w35

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 stencil-only constant block does not vanish under TT; the hinge term cancels the recorded residual, and the assembled block is fixed as hinge minus that residual.

The raw Jacobian coefficient is the single-entry radical form $J_{fg}/(2\sqrt{a^*_f})$ used by early bucket checks. Independently, the full literal rational stencil table assigns a phase-independent rational to every slot pair $(f,g)\in\mathrm{Fin},6\times\mathrm{Fin},6$, derived offline from closed forms of the flat-angle Jacobian and Freudenthal squared edge lengths. The evaluation theorem identifies every raw coefficient with the corresponding table entry as a real.

One-line wrapper. Rewrite the goal by the evaluation theorem at indices $(3,5)$, which replaces the raw radical coefficient by the cast of the literal rational stencil weight at bucket $\langle 3,5\rangle$. Then norm_num on that table entry reduces the rational (which is $0$) to the real zero.

Feeds zeroMode_free_coefficients, the algebraic identity that the six-tetrahedron raw-table contraction in seven free coefficients is the perfect square $(c_0+c_1+c_2-c_3-c_4-c_5+c_6)^2/2$. That identity is the exact shape of the assembled constant block: it vanishes precisely on the alternating-sum hyperplane inhabited by geometric edge-class coefficient vectors.

In the Gate C-A3 story this is one of the thirty-six table atoms needed before the free-coefficient square and the TT-witness cancellation can be stated in-kernel. It does not itself invoke the hinge term or the eight-tick/D=3 forcing chain; it is pure stencil arithmetic supporting the zero-mode headline.