w24
plain-language theorem explainer
The raw Jacobian coefficient at edge-pair slots (2,4) equals the rational -1/8. Anyone assembling the Regge TT constant block or checking the free-coefficient perfect-square identity cites this table entry. The proof is a one-line rewrite through the full 36-entry evaluation lemma, then kernel rational arithmetic on the literal stencil-weight table.
Claim. The single-entry radical coefficient of the flat-angle Jacobian at tetrahedron edge indices $f=2$, $g=4$, normalized by twice the square root of the corresponding Freudenthal squared edge length, equals $-1/8$.
background
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 per-slot raw Jacobian coefficients
$$J_{fg}/(2\sqrt{a^*_f}),$$
where $J_{fg}$ is the flat-angle Jacobian and $a^*_f$ is the Freudenthal squared edge length on a regular tetrahedron.
Upstream, rawJacobianCoefficient_eval asserts that every one of the 36 slot pairs evaluates to an exact rational drawn from an independent literal table rationalStencilWeight (phase-independent, never defined through a fiber sum). That table was derived offline from the closed forms and kernel-checked entry by entry. The present lemma is the concrete $(2,4)$ cell of that table.
proof idea
One-line wrapper. Rewrite by the universal evaluation lemma rawJacobianCoefficient_eval 2 4, which replaces the radical expression by the corresponding rational stencil weight cast to $\mathbb{R}$. Then norm_num on the literal match-table ReggeTTBucketAggregation.rationalStencilWeight discharges the equality to $-1/8$.
why it matters
Feeds zeroMode_free_coefficients, the algebraic identity that the six-tetrahedron contraction of the raw table over seven free edge-class coefficients is exactly 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 on the alternating-sum hyperplane inhabited by every geometric edge-class coefficient vector, and nowhere else. Pinning each rational cell (here $(2,4)$) is therefore mandatory bookkeeping for the zero-mode headline assembledConstantBlock_eq_zero and for the witness-tie that the hinge cancels the recorded stencil-only residual. In the broader RS gravity lane this is pure Gate C-A3 scaffolding closure, not a T0–T8 forcing step.
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