w54
plain-language theorem explainer
The normalized flat-angle Jacobian entry at tetrahedron edge-slot pair (5,4) is exactly zero. Anyone expanding the six-type raw-table contraction for the assembled constant block needs this table cell. Proof is a one-line rewrite through the full 36-entry rational evaluation lemma, then numeric normalization of the independent stencil-weight table.
Claim. The raw Jacobian coefficient at edge indices $f=5$, $g=4$, namely $\mathrm{flatAngleJacobian}(5,4)/(2\sqrt{a^*_5})$, equals $0$ in $\mathbb{R}$.
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 quadratic vanishes once the hinge term is included with a relative minus sign.
The raw Jacobian coefficient is the single-entry radical form $\mathrm{flatAngleJacobian}(f,g)/(2\sqrt{a^*_f})$ used by early bucket checks. Upstream, rawJacobianCoefficient_eval states that all 36 such entries equal the corresponding cell of an independent literal rational table (rationalStencilWeight), derived offline from closed forms and kernel-verified entry by entry. That table is phase-independent and never defined through a fiber sum.
The present fact is one cell of that verified table: the (5,4) entry is the rational 0.
proof idea
One-line wrapper. Rewrite by the universal evaluation lemma rawJacobianCoefficient_eval at arguments 5 and 4, which replaces the radical coefficient by the cast of rationalStencilWeight on the bucket $\langle 5,4\rangle$. Then norm_num discharges the match in the full 36-entry rational stencil table, yielding $0:\mathbb{R}$.
why it matters
Feeds zeroMode_free_coefficients, the algebraic identity that the raw-table contraction over the six tetrahedron types, in seven free edge-class coefficients, equals the perfect square $(c_0+c_1+c_2-c_3-c_4-c_5+c_6)^2/2$. That identity is the exact shape 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 hinge-cancellation and witness-tie results in this module, the zero cells of the rational table underwrite the headline assembledConstantBlock_eq_zero (Gate C-A3). In the broader RS gravity lane this is discrete TT kinematics on the Freudenthal tetrahedron, not a continuum GR claim; it sits downstream of the forcing chain only insofar as the discrete geometry inherits the eight-tick / $D=3$ scaffolding.
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