e_313230
plain-language theorem explainer
Pointwise identity: the folded coupling numerator at multi-index (3,1,3,2,3,0) equals eight times the explicit integer kernel at that same point. Gravity analysts cite it as one of 256 concrete decides that assemble into the universal m2Num = 8·explicitZ statement. Proof is a single kernel decision on fixed Fin-4 indices.
Claim. For the multi-index $(a,b,c,d,i,j)=(3,1,3,2,3,0)$ with each coordinate in $\{0,1,2,3\}$, the summed coupling numerator equals eight times the explicit kernel value: $N(3,1,3,2,3,0)=8\,Z(3,1,3,2,3,0)$.
background
This module is chunk 13 of a 256-case kernel certification that the 4D Regge midpoint numerator equals eight times a closed-form integer table. The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing a local contribution at each tuple of Fin-4 indices. The explicit kernel $Z$ is a pattern-matched integer table on the same six Fin-4 arguments (sample entries include $4$, $-2$, and so on).
The ambient goal is an exact algebraic identity in the discrete gravity analysis: after all contributions are summed, $N=8Z$ holds at every multi-index. Upstream definitions supply only the raw table and the fold; no analytic closed form is assumed beyond the case table.
proof idea
One-line computational proof: decide evaluates both sides at the concrete indices $(3,1,3,2,3,0)$ and checks integer equality. No lemmas are invoked; the kernel reduces the fold and the pattern match to numerals and compares them.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ and is proved by exhaustive fin_cases on all six indices, each branch landing on one of these chunk decides. Without the full 256-point cover, the universal midpoint identity in the 4D Regge analysis does not close. The factor eight is the structural constant being certified case-by-case; the chunking exists only to keep each decide inside the kernel's budget.
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