w32
plain-language theorem explainer
The raw Jacobian coefficient at edge-pair indices (3,2) equals the rational 1/4. Zero-mode algebra in the Regge TT hinge-aware gate cites this as one of the thirty-six table entries that contract into the free-coefficient perfect square. Proof is a one-line rewrite through the evaluated rational stencil table, then numeric normalization.
Claim. The single-entry radical coefficient of the flat-angle Jacobian at edge indices $f=3$ and $g=2$, divided by twice the square root of the Freudenthal squared edge length at $f$, equals $1/4$ as a real number.
background
In the Regge TT symbol program, each raw Jacobian coefficient for a pair of edge slots $(f,g)$ is the flat-angle Jacobian entry divided by twice the square root of the Freudenthal squared edge length. Independently of any fiber sum, the full 36-entry table of these coefficients reduces to exact rationals, recorded as a bare match of literal rational stencil weights.
This module is Gate C-A3 of the QG full-theory campaign (Paper C / Pillar 1, Lane C). It closes the hinge-aware zero-mode gate for the assembled constant block: the stencil-only residual at the reported TT witness is cancelled by the hinge term under the relative-minus assembly convention.
Upstream, the evaluation theorem states that every raw coefficient equals the corresponding rational stencil weight cast to $\mathbb{R}$, each entry obtained by radical-free normal form and kernel rational arithmetic on the flat integer tuple.
proof idea
One-line wrapper. Rewrite the goal by the evaluation theorem at indices $3$ and $2$, which replaces the raw coefficient by the rational stencil weight of bucket $\langle 3,2\rangle$ cast to $\mathbb{R}$. Then norm_num on that stencil-weight match discharges the equality to $1/4$.
why it matters
Feeds the free-coefficient identity: the raw-table contraction over the six tetrahedron types, in seven free coefficients, equals 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, and vanishes exactly on the alternating-sum hyperplane where every geometric edge-class coefficient vector lives.
Together with the sibling entry lemmas, this supplies the concrete rationals needed to expand and simplify that double sum. The parent result is a step toward the zero-mode headline of Gate C-A3 (assembled constant block vanishes under TT), which pins the hinge-minus-stencil assembly convention recorded by the sympy diagnostic.
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