e_111131
plain-language theorem explainer
Concrete kernel identity: the folded M2 numerator at indices (1,1,1,1,3,1) equals eight times the explicit Z table entry. Gravity analysts cite it as one of the 256 Fin-4 cases feeding the global m2Num = 8·explicitZ assembly. Proof is a single decide on the evaluated integers.
Claim. For the index sextuple $(1,1,1,1,3,1)\in(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the explicit integer table: $m_2^{\mathrm{num}}(1,1,1,1,3,1)=8\,Z_{\mathrm{expl}}(1,1,1,1,3,1)$.
background
In the 4D Regge exact-midpoint analysis, two integer kernels are compared pointwise on $(\mathbb{F}4)^6$. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the fold of a fixed coupling list: each term contributes an integer via a local contrib map, summed from zero. The closed form $Z{\mathrm{expl}}$ is a sparse pattern match on the six indices, returning small integers such as $4$, $-2$, or (by default) $0$.
The module is chunk 5 of a 256-case kernel certification whose sole claim is $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ at every index. Upstream, both sides are pure defs in the KernelCert module; no analytic hypothesis is required, only evaluation of finite data.
proof idea
One-line computational proof: decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are fixed at $1,1,1,1,3,1$, so the equality is a decidable integer comparison with no lemmas or rewriting.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathbb{F}4,; m_2^{\mathrm{num}}=8,Z{\mathrm{expl}}$ and discharges the universal claim by exhaustive fin_cases. That identity is the algebraic backbone of the 4D Regge midpoint M2TT certification in the Gravity analysis stack: it replaces a folded coupling sum by an explicit sparse table, enabling exact midpoint checks without residual summation. Within Recognition Science gravity work this is bookkeeping, not a forcing-chain step (T0–T8), but it is required scaffolding for any later continuum or continuum-limit claim that quotes the midpoint kernel.
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