w01
plain-language theorem explainer
The raw flat-angle Jacobian coefficient at edge-pair indices (0,1) is exactly zero as a real. Anyone expanding the six-tetrahedron free-coefficient contraction for the assembled constant block needs this slot. Proof is a one-line rewrite through the full rational evaluation table, then numeric normalization of the (0,1) entry.
Claim. The raw Jacobian coefficient at edge-pair indices $0$ and $1$ equals $0$ in $\mathbb{R}$. Equivalently, after radical-free normalization of the flat-angle Jacobian against the Freudenthal squared edge lengths, the $(0,1)$ stencil weight is the rational $0$.
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 sympy diagnostic found that the stencil-only constant residual is nonzero at the reported TT witness, while the assembled block (hinge minus that residual) vanishes; the kernel pins that relative-minus convention.
The raw Jacobian coefficient of a pair of edge slots $(f,g)\in\mathrm{Fin},6$ is the single-entry radical form $\mathrm{flatAngleJacobian}(f,g)/(2\sqrt{a^*_f})$, used as the early bucket falsifier input. Independently, the full literal rational stencil table records all 36 slot-pair weights as bare rationals (phase-independent, never defined through a fiber sum). The evaluation theorem states that every raw coefficient equals the corresponding table entry cast to $\mathbb{R}$.
In that table the $(0,1)$ entry is $0$. The present fact is the corresponding real equality for that single slot.
proof idea
One-line wrapper. Rewrite the goal by the full evaluation theorem at indices $(0,1)$, which replaces the raw coefficient by the rational stencil weight of bucket $\langle 0,1\rangle$ cast to $\mathbb{R}$. Then norm_num on that table definition discharges the equality, since the match arm for $(0,1)$ is the literal rational $0$.
why it matters
Feeds the free-coefficient identity for the assembled constant block: the six-tetrahedron double sum of raw coefficients against seven free edge-class coefficients collapses to 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 zero mode: the contraction does not vanish for free coefficients, but vanishes on the alternating-sum hyperplane where every geometric edge-class vector lives.
Together with the sibling slot equalities, this closes the stencil half of Gate C-A3 (hinge-aware zero mode). The module headline is that the Schlaefli-reduced assembled constant block is zero at the TT witness, matching the diagnostic report that hinge cancels the recorded stencil-only residual. No Recognition forcing-chain landmark (T5–T8, RCL, phi) is directly at stake; the stake is the Regge TT discrete gravity zero mode in the QG finishing charter.
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