e_310000
plain-language theorem explainer
At multi-index (3,1,0,0,0,0) the Regge midpoint mass-squared numerator equals eight times the explicit Z-kernel value. Assemblers of the 4D TT midpoint identity cite the global pointwise equality; this supplies one decide-closed cell in chunk 13. The proof is a single kernel decision on concrete Fin-4 integers.
Claim. For indices $a=3$, $b=1$, $c=0$, $d=0$, $i=0$, $j=0$ in $\mathrm{Fin}\,4$, the folded coupling sum $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, the mass-squared numerator is a fold over a fixed coupling list: each term contributes an integer depending on six $\mathrm{Fin},4$ indices, and the numerator is that running total. The companion map is a sparse integer table on the same six indices, recording the closed-form $Z$-kernel (entries such as $4$, $-2$, and so on on its nonzero support).
The local module is chunk 13 of a 256-way kernel-decide sweep whose goal is the pointwise identity numerator $= 8\cdot Z$. Both maps are defined upstream in the KernelCert module; this file only discharges concrete index tuples.
proof idea
One-line computational proof: decide evaluates both sides at the concrete $\mathrm{Fin},4$ sextuple $(3,1,0,0,0,0)$ and checks integer equality. No lemmas are applied beyond unfolding the folded numerator and the explicit $Z$ table.
why it matters
Feeds the assembly theorem that asserts the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhaustive case split on the six indices. That global equality is the algebraic backbone of the exact midpoint $M_2$ TT identity in 4D Regge gravity analysis. Matching the folded coupling numerator to eight times the explicit kernel certifies the discrete curvature bookkeeping used on the gravity side of the Recognition framework. Chunk 13 is one block of the decide cells that close the split.
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