e_333003
plain-language theorem explainer
For the six-index tuple (3,3,3,0,0,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one of the 256 kernel point-checks that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=b=c=3$, $d=0$, $i=0$, $j=3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(3,3,3,0,0,3)=8\,Z(3,3,3,0,0,3)$.
background
In the Regge exact-midpoint 4D analysis, the numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution term evaluated at six Fin-4 indices. The companion map $Z$ (explicitZ) is a piecewise integer table on those same six indices, recording the closed-form kernel values (e.g. 4, −2, and related patterns on the listed patterns).
This module is chunk 15 of the 256-point kernel certification that $m_2^{\mathrm{num}}=8,Z$ holds at every index sextuple. The local setting is pure finite enumeration over $\mathrm{Fin},4^6$, not continuum gravity: each chunk discharges a block of concrete equalities so the assembler can recombine them.
proof idea
One-line kernel proof: decide evaluates both sides at the concrete sextuple $(3,3,3,0,0,3)$. The left side runs the fold that defines the numerator; the right side looks up (or computes) $8$ times the explicit integer table entry. No lemmas beyond decidable integer arithmetic are required.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8,Z$ by exhaustive fin_cases on all six indices. Each chunk such as this one supplies a decided equality the case split can invoke, closing the numerator–kernel identity used in the Regge midpoint $M_2$ TT analysis. Within Recognition gravity, that identity is bookkeeping infrastructure for the discrete curvature/coupling side, not a forcing-chain landmark (T5–T8) by itself.
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