w34

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 Regge TT Bloch symbol. The assembled constant block mixes a hinge contribution with a stencil-only residual; the stencil side is built from raw Jacobian coefficients of the flat-angle map on a Freudenthal tetrahedron.

Each raw coefficient is the radical-normalized entry $J_{fg}/(2\sqrt{a^_f})$, where $J$ is the flat-angle Jacobian and $a^$ are squared edge lengths. Independently, a literal rational table (phase-free, not defined via any fiber sum) records the closed-form value of every one of the 36 slot pairs. An upstream evaluation theorem identifies every raw coefficient with the corresponding table entry cast to $\mathbb{R}$.

The present fact is the concrete (3,4) cell of that identification, needed when the six tetrahedron types are contracted against free edge-class coefficients.

One-line wrapper. Rewrite the goal by the universal evaluation lemma rawJacobianCoefficient_eval at arguments 3 and 4, which replaces the radical form by the rational stencil table entry. Then norm_num on that table entry discharges the equality to $-1/4$.

Feeds the free-coefficient identity zeroMode_free_coefficients, which expands the six-type raw-table contraction into the perfect square $(c_0+c_1+c_2-c_3-c_4-c_5+c_6)^2/2$. That algebraic shape is the exact form of the assembled constant block: it vanishes precisely on the alternating-sum hyperplane inhabited by geometric edge-class coefficient vectors, and is the algebraic engine behind the zero-mode headline that the Schlaefli-reduced assembled constant block is zero at the reported TT witness.

In the Gate C-A3 story this is one of the 36 table cells that must be pinned before the hinge residual can cancel the stencil-only residual under the kernel-fixed relative-minus assembly convention. It is pure gravity-analysis bookkeeping inside the Regge TT campaign, not a forcing-chain (T0–T8) step.