w15

This module closes Gate C-A3 of the QG full-theory campaign (Paper C / Pillar 1, Lane C): the hinge-aware zero mode of the assembled Regge TT constant block. The sympy diagnostic found that the stencil-only constant residual at the reported TT witness is nonzero, while the assembled block (hinge minus that residual) vanishes; the kernel pins that relative-minus convention.

The raw Jacobian coefficient is the single-entry radical form $J_{fg}/(2\sqrt{a^*_f})$ used in early bucket checks. Upstream, the full 36-entry evaluation theorem states every such coefficient equals the corresponding entry of an independent literal rational table (phase-independent match, never defined through a fiber sum). That table records the $(1,5)$ slot as $-1/8$.

Sibling private lemmas pin the other nonzero table entries needed for the free-coefficient contraction.

One-line wrapper. Rewrite by the universal evaluation theorem that equates every raw Jacobian coefficient to the cast of the literal rational stencil weight at the corresponding bucket. Then norm_num against that weight table discharges the concrete equality to $-1/8$.

Feeds the free-coefficient identity for the zero mode: the raw-table contraction over the six tetrahedron types equals $(c_0+c_1+c_2-c_3-c_4-c_5+c_6)^2/2$. That perfect-square shape is the algebraic skeleton of the assembled constant block; it vanishes exactly on the alternating-sum hyperplane where geometric edge-class coefficient vectors live.

Without each rational entry pinned (including this $(1,5)$ cell), the seven-coefficient expansion cannot be simplified in-kernel. The entry is scaffolding for the headline zero-mode theorem of Gate C-A3, not a physics claim on its own. It sits inside the Regge TT analysis that supports the discrete gravity side of the Recognition campaign, downstream of the forcing chain only indirectly via the broader QG finishing charter.