w42
plain-language theorem explainer
The normalized flat-angle Jacobian entry at Freudenthal edge indices (4,2) equals the rational −1/8. Anyone expanding the six-tetrahedron raw-table contraction for the Regge TT constant block cites this slot. Proof is a one-line rewrite through the 36-entry evaluation theorem plus kernel rational arithmetic on the literal stencil table.
Claim. For Freudenthal tetrahedron edge indices $f=4$ and $g=2$, the raw Jacobian coefficient $\mathrm{flatAngleJacobian}(4,2)/(2\sqrt{a^*_4})$ equals $-1/8$ as a real number.
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 stencil-only residual at the reported TT witness does not vanish; the hinge term cancels it under the relative-minus assembly convention, so the assembled $k=0$ block is zero on TT.
The raw Jacobian coefficient is the single-entry radical form $\mathrm{flatAngleJacobian}(f,g)/(2\sqrt{a^_f})$, with $a^$ the Freudenthal squared edge lengths. Upstream, rawJacobianCoefficient_eval proves every one of the 36 pairs equals the corresponding entry of an independent literal rational table rationalStencilWeight (phase-independent match, never defined via fiber sums). That table records the $(4,2)$ slot as $-1/8$.
These rationals feed the free-coefficient contraction over six tetrahedron types and seven displacement classes that shapes the assembled constant block.
proof idea
One-line wrapper. Rewrite by the universal evaluation theorem rawJacobianCoefficient_eval at indices $(4,2)$, which replaces the radical form by the cast of rationalStencilWeight on the bucket $\langle 4,2\rangle$. Then norm_num discharges the equality against the literal match clause $-1/8$ in the full rational stencil table.
why it matters
Feeds zeroMode_free_coefficients, the algebraic identity that the raw-table double sum over six tet types equals $(c_0+c_1+c_2-c_3-c_4-c_5+c_6)^2/2$. That perfect-square shape is the exact form 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 slot evaluations, this entry lets the free-coefficient theorem expand by pure rational arithmetic, closing the stencil half of the hinge-aware zero-mode gate. The module then ties hinge cancellation to the recorded stencil residual and proves the assembled constant block is zero at the TT witness (Gate C-A3 headline). No T0–T8 forcing step is touched; this is discrete gravity / Regge calculus infrastructure inside the RS gravity lane.
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