e_131102
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at indices (1,3,1,1,0,2) equals eight times the explicit integer table entry. Gravity analysts cite it as one cell of the 4D midpoint M2 kernel certification. The proof is a single kernel decide on concrete integers.
Claim. For the index sextuple $(1,3,1,1,0,2)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit table value: $m_2^{\mathrm{num}}(1,3,1,1,0,2)=8\,Z_{\mathrm{expl}}(1,3,1,1,0,2)$.
background
In the 4D Regge exact-midpoint analysis, the numerator of the M2 TT kernel is assembled by folding a fixed coupling list: each term contributes an integer depending on six indices in $\mathbb{F}4$, and $m_2^{\mathrm{num}}$ is that fold starting from zero. Parallel to it sits an explicit integer table $Z{\mathrm{expl}}$ on the same six indices, given by a finite pattern-match (e.g. $(0,0,1,1,2,2)\mapsto 4$, off-diagonal pairs $\mapsto -2$, and so on).
The local module is chunk 7 of a 256-cell kernel certification whose sole claim is the scalar identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ at every index sextuple. Upstream definitions supply only the fold and the table; no analytic closed form is assumed beyond those defs.
proof idea
One-line computational proof: decide evaluates both sides at the concrete Fin-4 sextuple $(1,3,1,1,0,2)$ and checks integer equality. No lemmas are invoked beyond the reducibility of the fold defining the numerator and the pattern-match defining the explicit table.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all $a,b,c,d,i,j:\mathbb{F}_4$ by exhausting cases. That universal equality is the certified bridge between the folded coupling numerator and the closed explicit table in the 4D Regge midpoint M2 TT kernel. Within Recognition gravity analysis it is bookkeeping infrastructure, not a forcing-chain step: it locks one of 256 cells so the global factor-of-eight relation can be quoted without residual kernel obligations.
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