has only the zero solution because the compressed Casimir is positive on non-scalar modes. □

Proposition 11 

(Lower-order mixed stability). Fix a finite set of candidate supports. Let $m_{*}>0$ be the smallest positive quadratic coefficient of (101) on their non-scalar active summands, and put ${\lambda }_{*}=min\{{\lambda }_W,{\lambda }_C\}$. There exists ${\epsilon }_{*}>0$, depending only on $m_{*}$ and ${\lambda }_{*}$, such that any lower-order mixed quadratic perturbation of operator norm at most ${\epsilon }_{*}$ preserves the support selected by the associated-graded comparison.

Proof. 

After Toeplitz projection the comparison is finite-dimensional. The quadratic part of (101) is positive on non-scalar active summands, with lower bound $m_{*}$. For sufficiently small mixed perturbation the quadratic form remains positive by the elementary Young inequality. A passive enlargement has zero forced response by Proposition 10 and increases the support cost by at least ${\lambda }_{*}$. By decreasing ${\epsilon }_{*}$ if necessary, the possible lower-order energy gain is kept below this cost. Hence the minimizing support is unchanged. □

A finite filtered two-channel support is called energy-selected when it minimizes (101) among the admissible Borel-Weil supports.

Corollary 2 

(No passive visible enlargement). For an energy-selected finite filtered support, no visible Borel-Weil summand beyond the active filtered source is retained in the ground state, provided the lower-order mixed terms are in the stability range of Propositions 5 and 11.

Proof. 

An unforced visible summand has zero active response by Proposition 10. The positive support terms in (100) still increase $Tr{P}_W$ or $Tr{P}_C$. Proposition 11 keeps this comparison stable under the allowed lower-order mixed perturbations. □

Proposition 12 

(Minimal two-channel support). Let the principal support Hessian be the associated-graded Hessian of the frozen split-conserved collar, with coefficient-channel separation as in Proposition 4. If the phase-current source family has highest nonzero normal order one and the Codazzi-gap source family has highest nonzero normal order two, then the energy-selected finite filtered two-channel support satisfies $q_W=2$ and $q_C=3$.

Proof. 

The support comparison is made on the separated coefficient channels $H_W$ and $H_C$. By Lemma 3, the first block which sees a type $V_{\ell }$ is $E_{\ell +1}$. The phase-current source family has highest nonzero type $V_1$, hence a visible support must have $q_W\ge 2$. The Codazzi-gap source family has highest nonzero type $V_2$, hence a quadrupole-capable support must have $q_C\ge 3$. The block $E_3$ also sees the degree-one Codazzi companion by (90).

For $q_W>2$ or $q_C>3$, the additional visible summands are unforced by the filtered source. Corollary 2 excludes such passive enlargement in the energy-selected ground state. Thus the first visible supports are the selected supports. □

Theorem 2 

(Minimal rank-one ARC link module). Let $C\simeq E_{q_C}$ and $W\simeq E_{q_W}$ be the non-abelian filtered link-index blocks assigned respectively to the Codazzi-gap source family and to the phase-current source family. Assume that closed local observables are singlet-valued and that the branch carries a non-degenerate second-jet link source. If the physical non-singlet boundary support is energy-selected by (101) among finite filtered two-channel supports, with lower-order mixed terms in the range of Proposition 5, then

$$q_C=3,q_W=2.$$

(104)

Equivalently, the minimal finite link-index module is

$$C_{\Gamma }=ker{D}_3^{+},W_{\Gamma }=ker{D}_2^{+}.$$

(105)

On the color-generating stratum of Corollary 1, after complexification of the Toeplitz images,

$$\left(T_3\left({\mathcal{H}}_1\right)\oplus T_3\left({\mathcal{H}}_2\right)\right){\otimes }_{\mathbb{R}}\mathbb{C}={End}_0\left(E_3\right).$$

(106)

The degree-two color response and the off-diagonal degree-one weak response are expressed by

$$F_C^{(\ell =2)}\ne 0,\mathcal{L}\langle F_C^{(\ell =1)},F_C^{(\ell =2)}\rangle =\mathfrak{su}\left(3\right),$$

(107)

and

$$F_W^{\mathrm{offdiag}}\ne 0,\mathcal{L}\langle F_W\rangle =\mathfrak{su}\left(2\right).$$

(108)

Proof. 

The principal low trace is supplied by Proposition 7. Proposition 8 identifies the non-scalar associated-graded pieces of the phase-current and Codazzi-gap source families. Proposition 9 gives the filtered assignment of the primitive families. Proposition 4 gives the associated-graded channel separation, and Proposition 12 gives the energy-selected two-channel minimality. This gives (104) and (105).

For $q=3$, (90) gives $V_1\oplus V_2={End}_0\left(E_3\right)$ after complexification. Lemma 3 identifies these summands with the Toeplitz images of ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$. This gives (106). The genericity statement is Corollary 1, and the non-empty generating locus is supplied by Lemma 4. □

The notation C and W is fixed by these low-link roles: C denotes the first Codazzi-gap quadrupole-generating block and W denotes the first phase-current dipole-only block. The stabilizer, exterior algebra, and one-Higgs channels are applied to the reconstructed object $C_{\Gamma }\oplus W_{\Gamma }$ in Section 4.

4. Internal Reconstruction and the Locked Low Operator

The support theorem of Section 3 fixes the finite link module before the exterior-algebra package is applied. The weak block is selected by the phase-current dipole channel, while the color block is selected by the Codazzi-gap quadrupole channel and contains the full low-mode color algebra. The projected connection on the moving zero-mode bundle gives the gauge connection of the finite module, and the locked odd operator gives the local low spectral sector. Write

$$C=C_{\Gamma },W=W_{\Gamma },V=C\oplus W,{dim}_{\mathbb{C}}C=3,{dim}_{\mathbb{C}}W=2.$$

(109)

The group of orthonormal zero-mode bases is $U\left(C\right)\times U\left(W\right)$. Fixing the total top form gives the unimodular subgroup

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

(110)

For the minimal module this is

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

(111)

The quotient is the usual global Standard Model form detected by line operators [62]. If $y_C$ and $y_W$ are the abelian weights, top-form neutrality gives $3y_C+2y_W=0$. With minimal weak normalization $y_W=1$,

$$C=(3,1,-2/3),W=(1,2,+1).$$

(112)

This is the convention used below. The transverse connection (86) is then the unbroken electromagnetic connection after vacuum selection.

4.1. Basis Group, Exterior Package, and Projected Connection

The matter representation is the standard $3+2$ exterior-algebra package applied to the reconstructed chiral object V with assignments (112). With the unimodular top form fixed, exterior duality gives ${\left({\Lambda }^{p}V\right)}^{*}\simeq {\Lambda }^{5-p}V$. Since ${dim}_{\mathbb{C}}V=5$, even and odd exterior parity are exchanged by duality. The even part has dimension 16 and gives the left-handed one-generation module familiar from the $SU\left(5\right)$/Spin$\left(10\right)$ organization [21]. In conventional unified models the enlarged gauge sector carries the additional vector multiplets and proton-decay channels [63,64]. In the ARC branch sector the $3+2$ module is obtained from the link before such an enlargement is introduced:

$${\Lambda }^{\mathrm{even}}V={\nu }_L^c\oplus u_L^c\oplus Q_L\oplus e_L^c\oplus d_L^c\oplus L_L.$$

(113)

Here ${\nu }_L^c$ denotes the conjugate of the right-handed neutrino. Related Clifford-ideal and ladder-operator constructions obtain the same particle-content package from complex Clifford algebras [22,65]. The family-oriented Clifford-ideal setting gives another algebraic comparison point [23]. The input used here is the reconstruction of $V=C\oplus W$ from the link; the exterior-algebra package is then applied with fixed exterior parity. This use of exterior parity is compatible with differential-form descriptions of fermions [29].

The identity in (113) is the standard bookkeeping decomposition of ${\Lambda }^{\mathrm{even}}V$ under the splitting $V=C\oplus W$. Using (112) and the fixed top form gives the displayed components and hypercharges.

The weak factor $W=(1,2,+1)$ has the quantum numbers of the minimal electroweak order parameter. For $\Phi \in W$, exterior multiplication and contraction define the Clifford-odd operator

$$c(\Phi +{\Phi }^{†})=\epsilon (\Phi )+\iota \left({\Phi }^{†}\right).$$

(114)

This operator changes exterior parity. Its one-Higgs singlet bilinear channels, after charge conjugation and diagonal locking, are

$$Q_L⟷u_L^c,Q_L⟷d_L^c,$$

(115)

and

$$L_L⟷{\nu }_L^c,L_L⟷e_L^c.$$

(116)

No other bilinear channel with a single $\Phi$ or ${\Phi }^{†}$ is invariant under (110).

Proposition 13 

(One-Higgs invariant channels). For the finite module (113) and the weak factor $\Phi \in W$, the one-Higgs Clifford-odd map (114) gives precisely the quark and lepton channels displayed in (115) and (116).

Proof. 

Exterior multiplication by $\Phi$ and contraction by ${\Phi }^{†}$ change the W-degree by one and preserve the color contraction required by the singlet bilinear. With the hypercharge convention (112), the invariant pairings are exactly those in (115) and (116). The remaining one-Higgs pairings have nonzero total hypercharge, an unmatched weak index, or an open color index. □

The anomaly cancellation is used as a consistency check on the minimal sector; the selection of (104) comes from the link-support argument. The hypercharges in the sums below are induced from (112) on the exterior module (113). With $A\left(3\right)=1$, $A\left(\overline{3}\right)=-1$, and $T\left(3\right)=T\left(\overline{3}\right)=T\left(2\right)=1/2$, the independent local sums are

$$\begin{array}{cc}\hfill {\mathcal{A}}_{SU{\left(3\right)}^3}& =2A\left(3\right)+A\left(\overline{3}\right)+A\left(\overline{3}\right)=0,\hfill \end{array}$$

(117)

$$\begin{array}{cc}\hfill {\mathcal{A}}_{SU{\left(3\right)}^{2}U\left(1\right)}& =2T\left(3\right){\displaystyle \frac{1}{3}}+T\left(\overline{3}\right)\left(-{\displaystyle \frac{4}{3}}\right)+T\left(\overline{3}\right){\displaystyle \frac{2}{3}}=0,\hfill \end{array}$$

(118)

$$\begin{array}{cc}\hfill {\mathcal{A}}_{SU{\left(2\right)}^{2}U\left(1\right)}& =3T\left(2\right){\displaystyle \frac{1}{3}}+T\left(2\right)(-1)=0,\hfill \end{array}$$

(119)

$$\begin{array}{cc}\hfill {\mathcal{A}}_{U{\left(1\right)}^3}& =6{\left({\displaystyle \frac{1}{3}}\right)}^3+3{\left(-{\displaystyle \frac{4}{3}}\right)}^3+2^3+3{\left({\displaystyle \frac{2}{3}}\right)}^3+2{(-1)}^3=0,\hfill \end{array}$$

(120)

$$\begin{array}{cc}\hfill {\mathcal{A}}_{{\mathrm{grav}}^{2}U\left(1\right)}& =6{\displaystyle \frac{1}{3}}+3\left(-{\displaystyle \frac{4}{3}}\right)+2+3{\displaystyle \frac{2}{3}}+2(-1)=0.\hfill \end{array}$$

(121)

The right-handed neutrino is hypercharge neutral in this convention.

Projected Zero-Mode Connection

The finite module (109) plays the internal role of the low link sector. In almost-commutative geometry the finite algebra is part of the spectral triple data [24], and the spectral action supplies the corresponding dynamics [25]. In the present construction the finite module is reconstructed before the exterior-algebra package is applied. This is the comparison level of geometric matter models in which particle labels are recovered from geometry and topology [28].

The gauge connection is the projected connection on a moving zero-mode bundle. Let ${\mathcal{M}}_{\Gamma }$ denote a local parameter space of regular deformations of the branch worldline and of its link data for which the positive link spaces have constant dimension. For $q=2,3$, the spaces $E_q=ker{D}_q^{+}$ form finite-rank Hermitian bundles over ${\mathcal{M}}_{\Gamma }$. If $P_q$ is the corresponding spectral projection, the projected connection is

$${\nabla }^{B,q}=P_q\mathrm{d}{P}_q.$$

(122)

In a local orthonormal zero-mode basis $e_a^{\left(q\right)}$, this reads

$${\left(A_{\mu }^{B,q}\right)}_{ab}={〈e_a^{\left(q\right)},{\partial }_{\mu }{e}_b^{\left(q\right)}〉}_{\mathrm{link}}.$$

(123)

For the minimal module this gives the projected C and W connections. This is the Berry-Wilczek-Zee connection of the degenerate link-zero-mode bundle [66,67]. The finite labels are the zero-mode coordinates of the moving link. Dynamical-principal-bundle formulations provide a natural comparison class for gauge variables [68].

Proposition 14 

(Berry curvature of the link zero-mode bundle). For a regular family of link data with constant $dim{E}_q$, the curvature of (122) is

$$F^{B,q}=P_q\mathrm{d}{P}_q\wedge \mathrm{d}{P}_q{P}_q.$$

(124)

In a local zero-mode basis this is

$$F^{B,q}=\mathrm{d}{A}^{B,q}+A^{B,q}\wedge A^{B,q}.$$

(125)

For the minimal link module, $F^B=F^{B,3}\oplus F^{B,2}$ is valued in $\mathfrak{u}\left(C\right)\oplus \mathfrak{u}\left(W\right)$. After the total top form is fixed, the admissible curvature satisfies

$${Tr}_C{F}^{B,3}+{Tr}_W{F}^{B,2}=0.$$

(126)

Thus its Lie algebra is $\mathfrak{s}\left(\mathfrak{u}\right(3)\oplus \mathfrak{u}(2\left)\right)$.

Proof. 

The identity (124) is the curvature of the projected connection on the image of $P_q$. Equation (125) is its local form in the orthonormal basis used in (123). The last statement is the infinitesimal form of the top-form condition defining (110). □

The curvature (125) is the Yang-Mills field-strength datum of the moving zero-mode bundle. Its kinetic normalization belongs to the branch Hessian or spectral matching problem and is not used in the structural reconstruction. The bosonic Schur Hessian in Section 5.1 uses the same finite link data through the reduced even Hessian of a weak-sector projector.

4.2. Locked Low Operator and Codazzi-Callias Mass

The exterior-algebra construction fixes the local representation content through (113). The spinorial branch frame fixes the null pair $U\pm N$ through the optical branch bilinears. These data are combined on the associated bundle

$$E_F=S_{\mathrm{branch}}\otimes {\Lambda }^{•}V,V=C\oplus W.$$

(127)

The finite part is the minimal link module of Theorem 2. The branch spinor part is supplied by the optical ARC geometry of Section 2.

Let $N_C$ and $N_W$ be the exterior number operators on ${\Lambda }^{•}C$ and ${\Lambda }^{•}W$, and put $N=N_C+N_W$. The primitive finite parities are ${\Gamma }_C={(-1)}^{N_C}$ and ${\Gamma }_W={(-1)}^{N_W}$, so that ${\Gamma }_V={(-1)}^N={\Gamma }_C{\Gamma }_W$. Let ${\gamma }_{\mathrm{branch}}^5$ be the chirality operator defined by the branch tetrad. The diagonal locking involution is

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

(128)

The branch Dirac term is odd with respect to the branch chirality factor, and the weak Clifford map (114) is odd with respect to total exterior parity. Charge conjugation uses the exterior duality induced by the fixed top form. This fixes (128) up to the overall sign among primitive product-type signs. The resulting role of the locking condition is analogous in aim to spectral-geometric Standard Model models in which the finite geometry removes the fermion doubling [69]. The global bundle construction is the standard spin^c one [70], applied here to the branch spinor factor and the finite exterior module.

Let $P_{+}$ and $P_{-}$ be the projectors onto the two locked sectors. The mirror projector is denoted by ${\Pi }_{\mathrm{mir}}=1-{\Pi }_{\mathrm{lock}}$. The gap-selected branch-helicity condition is

$${\Pi }_{\mathrm{mir}}{K}_{\mathrm{lock}}{\Pi }_{\mathrm{mir}}\ge m_{\Gamma }{\Pi }_{\mathrm{mir}},K_{\mathrm{lock}}=K_F+m_{\Gamma }{\Pi }_{\mathrm{mir}},m_{\Gamma }=m_0{\left|{\Delta }_C\right|}_{\Gamma }.$$

(129)

Equivalently,

$$\sigma \left({\Pi }_{\mathrm{mir}}{K}_{\mathrm{lock}}{\Pi }_{\mathrm{mir}}\right)\subset [m_{\Gamma },\infty ).$$

(130)

The comparison class is the localization of chiral modes by a defect of a Dirac operator, as in the Jackiw-Rebbi mechanism [71] and in domain-wall fermions [72].

The branch Dirac operator on (127) is

$$D_E={\gamma }^{\mu }{\nabla }_{\mu }^E,{\nabla }_{\mu }^E={\nabla }_{\mu }^S\otimes 1+1\otimes {\nabla }_{\mu }^{\Lambda V}.$$

(131)

By (128), $D_E$ is odd with respect to the locked grading. The odd operator used in the locked sector is

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

(132)

Here $Q_{\mathrm{hol}}$ is the holonomy response, and $Q_C$ is the middle spinorial projection of the linearized Codazzi response. The only first-order spacetime part of (132) is (131). The weak Clifford map, the holonomy response, and the Codazzi response are zero-order in the branch directions. Hence the effective first-order principal symbol is the branch Clifford symbol restricted to the finite low-cluster coefficient bundle:

$${\sigma }_1\left(Q_F\right)(x,\xi )=P_{+}\gamma \left(\xi \right)P_{-}.$$

(133)

The corresponding positive operator is

$$K_F=Q_F^{†}{Q}_F,{\sigma }_2\left(K_F\right)(x,\xi )={\left|\xi \right|}_k^{2}I.$$

(134)

The gap term in (129) is a boundary-domain term and does not change the local exterior representation content fixed above.

The branch-helicity splitting is obtained from the two principal spinors of the optical branch. If ${\delta }_C$ denotes the spinorial form of the linearized Codazzi response, its extreme and middle projections are

$${\delta }_C^{\mathrm{ext}}=P_{\mathrm{ext}}{\delta }_C{P}_{\mathrm{ext}},{\delta }_C^{\mathrm{mid}}=P_{\mathrm{mid}}{\delta }_C{P}_{\mathrm{mid}}.$$

(135)

Equations (49) and (50) give the branch-geometric origin of the extreme projection. The middle projection contains the two branch-helicity components of a spin-$1/2$ mode. The corresponding middle Codazzi contribution in (132) is denoted

$$Q_C=P_{\mathrm{mid}}{\delta }_C{P}_{\mathrm{mid}}.$$

(136)

The leading locked contribution is obtained by applying the middle-spinor projection to the zeroth-order Codazzi part of the normal operator (74). The following statement records the part of this projection which is fixed by the branch Clifford structure.

Proposition 15 

(Leading Clifford-odd Codazzi component). On the locked low collar, let $Q_C^{odd}$ denote the zeroth-order part of the middle-spinor Codazzi response which anticommutes with the branch Clifford symbol. If the corresponding locked intertwiner space is one-dimensional, then

$$Q_C=a{\Delta }_C{\gamma }_{\mathrm{branch}}^5\otimes J+R_C,J^{†}J=1,$$

(137)

where a is a real normalization constant and $R_C$ contains the Clifford-even, lower-order, and closed-branch zeroth-order remainders.

Proof. 

The only first-order branch symbol in (132) is (133). The zeroth-order component which acts as a Callias mass anticommutes with this branch Clifford symbol. On the branch spinor factor this fixes the Clifford part to be proportional to ${\gamma }_{\mathrm{branch}}^5$. The Codazzi contribution is measured, at leading frozen order, by the two-eigenvalue splitting of Theorem 1; hence its scalar coefficient is proportional to ${\Delta }_C$ from (54). After the locked finite projection, the remaining coefficient is an intertwiner J on the finite low space. If this intertwiner space is one-dimensional, the Clifford-odd component has the form shown in (137). The remaining zeroth-order terms are collected in $R_C$. □

When $a\ne 0$ and $R_C$ is controlled, the Codazzi gap contributes a Callias-type mass term to the locked odd operator. In the warped collar model of Appendix A,

$${\Delta }_C=-{\displaystyle \frac{2F_0}{3-s}},0<s<1.$$

(138)

For $F_0\ne 0$,

$$|{\Delta }_C|\ge {\displaystyle \frac{2|F_0|}{3}}.$$

(139)

Proposition 16 

(Codazzi-gap lower bound). Assume that the projection (137) holds with $a\ne 0$. Let $\parallel R_C+Q_{\mathrm{hol}}+c(\Phi +{\Phi }^{†})\parallel \le R_0$, let the zeroth-order curvature and derivative losses in the square of (132) be bounded below by $-(R_1+R_2)I$, and put $m_{\Delta }=2\left|a{F}_0\right|/3$. If

$${(m_{\Delta }-R_0)}^2>R_1+R_2,$$

(140)

then, on the locked collar,

$$K_F\ge {\nabla }_E^{*}{\nabla }_E+{\mu }^{2}I,{\mu }^2={(m_{\Delta }-R_0)}^2-R_1-R_2>0.$$

(141)

Proof. 

By Proposition 15, the leading term in (137) anticommutes with the branch Dirac symbol. In the square of (132), the first-order cross-term is reduced to a zeroth-order derivative contribution. The positive term from (137) is bounded below by $m_{\Delta }^{2}I$ using (139). The bounded remainder gives the shift $R_0$, and the remaining zeroth-order losses give $R_1+R_2$. This gives (141). □

A local curvature reading of a mass parameter is visible already in the test branch

$$d{s}_k^2=d{t}^2-d{x}^2-e^{2\chi (t,x)}(d{y}^2+d{z}^2).$$

(142)

For $\chi ={\chi }_0+\epsilon f(t,x)$ one obtains

$$R^{\left(1\right)}=-4\epsilon (f_{,tt}-f_{,xx}).$$

(143)

If $f_{,tt}-f_{,xx}+m_{\mathrm{eff}}^{2}f=0$, with $f\ne 0$, then

$$m_{\mathrm{eff}}^2={\displaystyle \frac{R^{\left(1\right)}}{4\epsilon f}}.$$

(144)

Thus a local mass parameter is read from the scalar-curvature response of the anisotropic branch. Related Gabor-regularized metric models show explicitly how finite-resolution smoothing can induce effective curvature and stress-energy [73].

Writing $\Omega =({\omega }_{+},{\omega }_{-})$ in the two branch-helicity lines, the local fermion operator has the Dirac form

$$Q_F=\left(\begin{array}{cc}0& Q_{-}\\ Q_{+}& 0\end{array}\right),Q_F\Omega =m\Omega .$$

(145)

Equivalently,

$$Q_{-}{Q}_{+}{\omega }_{+}=m^2{\omega }_{+},Q_{+}{Q}_{-}{\omega }_{-}=m^2{\omega }_{-}.$$

(146)

The mass is the singular value of the odd Codazzi-Higgs-holonomy operator on the locked low sector. The exterior algebra fixes the allowed finite channels, while $Q_{\mathrm{hol}}$, $Q_C$, and the eigenvalues of (134) remain branch spectral data.

Let $P_{\mathrm{low}}$ be the smooth projection onto the isolated physical branch-helicity cluster in the local fibrewise low-cluster reduction. On parameter patches where the gap persists it is represented by the corresponding Riesz projection. The projected first-order operator on this finite-rank coefficient bundle is

$$D_{\mathrm{eff}}=P_{\mathrm{low}}{Q}_F{P}_{\mathrm{low}}.$$

(147)

The corresponding mass operator is

$$M_{\mathrm{eff}}=P_{\mathrm{low}}{K}_F^{1/2}{P}_{\mathrm{low}}.$$

(148)

A low branch excitation is a spinor-valued coefficient field on the locked finite module, and a change of basis in the zero-mode bundle is read as a gauge transformation. The corresponding effective action has the standard first-order form

$$S_{\mathrm{low}}=\int {\overline{\psi }}_{\mathrm{low}}\left(i{\gamma }^{\mu }{D}_{\mathrm{eff},\mu }-M_{\mathrm{eff}}\right){\psi }_{\mathrm{low}}d{\mathrm{vol}}_k+S_{\mathrm{higher}}.$$

(149)

Here $D_{\mathrm{eff},\mu }$ contains the branch spin connection and the projected link connection. The term $S_{\mathrm{higher}}$ denotes operators suppressed by the gap and by the scales at which the frozen branch and link data are resolved.

Proposition 17 

(Locked gap stability). Let $K_F$ have an isolated low cluster below Δ on the locked bundle and let the mirror complement satisfy (130). If a self-adjoint perturbation R satisfies $\parallel R\parallel <\Delta /2$ and $\parallel R\parallel <m_{\Gamma }/2$ on the corresponding spectral subspaces, then the locked low cluster and the mirror gap persist.

Proof. 

The statement is the usual spectral stability estimate for bounded self-adjoint perturbations of an isolated spectral set, in the sense of Kato [74]. The Hausdorff displacement of the spectrum is bounded by $\parallel R\parallel$. Hence a gap larger than $2\parallel R\parallel$ cannot close. □

Quantization of (149) is the standard CAR quantization of the low coefficient field. The finite module, locking, one-Higgs representation channels, Dirac principal symbol, and leading Clifford-odd Codazzi form are structural. The nonzero value of a, the remainder bound on $R_C$, the numerical fermion masses, mixing matrices, and fermionic threshold data require the global ARC spectral problem.

5. Schur Hessian, Projective Color, and Global Completion

The preceding sections give the local reconstruction. The Codazzi closure supplies the optical branch, the compact-leaf obstruction supplies the projective link, the equivariant normal operator supplies the coefficient-channel separation, the rank-one Borel-Weil tower supplies the finite support, and the locked operator supplies the local low spectral sector. The remaining structures considered here are the finite Schur Hessian associated with the same normal operator, the projective color sector, the central family response, and the completion of the local collar to a finite-action ARC branch. Running electroweak matching and threshold conversion belong to the global matching problem.

The regular edge-normal consequences used below are collected first.

Proposition 18 

(Regular edge-normal consequences). Let an optical ARC collar be edge-regular, split-conserved at frozen principal order, and $SU\left(2\right)$-equivariant on the frozen link. Assume that the transverse-frame resolution supplies an integral spin^c determinant line on $S_{\Gamma }^2\simeq {\mathbb{CP}}^1$, that the collar coefficients are $C^1$ in the radial variable, and that a finite family of Borel-Weil candidate supports has been fixed. Then the projected boundary operator is of spin^c Dirac type up to zeroth order, and the collar-lower-order mixed terms satisfy (84) after finite Toeplitz compression. If the heavy Calderon block is invertible on the complement of the low support, the finite low Hessian is the Schur complement (154). If the locked Codazzi estimate (141) holds and the perturbation is below the stability threshold of Proposition 17, the mirror exterior-parity sector remains separated from the low cluster.

Proof. 

The spin^c boundary normal form is Lemma 2. The mixed estimate is Proposition 5; the $C^1$ radial regularity gives the required $O\left(\rho \right)$ collar remainder after compression to a fixed finite support family. The Schur statement is the finite-dimensional elimination of the massive Calderon variables in (153). The last statement follows from the lower bound in Proposition 16 and the spectral stability estimate of Proposition 17. □

5.1. Bosonic Schur Hessian and Protected Determinant Entry

The bosonic comparison uses the reconstructed module (109), the top-form preserving group (110), the hypercharge convention (112), and the weak order parameter contained in (114). The weak order parameter is treated as a branch projector in W. Its four real deformations split into three angular directions and one radial direction. Together with the determinant abelian direction, these give the light bosonic variables used in the reduced even Hessian.

The reduced Hessian has three scalar entries: an angular weak-projector stiffness A, a determinant split B, and a radial breathing stiffness C. In the canonically normalized basis $(W^1,W^2,W^3,B,h)$, the neutral block has one null direction fixed by (112):

$$M_{\mathrm{bos}}^2={\displaystyle \frac{v^2}{4}}\left(\begin{array}{ccccc}A& 0& 0& 0& 0\\ 0& A& 0& 0& 0\\ 0& 0& A& -\sqrt{AB}& 0\\ 0& 0& -\sqrt{AB}& B& 0\\ 0& 0& 0& 0& 4C\end{array}\right).$$

(150)

The form (150) is the finite low Schur Hessian associated with the weak-sector projector. Its dimensionless neutral reading is the on-shell link angle (A21). The mass values in Appendix B are then obtained only after the scale v is inserted.

The leading closed-cell weights are fixed by the minimal $3+2$ support. The weak projector contributes three angular directions and one radial direction, while the unimodular color compensation contributes ${dim}_{\mathbb{C}}C=3$. The closed-cell weight is

$$D_{\mathrm{cell}}=2{dim}_{\mathbb{C}}W+{dim}_{\mathbb{C}}C=7.$$

(151)

Proposition 19 

(Protected determinant entry). For the minimal support $C\oplus W=E_3\oplus E_2$ with the top-form constraint (110) and the closed-cell normalization (151), the determinant entry of the reduced bosonic Hessian is

$$B_0={\displaystyle \frac{{dim}_{\mathbb{C}}C{dim}_{\mathbb{C}}W}{D_{\mathrm{cell}}^2}}={\displaystyle \frac{6}{49}}.$$

(152)

Proof. 

The determinant direction is the relative abelian direction left after the total top form has been fixed. For the $3+2$ module, its closed-cell weight is proportional to the product of the two determinant block sizes, ${dim}_{\mathbb{C}}C{dim}_{\mathbb{C}}W$. The normalization is the square of the closed-cell weight (151). Since ${dim}_{\mathbb{C}}C=3$ and ${dim}_{\mathbb{C}}W=2$, (152) follows. □

Thus $B_0$ is fixed by the finite $3+2$ module and the top-form constraint. No frozen heavy-block normalization enters this determinant entry.

The angular and radial entries depend on the frozen Calderon-Schur normalization. Let the low variables be denoted by x, and let the orthogonal massive Calderon variables be denoted by y. At the frozen link, the quadratic form has the block shape

$$\mathcal{H}(x,y)={\displaystyle \frac{1}{2}}\langle x,H_{LL}x\rangle +\langle x,H_{LH}y\rangle +{\displaystyle \frac{1}{2}}\langle y,H_{HH}y\rangle .$$

(153)

Eliminating y gives

$$H_{\mathrm{eff}}=H_{LL}-H_{LH}{H}_{HH}^{-1}{H}_{HL}.$$

(154)

Equivalently, (154) is the second-order part of the frozen normal determinant. The branch-level Schur benchmark below uses the frozen Calderon-Schur scalar unit for one real channel. With the standard four-dimensional logarithmic unit $1/\left(16{\pi }^2\right)$ and the quadratic Schur factor $1/4$, this unit is

$$u={\displaystyle \frac{1}{4}}{\displaystyle \frac{1}{16{\pi }^2}}={\displaystyle \frac{1}{64{\pi }^2}}.$$

(155)

This is the local normalization used for the finite Schur benchmark; covariant derivative expansion calculations give the comparison normalization [75], while precision threshold matching gives the electroweak comparison class [76].

The sign and multiplicity of the finite correction are fixed by the projected channel type. The weak angular sector is the $V_1$ part of ${End}_0\left(E_2\right)$ and has three equivalent real directions. Each constrained angular projector direction contributes $-u/2$ to the common angular stiffness. The radial breathing channel is a single scalar channel and contributes $+u$. The determinant split is unchanged by this angular Schur correction because the weak angular generators are traceless and the determinant variable is the trace/top-form direction. Thus

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

(156)

The protected part of (156) is the determinant entry (152). In particular, the determinant split in the neutral block is fixed by the $3+2$ determinant weight and is independent of the scalar Schur unit (155). The numerical branch-level comparison is left to Appendix B.

5.2. Projective Color, Families, and Global Completion

Closed local color observables are singlet-valued in the branch tensor sector. Thus the closed color holonomy is naturally projective and is represented by a $PSU\left(C\right)$ link holonomy. In the minimal support of Theorem 2, $C=E_3$, so the relevant group is $PSU\left(3\right)$. Principal bundles over $S^2$ are classified at this level by the fundamental group of the structure group [77]. Since ${\pi }_1\left(PSU\left(3\right)\right)\simeq {\mathbb{Z}}_3$, the central projective-color sectors over the link are modeled on

$$H^2(S_{\Gamma }^2,{\mathbb{Z}}_3)\simeq {\mathbb{Z}}_3.$$

(157)

Characteristic and secondary classes give the corresponding global comparison level [78]. The same global form of the gauge group is detected by line operators [62].

Let ${\mathcal{L}}_{\Gamma }$ denote the set of central projective-color sectors associated with the link. It is an affine torsor for the group in (157):

$$H^2(S_{\Gamma }^2,{\mathbb{Z}}_3)⤻{\mathcal{L}}_{\Gamma }.$$

(158)

After a choice of auxiliary origin, a sector may be labelled by an element $c_{\Gamma }\in H^2(S_{\Gamma }^2,{\mathbb{Z}}_3)$. The closed projective color branch does not select such an origin. The canonical finite Hilbert space attached to this torsor is

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

(159)

After a choice of auxiliary origin in ${\mathcal{L}}_{\Gamma }$, this is identified with the regular representation $\mathbb{C}\left[{\mathbb{Z}}_3\right]$.

Proposition 20 

(Canonical central-sector quantization). The central-sector space (159) is the regular representation of ${\mathbb{Z}}_3$. In particular,

$${dim}_{\mathbb{C}}{\mathcal{H}}_{\mathrm{cen}}=3.$$

(160)

Proof. 

By (158), $H^2(S_{\Gamma }^2,{\mathbb{Z}}_3)$ acts freely and transitively on ${\mathcal{L}}_{\Gamma }$. Since this group is isomorphic to ${\mathbb{Z}}_3$, the set ${\mathcal{L}}_{\Gamma }$ is a finite ${\mathbb{Z}}_3$-torsor. The canonical Hilbert space of a finite torsor is the space of square-summable functions on it. Choosing an auxiliary origin identifies ${\mathcal{L}}_{\Gamma }$ with ${\mathbb{Z}}_3$, and then ${\ell }^2\left({\mathcal{L}}_{\Gamma }\right)$ becomes $\mathbb{C}\left[{\mathbb{Z}}_3\right]$ with the left regular action. The dimension statement follows. □

Let ${\omega }_3=e^{2\pi i/3}$. After choosing an auxiliary cyclic labeling of ${\mathcal{L}}_{\Gamma }$, the regular representation is written with clock and shift matrices

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

(161)

They obey

$$ZS={\omega }_{3}SZ,Z^3=S^3=1.$$

(162)

The gap-scaled central response is represented by the finite clock-shift energy

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

(163)

The natural scale is ${\nu }_{\Gamma }\sim {\left|{\Delta }_C\right|}_{\Gamma }{e}^{-S_{\Gamma }}$ when the adjacent central transitions are induced by a finite boundary tunnelling action $S_{\Gamma }$. The case ${\nu }_{\Gamma }=0$ corresponds to a central response blocked by an additional symmetry.

The self-adjoint central-degree family response has the form

$$\begin{array}{cc}\hfill Y_F& =a_{0}1+a_{1}Z+a_1^{*}{Z}^{†}+b_{1}S+b_1^{*}{S}^{†}+b_{2}ZS+b_2^{*}{S}^{†}{Z}^{†}\hfill \\ \hfill & +b_3{Z}^{†}S+b_3^{*}{S}^{†}Z.\hfill \end{array}$$

(164)

The adjacent response is called non-degenerate if the clock component is nonzero and at least one of the adjacent-shift coefficients $b_1,b_2,b_3$ is nonzero. The sector with all adjacent-shift coefficients equal to zero is called central-transition blocked. A model unblocked response is

$$Y_F^{model}=Z+Z^{†}+ϵ(S+S^{†}),ϵ\ne 0.$$

(165)

It contains the central phase and an adjacent shift, and therefore represents the open clock-shift sector used below.

Lemma 5 

(Generic central-degree response). Let ${\mathcal{B}}_{\mathrm{cen}}$ be a finite-dimensional space of admissible central-degree boundary traces for the locked low cluster. Assume that the projected trace maps from ${\mathcal{B}}_{\mathrm{cen}}$ to the clock component and to the adjacent-shift component in (164) are linear and not identically zero. Then the non-degenerate central-degree response is open and dense in ${\mathcal{B}}_{\mathrm{cen}}$.

Proof. 

The vanishing of the clock component is the kernel of a nonzero linear map, hence a proper closed subspace of ${\mathcal{B}}_{\mathrm{cen}}$. The vanishing of all adjacent-shift components is the kernel of the nonzero adjacent-shift projection, hence also a proper closed subspace. The complement of their union is open and dense. □

Proposition 21 

(Central response algebra). On the canonical central-sector space (159), if the central-degree boundary response is non-degenerate, then the finite algebra generated by the central phase and any nonzero adjacent-shift component is $M_3\left(\mathbb{C}\right)$.

Proof. 

By Proposition 20, the central-sector space is the regular representation of ${\mathbb{Z}}_3$. In the cyclic labeling used in (161), the operators Z and S obey the finite Heisenberg relation (162). A nonzero adjacent-shift component in (164) is S, $ZS$, or $Z^{†}S$, up to a nonzero scalar and adjoint. Since Z is present as the central phase, any of these components gives S inside the generated algebra. The matrix units are obtained explicitly as

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

(166)

Hence the generated algebra is $M_3\left(\mathbb{C}\right)$. By Lemma 5, this is the generic unblocked central-degree response. □

Let the sector-independent part of the closed fermion operator be

$$K_0=1_{{\mathcal{H}}_{\mathrm{cen}}}\otimes K_{*},\sigma \left(K_{*}\right)\subset \left\{0\right\}\cup [\Delta ,\infty ),dimker{K}_{*}=r,\Delta >0.$$

(167)

Proposition 22 

(Generic locked source simplicity). Assume that the locked source sector has scalar commutant, so that no residual symmetry protects a multiplicity in $ker{K}_{*}$. Then the case $r=1$ is open and dense among admissible self-adjoint locked source perturbations which preserve the gap in (167).

Proof. 

For a finite-dimensional isolated spectral cluster, multiplicity of the lowest eigenvalue is equivalent to the vanishing of the corresponding spectral discriminant. This is a closed algebraic condition in a local self-adjoint perturbation space. If the commutant is scalar, the degeneracy is not symmetry-protected, so the discriminant is not identically zero. Its complement is open and dense. The gap is preserved under sufficiently small bounded perturbations by Proposition 17. □

The unperturbed lowest cluster is ${\mathcal{H}}_{\mathrm{cen}}\otimes ker{K}_{*}$. A finite family response generated by the boundary algebra is an operator $Y_F=Y_F^{†}\in M_3\left(\mathbb{C}\right)$ after an auxiliary cyclic labeling of ${\mathcal{L}}_{\Gamma }$ has been chosen. If $\sigma \left(Y_F\right)\subset [0,\Delta )$ and $P_{*}$ is the projection onto $ker{K}_{*}$, then

$$K_{F,\mathrm{low}}=1_{{\mathcal{H}}_{\mathrm{cen}}}\otimes K_{*}+Y_F\otimes P_{*}.$$

(168)

It has exactly $3r$ eigenvalues below $\Delta$, counted with multiplicity:

$$rank{P}_{[0,\Delta )}\left(K_{F,\mathrm{low}}\right)=3r.$$

(169)

Thus the generic simple-source case has three lowest family modes when $r=1$.

The one-Higgs channels of Proposition 13 determine the allowed matrix blocks. Their numerical matrices are obtained by compression of the locked odd operator to the isolated family cluster. If ${\psi }_i^A$ and ${\psi }_j^B$ are orthonormal low modes in the exterior sectors A and B, then

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

(170)

Here $(A,B)$ is one of the invariant pairs in (115) and (116). The representation-theoretic support of the Yukawa operator is fixed by the exterior algebra, while the entries in (170) depend on the global branch solution, the holonomy response, the Codazzi response, and the chosen vacuum branch. Fermion masses, mixing matrices, and neutral-sector eigenvalues therefore enter the closed ARC spectral problem through (134) and (170).

The local construction is formulated on the blown-up collar of a single optical defect. A global branch is a solution of the closed ARC equations on $X_e=[M;\Gamma ]$ whose boundary data restrict to the edge-regular collar used above. The global problem is written schematically as

$$\mathcal{F}\left(u\right)=0,u=(k,Y,A,\varphi ,\chi ,f_{\zeta },{\rho }_{\zeta },\mathrm{spinorial}\mathrm{link}\mathrm{data}).$$

(171)

The map $\mathcal{F}$ contains the Rainich algebraic constraint, the Codazzi closure, the translational-current equation, the material branch balance, the spinorial boundary data with the Dirac-type normal form, and the locked spectral gap condition. The local collar model of Appendix A supplies a non-empty reference class with ${\Delta }_C\ne 0$.

The scalar multiplier obstruction is the de Rham class $\left[{\theta }_B\right]\in H_{\mathrm{dR}}^1(M\setminus \Gamma )$ from Proposition 1. In the single-defect local topology (61), it vanishes once $\mathrm{d}{\theta }_B=0$. For global ambient topology, the admissible class is restricted by the corresponding periods. The integral link class remains the $H^2$ class in (88).

Let $u_0$ be an edge-regular branch with the same projective link and with an isolated locked low cluster. The linearization

$$L=D{\mathcal{F}}_{u_0}$$

(172)

is considered on weighted edge Sobolev or Hölder spaces compatible with the blow-up. After gauge-fixing, the Codazzi part has the elliptic principal symbol of Lemma 1. The spinorial boundary component is controlled by Assumption 1, and its Dirac-type normal form is Lemma 2. With a Fredholm edge realization of L, the local completion problem is reduced to a finite-dimensional Kuranishi map

$$\kappa :kerL⟶cokerL.$$

(173)

The nearby finite-action branches are represented by ${\kappa }^{-1}\left(0\right)$, modulo the residual gauge action. Thus membership in ${\kappa }^{-1}\left(0\right)$ is the global completion condition for the local collar data used in the support theorem. This is the usual finite-dimensional reduction for nonlinear Fredholm problems. The APS framework gives the boundary comparison class for manifolds with boundary or cylindrical ends [79]; the index-theoretic background is the standard one [80].

The structural theorem below records the part of the construction which depends only on the completed collar class and is independent of the global numerical spectrum.

Definition 3 

(Structurally admissible ARC defect). A stationary finite-action optical ARC branch with a transverse-frame resolved codimension-three defect Γ is called structurally admissible if:

(i)

the blown-up collar is edge-regular and carries the spinorial boundary data of Assumption 1;

(ii)

the projected normal two-jet source is non-degenerate in the sense used in Proposition 8;

(iii)

the physical non-singlet support is energy-selected by (101) among finite filtered two-channel supports, with lower-order mixed terms in the stability range of Proposition 5;

(iv)

the locked low sector has an isolated spectral cluster, and the mirror sector satisfies (130).

It is called color-regular if the non-degenerate second-jet source lies in the color-generating stratum of Corollary 1. It is called family-admissible if, in addition, the projective color link has the central-sector torsor (158) with non-degenerate central-degree response.

Theorem 3 

(Structural link support). Every structurally admissible ARC defect has finite internal link module $E_3\oplus E_2$.

Proof. 

The optical branch and the link are supplied by Theorem 1, Definition 1, and the blown-up collar construction. The spin^c link spaces are those of (1). Proposition 4 supplies the principal coefficient-channel separation, and Proposition 5 gives the lower-order stability range. The minimal support follows from Theorem 2. □

Corollary 3 

(Exterior-algebra multiplet package). Every color-regular structurally admissible ARC defect has zero-mode basis group $S\left(U\right(3)\times U(2\left)\right)$. The even exterior algebra gives the one-generation package (113), and the allowed one-Higgs channels are those of Proposition 13. If the defect is family-admissible and the locked source is generically simple, the lowest family cluster consists of three modes.

Proof. 

The basis group is (110), with the global form recorded in (111). The exterior package is (113). The one-Higgs channels are Proposition 13. In the family-admissible case, Proposition 20 supplies the three-dimensional central-sector space, Proposition 21 gives the finite clock-shift family algebra under the non-degeneracy condition, and Proposition 22 with (169) gives the generic simple three-mode family cluster. □

Several quantities remain outside Theorem 3 and Corollary 3. The full ARC branch dynamics must determine the branch geometry, the vacuum branch, the spectral gap, the holonomy response, the Codazzi response, and the low eigenvectors entering (170). Threshold corrections, confinement scales, CKM and PMNS matrices, neutral-sector eigenvalues, and higher spectral clusters require the corresponding global spectral problem. The line-operator global form (111) is fixed at the finite-module level, while possible refinements by generalized global structure follow the usual gauge-theory classification [81].

6. Discussion

The reconstruction is organized by the equivariant normal operator of the blown-up optical defect. The branch part supplies the closed Codazzi geometry, the projective spinor link, and the edge collar. The normal operator supplies the principal coefficient-channel separation in Proposition 4. The Borel-Weil tower (1), the normal second jet, and the Toeplitz visibility rule then give the minimal support (105). The reconstructed module carries the basis group (110), the exterior package (113), the locked odd operator (132), and the family overlap matrices (170). The finite Schur test of Appendix B gives the independent scale-free reading (A22).

The role of the twistor language is local and rank-one. The optical Codazzi structure gives the shear-free projective-spinor link, and the branch worldline has the link sphere (60). Thus the twistor object used here is the projective line associated with the defect, in the sense of the standard spinor and twistor formalism [11,12]. Curved twistor constructions give a broader comparison class [43]. The present construction uses only this projective line, its spin^c quantization, and the finite normal-operator compression.

The rank-one feature is restrictive. For a general compact Kähler link one could replace (1) by the corresponding holomorphic quantization spaces. In the ARC worldline case the link is fixed by the four-dimensional optical geometry. The second normal jet has principal non-scalar types $V_1$ and $V_2$, and the ${\mathbb{CP}}^1$ Toeplitz visibility rule of Lemma 3 gives the first blocks on which these types appear. The phase-current source gives $E_2$, while the Codazzi-gap source gives $E_3$. This is the point at which the dimensions 2 and 3 enter. The Borel-Weil input is the standard rank-one case [16], and the finite Toeplitz comparison is the standard ${\mathbb{CP}}^1$ quantization [18].

The finite internal space is a boundary and link object. Closed local branch tensors are singlet-valued in the finite factor, while the non-singlet data live in the link zero-mode bundle, in its moving basis, and in the boundary response. Dynamical-principal-bundle formulations give a useful comparison class for gauge variables [68]. Recent reconstructions of internal symmetries in geometric and Kaluza-Klein-type settings give further comparison points [82,83]. In almost-commutative geometry the finite algebra is part of the spectral triple data [24], and the spectral action supplies the corresponding dynamics [25]. In the present sector the finite module is reconstructed before the exterior-algebra package is applied. The comparison with geometric matter models is therefore closest at the structural level [28].

The gauge connection is the connection on a moving zero-mode bundle. For a regular family of link data, the projected connection (122) gives the Berry-Wilczek-Zee connection of the degenerate link-zero-mode space [66,67]. Its curvature is (124), and in a local zero-mode basis it has the Yang-Mills form (125). The group acting on the basis is the compact zero-mode basis group (110); after the total top form is fixed, the curvature takes values in the Lie algebra of (111). The normalization of the curvature norm remains a branch Hessian or spectral matching datum.

The bosonic Schur Hessian uses the same finite link data in a different compression. The weak order parameter is treated as a branch projector in the W block, and the reduced even Hessian is (150). The main dimensionless output of the finite-link test is the on-shell neutral angle (A21). The determinant entry (152) is fixed by the $3+2$ support and by the unimodular top-form constraint, while the Schur unit (155) and the remaining entries in (156) depend on the frozen heavy-block normalization. The vector and radial mass values are secondary benchmarks obtained after inserting the scale v, as recorded in Appendix B.

The fermionic mass mechanism is spectral. The exterior algebra fixes the allowed finite channels, and the diagonal locking fixes the branch-helicity sector. The leading Clifford-odd Codazzi component is fixed by Proposition 15. The lower bound of Proposition 16 applies when the coefficient a is nonzero and the remainder $R_C$ is controlled. The numerical masses are obtained from the positive operator (134) after projection to an isolated low cluster. In local branch-helicity form this is the square system (146). The comparison class for defect-localized chiral modes is supplied by the Jackiw-Rebbi mechanism [71] and by domain-wall fermions [72].

The Yukawa matrices are also spectral. The representation-theoretic support of each one-Higgs channel is fixed by Proposition 13. The numerical entries are the matrix elements (170). They depend on the global branch solution, the Codazzi response, the holonomy response, and the vacuum branch. Thus the construction fixes the channel structure, while numerical fermion masses and mixing matrices remain data of the closed ARC spectral problem.

The family mechanism is global. It is a refinement of the selected color block and is not used in the proof of the minimal support (105). The closed color holonomy is projective, and the projective color link has the central-sector torsor (158). Proposition 20 gives the canonical three-dimensional central space. When the central-degree response is non-degenerate, Proposition 21 gives the full finite family algebra $M_3\left(\mathbb{C}\right)$. The generic simple-source case then gives three lowest modes by Proposition 22 and (169). The central torsor is topological, while the splitting and ordering of the three low modes are controlled by the central response operator (164). The global classification level is the standard one for principal bundles [77] and for characteristic classes [78].

The edge formulation separates the local support mechanism from the global completion problem. The local single-defect topology removes the scalar multiplier period obstruction when $\mathrm{d}{\theta }_B=0$, while preserving the integral $H^2$ class of the link. The gauge-fixed Codazzi symbol gives the elliptic core of the collar problem. The spinorial boundary data of Assumption 1 and the normal form of Lemma 2 give the spin^c link operator used in (1), up to the zeroth-order term in (73). The full finite-action branch is governed by the global equation (171) and, near an edge-regular solution, by the Kuranishi map (173). The APS framework gives the boundary comparison class for manifolds with boundary or cylindrical ends [79]; the index-theoretic background is the standard one [80].

The absence of passive visible enlargement is a consequence of the energy-selected finite second-jet support problem. Larger Borel-Weil blocks are admissible as higher-jet sectors only after additional source data or global spectral data have been supplied. Within the support energy (101), an unforced visible summand has zero active response and positive support cost, as recorded in Corollary 2. Thus the theorem fixes the active second-jet support, while non-minimal sectors belong to the global completion problem.

The structural output is the following. A structurally admissible optical ARC defect carries a rank-one projective Kähler link. The blown-up collar gives the edge boundary setting, and the spinorial link data give the Dirac-type boundary operator of Lemma 2. The equivariant normal operator gives the coefficient-channel separation used in the support Hessian. The Borel-Weil blocks used for support selection are those of (1). The normal second jet has phase-current and Codazzi-gap non-scalar source families of orders one and two. Proposition 12 then gives the minimal $3+2$ link module. The Standard Model basis group and one-generation exterior module are obtained as the exterior-algebra reading of this support. The finite Schur block gives the on-shell neutral link angle (A22). The chiral, mass, and family structures are organized by the locked branch operator and by the projective color lift. Fermionic masses, flavor mixing, threshold corrections, running electroweak couplings, and higher clusters require the global ARC spectral and matching problem. The line-operator global form (111) is fixed at the finite-module level, while possible refinements by generalized global structure follow the usual gauge-theory classification [81].