w13

This module sits in the QG full-theory campaign (Paper C / Pillar 1, Lane C) and closes the hinge-aware zero-mode gate for the assembled Regge TT constant block. The stencil is built from per-tetrahedron flat-angle Jacobians normalized by edge-length square roots on the Freudenthal tet.

The raw coefficient at a pair of edge slots $(f,g)$ is the radical-normalized single entry $J_{fg}/(2\sqrt{a^*_f})$, not a fiber sum. Independently, a full 36-entry literal rational table records the closed-form values of those entries; the pair $(1,3)$ is listed as $-1/4$. An upstream evaluation theorem identifies every raw coefficient with the corresponding table entry cast to $\mathbb{R}$.

Locally, these coefficients feed the assembled hinge/edge-diagonal $O(1)$ block and the free seven-coefficient contraction that becomes a perfect square on the alternating-sum hyperplane where geometric edge-class vectors live.

One-line wrapper. Rewrite the goal by the universal evaluation theorem that equates every raw coefficient to its rational-table entry as a real. Specialize to indices $(1,3)$, then run kernel rational arithmetic on the literal match clause for that pair, which is $-1/4$.

Gate C-A3 needs every raw table entry pinned before the assembled constant block and the free-coefficient identity can be trusted. This entry is one of the six-by-six factors inside the double sum that collapses to $(c_0+c_1+c_2-c_3-c_4-c_5+c_6)^2/2$, the algebraic shape of the zero mode off and on the constraint surface.

Downstream, the free-coefficient theorem in this module uses the entry directly. Three 4D star-kernel stationarity theorems (homothety directional derivative of the full star equals zero) also depend on the same coefficient table through the hinge-aware assembly. Together they lock the relative-minus convention that cancels the recorded stencil-only residual against the hinge block at the TT witness, so the assembled $k=0$ quadratic vanishes.