e_201300
plain-language theorem explainer
For the Fin-4 multi-index (2,0,1,3,0,0), the folded Regge midpoint numerator equals eight times the explicit integer kernel table. Gravity analysts building the 4D exact-midpoint M2 TT identity cite this as one cell in the chunked kernel check. The proof is a single decide on concrete integer equality after the six indices are fixed.
Claim. At multi-index $(2,0,1,3,0,0)\in(\mathbb{F}_4)^6$, the midpoint mass-matrix numerator equals eight times the explicit integer kernel: $N_{M_2}(2,0,1,3,0,0)=8\,Z(2,0,1,3,0,0)$.
background
In the 4D Regge-calculus analysis of the exact midpoint M2 TT identity, two integer kernels on six indices in $\mathbb{F}_4$ are compared. The numerator is the fold of a fixed coupling list: sum local contributions at $(a,b,c,d,i,j)$. The explicit kernel is a sparse case table of small integers (entries such as $4$, $-2$, and so on).
Both objects live in the KernelCert module. This module is chunk 8 of the 256 kernel decides that discharge the cellwise identity numerator $= 8\cdot$ explicit table. The local goal is only the single multi-index $(2,0,1,3,0,0)$.
proof idea
One-line computational proof: by decide. With the six Fin-4 arguments fixed at $2,0,1,3,0,0$, both the coupling fold and the explicit table reduce to concrete integers, so the decision procedure closes the equality from the definitions alone. No intermediate lemmas are invoked.
why it matters
Supplies one discharged cell to the assembly theorem that states the identity for every six-tuple in $(\mathbb{F}_4)^6$ by exhaustive fin-cases. That global equality is the algebraic backbone of the Regge exact-midpoint M2 TT identity in four dimensions, inside the Gravity analysis layer. The constant factor eight is the structural normalization between the folded coupling sum and the closed-form kernel; each chunk theorem is one verified cell of that normalization.
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