e_310323
plain-language theorem explainer
For the fixed Fin-4 indices (3,1,0,3,2,3), the folded midpoint Regge numerator equals eight times the tabulated explicit kernel entry. Gravity analysts certifying the 4D midpoint M2TT identity cite it as one cell of chunk 13 (256 kernel decides). The proof is a single decide on concrete integer arithmetic.
Claim. At indices $(3,1,0,3,2,3)\in\{0,1,2,3\}^6$, the folded coupling numerator equals eight times the explicit kernel value: $N(3,1,0,3,2,3)=8\,Z(3,1,0,3,2,3)$, both sides integers.
background
In the Regge-exact midpoint analysis for 4D gravity, two integer-valued maps on six Fin-4 indices are compared. The numerator $N(a,b,c,d,i,j)$ is obtained by folding a contribution function over a fixed coupling list. The kernel $Z$ is a sparse case table returning small integers (typically $0,\pm 2,\pm 4$).
The local module certifies the pointwise identity $N=8Z$ by splitting the $4^6$ index space into decide-chunks. This file is chunk 13 of that certification. Upstream, both maps are defined in the kernel-certificate module; downstream, the cells are reassembled into the universal quantified statement.
proof idea
One-line wrapper: by decide. Both sides are closed integer terms at the concrete indices $(3,1,0,3,2,3)$, so the kernel decision procedure checks equality with no lemmas beyond the definitions of the numerator fold and the explicit kernel table.
why it matters
Feeds the universal assembly theorem m2Num_eq_eight_explicitZ, which cases on all six Fin-4 indices and discharges each cell. That global identity is the numeric core of the Regge-exact midpoint M2TT 4D kernel certificate in the gravity analysis stack. It is infrastructure for discrete gravity inside Recognition Science, not a T0–T8 forcing step; it locks the algebraic factor of eight between the folded coupling numerator and the sparse kernel table used in the midpoint identity.
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