e_213313
plain-language theorem explainer
For the six-index tuple (2,1,3,3,1,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the exact midpoint M2TT identity in 4D Regge calculus. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the summed coupling numerator at $(2,1,3,3,1,3)$ equals eight times the explicit integer kernel at those same indices: $N(2,1,3,3,1,3)=8\,Z(2,1,3,3,1,3)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices appear. The numerator $N=m2Num$ is obtained by folding a fixed coupling list and summing local contributions at $(a,b,c,d,i,j)$. The comparison object $Z=explicitZ$ is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and so on).
The module is chunk 9 of a 256-case kernel certification that $N=8Z$ pointwise. Each chunk theorem fixes one concrete six-tuple; the present case is $(2,1,3,3,1,3)$. Upstream definitions supply only the fold for $N$ and the table for $Z$; no analytic closed form is assumed beyond those defs.
proof idea
One-line decide. Both sides evaluate to concrete integers once the six Fin-4 arguments are substituted: the fold that defines $N$ and the pattern match that defines $Z$ become ground arithmetic, which the kernel checks for equality after multiplying $Z$ by eight.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases over Fin 4. That global identity is the algebraic backbone of the exact midpoint M2TT relation in the 4D Regge gravity analysis. Without the chunk lemmas, the assembler has nothing to discharge at each index. The result is pure discrete kernel bookkeeping inside the gravity domain; it does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law, but it supports the discrete curvature side of the RS gravity stack.
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