w33
plain-language theorem explainer
The raw flat-angle Jacobian coefficient at edge-slot pair (3,3) equals the rational 1/4. Anyone expanding the assembled constant block over free edge-class coefficients cites this entry. The proof is a one-line wrapper: rewrite by the full 36-entry evaluation theorem, then close by norm_num on the literal rational stencil table.
Claim. The single-entry radical coefficient of the flat-angle Jacobian at indices $f=3$, $g=3$ equals $1/4$ in $\mathbb{R}$: $$J_{33}/(2\sqrt{a^*_3})=1/4.$$
background
This module sits in the QG full-theory campaign (Paper C / Pillar 1, Lane C): it closes the hinge-aware zero-mode gate for the Regge TT Bloch symbol. The assembled O(1) block is the Schlaefli-reduced cell stencil plus a hinge correction; the stencil-only residual at the reported TT witness is nonzero, and the hinge is proved to cancel it under the relative-minus assembly convention.
rawJacobianCoefficient f g is the single-entry radical form flatAngleJacobian f g / (2 * sqrt(freudenthalTetSqEdges f)). It is not a fiber sum. Upstream, rawJacobianCoefficient_eval identifies every one of the 36 pairs with a phase-independent literal rational from rationalStencilWeight, the full match-table demanded by the panel (derived offline from the closed forms, then kernel-checked entrywise).
The present lemma is the (3,3) specialization of that table. Downstream free-coefficient algebra contracts these rationals against seven displacement-class coefficients.
proof idea
One-line wrapper. Rewrite with rawJacobianCoefficient_eval 3 3, which replaces the radical coefficient by the cast of rationalStencilWeight at bucket ⟨3,3,…⟩. Then norm_num on ReggeTTBucketAggregation.rationalStencilWeight evaluates the match arm to the rational 1/4 and finishes in ℝ.
why it matters
Feeds zeroMode_free_coefficients, the algebraic identity that the six-tetrahedron raw-table contraction in seven free coefficients is exactly the perfect square (c0+c1+c2-c3-c4-c5+c6)^2/2. That identity is the exact shape of the assembled constant block: it does not vanish off the constraint surface, and vanishes precisely on the alternating-sum hyperplane inhabited by geometric edge-class vectors.
In the Gate C-A3 story this is bookkeeping for the zero-mode headline: once the free expansion is a square and the TT witness lies on the kernel hyperplane, the assembled k=0 block is zero. The entry is private scaffolding inside that expansion, not a standalone physics claim, but without the rational table pinned entry-by-entry the free-coefficient square cannot be certified in-kernel.
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