w20
plain-language theorem explainer
The raw single-entry Jacobian coefficient at edge-slot pair (2,0) is exactly zero. Anyone expanding the six-tetrahedron constant-block contraction cites this entry when clearing the free-coefficient quadratic form. The proof is a one-line rewrite onto the literal rational stencil table, then kernel arithmetic.
Claim. The raw single-entry Jacobian coefficient at slot indices $f=2$ and $g=0$ equals $0$ in $\mathbb{R}$. Equivalently, after radical-free normalization it matches the literal rational stencil weight of the bucket $\langle 2,0\rangle$, which 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 at the reported TT witness is nonzero, while the assembled block (hinge minus that residual) vanishes; the kernel pins that relative-minus convention.
The raw coefficient is the single-entry radical form flatAngleJacobian(f,g) / (2 sqrt(freudenthalTetSqEdges f)). Upstream, rawJacobianCoefficient_eval evaluates all 36 entries to exact rationals by matching an independent literal table rationalStencilWeight (phase-independent, never defined through a fiber sum). That table is the panel-demanded bare match of closed-form values, kernel-checked entry by entry.
Entry (2,0) is one of the vanishing slots in that table. The free-coefficient identity later contracts all six tetrahedron types against these coefficients.
proof idea
One-line wrapper. Rewrite the left-hand side by rawJacobianCoefficient_eval 2 0, which replaces the radical form by the cast of rationalStencilWeight ⟨2,0,…⟩. Then norm_num on that match arm yields the rational zero, hence the real zero.
why it matters
Feeds zeroMode_free_coefficients, the algebraic shape of the assembled constant block: the raw-table contraction over six tetrahedron types in seven free coefficients equals (c0+c1+c2-c3-c4-c5+c6)^2/2. That perfect square vanishes exactly on the alternating-sum hyperplane where geometric edge-class coefficient vectors live; it does not vanish for unconstrained free coefficients.
Gate C-A3 needs every table entry explicit so the free-coefficient expansion is a pure rational identity, not a residual numerical claim. This vanishing slot is one of the thirty-six atoms of that expansion. Together with the hinge-cancellation and witness-tie results in the same module, it supports the headline that the Schlaefli-reduced assembled constant block is the zero mode at the reported TT witness.
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