w23
plain-language theorem explainer
The single raw Jacobian stencil entry at edge-pair indices (2,3) equals the rational 1/4. Anyone expanding the assembled constant block or the free-coefficient zero-mode identity cites this as a table lookup. The proof is a one-line wrapper: evaluate via the closed-form rational table, then discharge by kernel arithmetic.
Claim. The raw single-entry Jacobian coefficient at tetrahedron edge indices $f=2$ and $g=3$ equals $1/4$ in $\mathbb{R}$: $J_{\mathrm{raw}}(2,3)=1/4$.
background
This module sits in the QG full-theory campaign (Paper C / Pillar 1, Lane C), closing the hinge-aware zero-mode gate for Regge TT. The assembled O(1) block combines a hinge contribution with the stencil-only constant residual; the stencil side is built from raw Jacobian coefficients of the flat-angle map on a Freudenthal tetrahedron.
Each raw coefficient is the radical-normalized entry $J_{fg}/(2\sqrt{a^f})$, where $J{fg}$ is the flat-angle Jacobian and $a^$ are squared edge lengths. Upstream, rawJacobianCoefficient_eval proves every one of the 36 pairs equals a literal rational from an independent match table rationalStencilWeight (phase-independent, never defined through a fiber sum). The (2,3) slot of that table is the rational $1/4$.
Only the assembled block (hinge minus stencil residual) is claimed to vanish on TT; individual raw entries need not be zero and are used as algebraic building blocks.
proof idea
One-line wrapper. Rewrite by the universal evaluation theorem rawJacobianCoefficient_eval at $(f,g)=(2,3)$, which replaces the radical form by the cast of rationalStencilWeight on the corresponding bucket. Then norm_num on that match table discharges the equality to $1/4$ by kernel rational arithmetic. No case split beyond what the evaluation lemma already performed.
why it matters
Feeds the free-coefficient identity zeroMode_free_coefficients, which contracts the full $6\times 6$ raw table over six tetrahedron types and seven displacement-class coefficients and obtains the perfect square $(c_0+c_1+c_2-c_3-c_4-c_5+c_6)^2/2$. That algebraic shape is the exact form of the assembled constant block: it vanishes precisely on the alternating-sum hyperplane inhabited by geometric edge-class vectors, which is the zero-mode headline of Gate C-A3.
In the broader finishing charter this is bookkeeping, not a new physical claim: each rational table entry must be pinned so the contraction can be expanded without floating-point or sympy residue. It supports the witness-tie story (hinge cancels the recorded stencil residual under the relative-minus assembly convention) by making the stencil side fully kernel-rational.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.