fullStarClassKernel
plain-language theorem explainer
Closed-form full-star deficit class kernel for the type-(2,2) Regge hinge on the 4D periodic lattice: a 15-slot stencil of coefficients in {+1,-1}. Gravity analysts cite it when assembling Bloch/orbit quadratics and local incidence sums for the (2,2) Freudenthal star. The body is a pure pattern match on Fin 15; no proof obligations.
Claim. The full-star deficit class kernel for the type-$(2,2)$ hinge is the map $K_{22}:\{0,\ldots,14\}\to\mathbb{R}$ with values $(1,1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1)$ in stencil-class order. It is the coefficient vector of the linearized deficit of the full periodic Freudenthal star of the triangle hinge with vertex masks $\{0,3,15\}$ (difference masks $(3,12)$).
background
This module sits in the QG full-theory campaign for 4D Regge calculus on the periodic Freudenthal triangulation. After the type-(1,1) seed orbit kernel, it treats the type-(2,2) triangle hinge ${0, e_0+e_1, e_0+e_1+e_2+e_3}$ (masks ${0,3,15}$, difference masks $(3,12)$) and its full star of incident Kuhn simplices.
The ambient API is fixed upstream: the 15-class edge stencil, Freudenthal incidence, and Gram-projection cosine calculus. Sibling material enumerates the star (exactly four cube-translate/simplex pairs contain the hinge) and proves flat cosines all vanish, so the star angle sum is exactly $2\pi$.
Parallel kernels exist for other orbits: type (1,1) uses a sparse $\pm 1$ pattern; types (1,2) and (1,3) carry irrational entries ($\sqrt{2}/2$, $\sqrt{3}$). The present map is the (2,2) closed form on the same 15 classes.
proof idea
No proof: this is a definition by exhaustive pattern match on Fin 15. Each class index is assigned a constant in ${+1,-1}$. The values are the committed closed form of the full-star linearized deficit for the (2,2) orbit, recorded after the star enumeration and flatness gates in the same module. Downstream lemmas treat it as a fixed coefficient vector in class-dot products and orbit means.
why it matters
Delivers item 4 of the module's deliverable A: the full-star deficit class kernel for the (2,2) hinge, the next kernel-checked increment after the (1,1) seed. Parent consumers in the Bloch layer (factorized slot terms, transported deficit reindexing, orbit-mean local kernels, star-assembly identities) multiply or sum against such kernels to build zero-momentum quadratics and phased class dots.
In the Recognition gravity stack this is discrete curvature data on the eight-tick / $D=3$ lattice geometry, feeding later Hessian and continuum-limit arguments. It does not by itself close flat Hessian assembly over all hinges, prove $S_{\mathrm{RS}}$ converges to Einstein–Hilbert in 4D, or flip gap_action_recovery. Other hinge orbits remain previously closed or open.
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