w41
plain-language theorem explainer
The raw Jacobian coefficient at edge-pair indices (4,1) equals the rational 1/4. Gravity analysts closing the Regge TT hinge-aware zero-mode gate cite this as one of the thirty-six table entries that feed the free-coefficient contraction of the assembled constant block. The 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 tetrahedron edge indices $f=4$ and $g=1$, divided by twice the square root of the corresponding Freudenthal squared edge length, equals $1/4$ in $\mathbb{R}$.
background
This module sits in the QG full-theory campaign (Paper C / Pillar 1, Lane C) and closes the hinge-aware zero-mode gate for Regge TT. The assembled constant block combines a hinge contribution with the stencil-only residual; the stencil side is built from per-edge raw Jacobian coefficients contracted against edge-class polarization weights.
The raw coefficient at a pair of edge indices $(f,g)$ is the flat-angle Jacobian entry divided by twice the square root of the Freudenthal squared edge length. Independently, a full 36-entry literal rational table records the radical-free normal form of every such coefficient. Upstream, the evaluation theorem states that each raw coefficient equals the corresponding table entry cast to $\mathbb{R}$.
Only the assembled $k=0$ block is under proof here; the sympy diagnostic residual and hinge cancellation supply context, not hypotheses.
proof idea
One-line wrapper. Rewrite the goal by the upstream evaluation theorem at indices $(4,1)$, which replaces the raw coefficient by the rational stencil weight of bucket $\langle 4,1\rangle$ cast to $\mathbb{R}$. Then norm_num on that table definition discharges the numeric equality $1/4$.
why it matters
Feeds the parent theorem on zero-mode free coefficients, which expands the six-tetrahedron double sum of raw coefficients against seven free edge-class weights and identifies the contraction as 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.
In the Gate C-A3 story, the stencil table entries are the concrete numbers behind the Schlaefli-reduced assembled block whose vanishing is the zero-mode headline. Pinning entry $(4,1)$ is routine bookkeeping, but every free-coefficient expansion step depends on having each of the thirty-six values kernel-checked. No Recognition forcing-chain landmark (T5–T8, RCL, phi) is directly at stake; the stake is the Regge TT continuum limit residual cancellation.
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