e_311020
plain-language theorem explainer
For the six Fin-4 indices (3,1,1,0,2,0), the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one atomic cell in the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=3$, $b=1$, $c=1$, $d=0$, $i=2$, $j=0$ in $\mathrm{Fin}\,4$, the summed coupling numerator equals eight times the explicit integer kernel: $N(3,1,1,0,2,0)=8\,Z(3,1,1,0,2,0)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six $\mathrm{Fin},4$ indices are compared. The numerator $N=m2Num$ is obtained by folding a fixed coupling list and accumulating local contributions at each multi-index. The comparison target $Z=explicitZ$ is a sparse closed-form table of small integers (entries such as $4$, $-2$, and defaults).
The module is chunk 13 of a 256-cell decide grid that discharges $N=8Z$ pointwise. Each cell fixes one concrete sextuple of indices so the equality becomes a pure integer computation. Upstream, $m2Num$ and $explicitZ$ are defined in the kernel certificate module; this chunk only evaluates them at one point.
proof idea
One-line decide proof. Both sides reduce to concrete Int values once the six indices are substituted into the fold definition of $m2Num$ and the pattern-match table of $explicitZ$; the kernel checks equality of those integers.
why it matters
Feeds the assembler theorem $m2Num_eq_eight_explicitZ$, which universalizes $N=8Z$ over all six $\mathrm{Fin},4$ indices by exhaustive fin_cases and invokes each chunk cell (including this one) as a leaf. That global identity is part of the exact midpoint M2TT certificate in the gravity analysis stack: it confirms the numerator of the discrete curvature coupling matches eight times the tabulated kernel, a bookkeeping step toward the Regge-calculus side of Recognition gravity rather than a forcing-chain (T0–T8) landmark.
Without the pointwise cells, the assembler cannot close; each decide is therefore load-bearing scaffolding for the 4D identity, not an independent physical claim.
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