orth_kernel_zero_off

This module builds the full periodic-lattice star deficit class kernel for a seed triangle hinge in 4D Regge calculus. Scope is the seed hinge ${0,e_0,e_0+e_1}$ and its Freudenthal star: four containing unit cubes and six incident 4-simplices. The campaign is kernel-checked only; flat Hessian assembly over all hinges and $S_{\mathrm{RS}}\to\mathrm{EH}_{4d}$ remain open.

The orthogonal deficit kernel is the piecewise map $\mathrm{Fin},10\to\mathbb{R}$ that sends indices $1,3,7,9$ to $-1/2,+1/2,+1/2,-1/2$ respectively and every other index to $0$. Those four support indices are the orthogonal flat-square edges of the star stencil. Off-support vanishing is the algebraic fact needed when star-member assemblers sum kernel values over the fifteen edge classes.

One-line wrapper: case-split on the ten values of $e:\mathrm{Fin},10$. On each branch, either the goal is definitional reflexivity (the catch-all clause of the kernel returns $0$) or the branch contradicts one of the four hypotheses $e\neq 1,3,7,9$. No external lemmas are invoked beyond the kernel definition itself.

Feeds the private evaluators for star members 2 and 3, which expand assembleStarMember into explicit four-term sums over classes ${2,6,10,14}$ with coefficients $\pm 1/2$. Those evaluators are steps toward deliverable A of the module: the full-star deficit class kernel on classes $(2,3,6,7,10,11,14)$ with values $(-1,-1,+1,-1,+1,+1,-1)$, plus the nonvacuity, swap-symmetry, scaling-decoy, and homothety-stationarity gates.

In the QG campaign this is the next kernel-checked increment after the two-simplex dihedral cosine calculus. It does not close gap-action recovery or reverse-engineer Einstein–Hilbert weights; it only certifies that the orthogonal channel is sparse on the ten-edge stencil.