e_313112
plain-language theorem explainer
For the six-index slot (3,1,3,1,1,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the exact midpoint M2 TT identity cite this as one of the 256 kernel cases. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}3$, $b{=}1$, $c{=}3$, $d{=}1$, $i{=}1$, $j{=}2$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
This module is chunk 13 of a 256-case kernel certification that the Regge midpoint M2 TT numerator equals eight times a closed-form table. The ambient setting is 4D discrete gravity analysis: six Fin-4 indices label pairs of edges or simplex directions in the midpoint identity.
The numerator m2Num is defined by folding a fixed coupling list and summing each term's contribution at the six indices. The companion table explicitZ is a total function Fin 4^6 → ℤ given by exhaustive pattern match (sample values include 4, −2, and so on). The claim under audit is the equality of these two integers at one concrete multi-index.
Upstream, both definitions live in the KernelCert module; this chunk only evaluates them at (3,1,3,1,1,2).
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the fold for m2Num and the pattern match for explicitZ are evaluated at the fixed Fin-4 arguments, so the kernel closes the equality with no further lemmas.
why it matters
Parent theorem m2Num_eq_eight_explicitZ asserts the identity for every six-tuple in Fin 4 by nested fin_cases, and each leaf is one of these chunk theorems. Without the (3,1,3,1,1,2) case the universal assembly fails.
In the Recognition gravity stack this certifies the exact algebraic midpoint form of the M2 TT identity in 4D Regge calculus, a discrete precursor to continuum curvature identities. It is pure integer bookkeeping, not a continuum limit, but it locks the combinatorial kernel that later continuum or continuum-limit arguments rely on.
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