w52
plain-language theorem explainer
The raw single-entry Jacobian coefficient at tetrahedron edge indices (5,2) is exactly zero as a real. Anyone expanding the six-type stencil contraction for the assembled constant block cites this entry. The proof is a one-line wrapper: rewrite by the full rational evaluation table, then close by numeric normalization of the literal weight.
Claim. The raw Jacobian coefficient at edge indices $f=5$, $g=2$ vanishes: $\mathrm{flatAngleJacobian}(5,2)/(2\sqrt{a^*_5})=0$ in $\mathbb{R}$, equivalently the corresponding entry of the literal rational stencil-weight table is the rational $0$.
background
This module sits in the QG full-theory campaign (Paper C / Pillar 1, Lane C) and closes the hinge-aware zero-mode gate for Regge TT. The assembled constant block is built from per-tet Hessian data contracted with edge-class coefficients; the stencil-only piece is recorded separately from the hinge correction, and their relative-minus assembly is what cancels at the TT witness.
The raw Jacobian coefficient is the single-entry radical form $\mathrm{flatAngleJacobian}(f,g)/(2\sqrt{a^*_f})$, not a fiber sum. Upstream, rawJacobianCoefficient_eval proves every one of the 36 pairs equals the corresponding entry of an independent literal rational table (rationalStencilWeight), derived offline from the closed forms and kernel-checked entrywise. That table is phase-independent and never defined through the radical expression itself.
Index pair $(5,2)$ is one cell of that $6\times 6$ table. Its vanishing is a concrete arithmetic fact used when the free-coefficient contraction is expanded by hand over all tetrahedron types and slot pairs.
proof idea
One-line wrapper. Rewrite the goal by the universal evaluation theorem rawJacobianCoefficient_eval 5 2, which replaces the radical coefficient by the cast of rationalStencilWeight at bucket $\langle 5,2\rangle$. Then norm_num on that literal match closes the equality to $0$ in $\mathbb{R}$. No case split or geometric argument is needed at this site; the work was already done in the 36-entry evaluation lemma.
why it matters
Feeds the parent theorem zeroMode_free_coefficients, which expands the six-type raw-table contraction in seven free 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 algebraic 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.
In the Gate C-A3 story this is bookkeeping that makes the zero-mode headline (assembledConstantBlock_eq_zero) and the hinge-cancellation chain mechanically checkable. It does not itself invoke the Recognition forcing chain (T5–T8) or the RCL; it is local Regge-calculus arithmetic supporting the TT zero mode in the gravity analysis lane.
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