w44

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 side is built from single-entry radical coefficients

$$J_{fg}/(2\sqrt{a^*_f}),$$

the raw Jacobian coefficients on the six Freudenthal edge slots of a tetrahedron.

Upstream, rawJacobianCoefficient_eval proves every one of the 36 entries equals a phase-independent literal rational from the full stencil-weight table (derived offline from closed forms of the flat-angle Jacobian and squared edge lengths, then kernel-checked entrywise). The table is independent of any fiber sum: pair $(4,4)$ is recorded as the rational $1/8$. The present lemma simply names that one diagonal entry for downstream contraction.

One-line wrapper. Rewrite by the universal evaluation lemma rawJacobianCoefficient_eval 4 4, which replaces the radical form by the cast of rationalStencilWeight at bucket $\langle 4,4\rangle$. Then norm_num on that literal match arm yields $1/8$ in $\mathbb{R}$.

Feeds zeroMode_free_coefficients, the algebraic identity that contracts the full raw table over the six tetrahedron types in seven free edge-class coefficients and obtains 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 does not vanish for free coefficients, but vanishes on the alternating-sum hyperplane where every geometric edge-class coefficient vector lives. Pinning the $(4,4)$ diagonal entry is one of the thirty-six rational facts that make the free-coefficient expansion a pure square, and thereby underwrite the zero-mode headline that the Schlaefli-reduced assembled constant block vanishes on TT witnesses. Local to Gate C-A3 of the Regge TT finishing charter; not a T0–T8 forcing step.