fullStarClassKernel
plain-language theorem explainer
The full-star deficit class kernel is the integer weight table on the 15 edge classes of the Freudenthal stencil: nonzero only on classes 2,3,6,7,10,11,14 with values (-1,-1,+1,-1,+1,+1,-1). Anyone building the (1,1)-orbit Regge deficit, Bloch fold, or M2 symbol on the periodic 4D lattice cites this table. It is a pure case-split definition encoding the committed star kernel for the seed hinge.
Claim. Define the full-star class kernel $K:\{0,\ldots,14\}\to\mathbb{R}$ by $K(2)=K(3)=K(7)=K(14)=-1$, $K(6)=K(10)=K(11)=+1$, and $K(d)=0$ for every other class index $d$.
background
In 4D Regge calculus on the integer lattice, each triangular hinge carries a deficit assembled from dihedral angles of the incident 4-simplices. This module fixes the seed hinge ${0,e_0,e_0+e_1}$ and its full periodic Freudenthal star: four containing unit cubes and six incident 4-simplices. Edge perturbations are organized into a 15-class stencil under lattice symmetry, so a linearized deficit is a vector in class space.
Sibling modules supply analogous kernels for other hinge orbits, with irrational entries built from $\sqrt{2}$ and $\sqrt{3}$. The present table is the integer-valued kernel for the $(1,1)$ orbit. The module deliverable states the full-star deficit class kernel on classes $(2,3,6,7,10,11,14)$ with values $(-1,-1,+1,-1,+1,+1,-1)$, together with flatness (star angle sum exactly $2\pi$) and algebraic gates (nonvacuity, swap symmetry, scaling decoys, homothety stationarity).
proof idea
Pure definition by cases on $\mathrm{Fin},15$. Seven nonzero entries are hard-coded as $\pm 1$; the other eight classes map to $0$. No lemmas fire at the definition site. Equality of this table to the sum of the six star-member contributions is proved separately (via the assembled-kernel identity and the member-sum theorem used downstream).
why it matters
This kernel is the seed deficit weight vector for the $(1,1)$ hinge orbit in the Bloch-fold and M2-symbol pipeline. Downstream, transported deficit reindexes it across the 24 lattice orientations; factorized slot terms multiply area and deficit phased dots against it; the local orbit-mean kernel for type $(1,1)$ is exactly one-sixth of it; and the star-assembly identity equates the sum over six star members to these values. Zero-momentum consistency of the factorized Bloch fold recovers the committed $(1,1)$ quadratic from the same table.
It is deliverable A.4 of the Regge 4D full-star campaign, the kernel-checked step after the dihedral-cosine layer. Scope remains the seed hinge orbit only: other lattice hinge orbits stay open, and this does not finish flat Hessian assembly, continuum Einstein-Hilbert recovery, or the gap-action flag.
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