e_132012
plain-language theorem explainer
For the Fin-4 multi-index (1,3,2,0,1,2), the folded coupling numerator equals eight times the explicit integer kernel entry. Analysts checking the 4D Regge exact-midpoint M2TT identity cite this as one certified kernel case among 256. The proof is a single decide on concrete integer arithmetic.
Claim. At indices $(a,b,c,d,i,j)=(1,3,2,0,1,2)$ with each coordinate in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel value $Z(1,3,2,0,1,2)$.
background
In the 4D Regge exact-midpoint stack, the M2TT identity is reduced to a pointwise integer check: a folded numerator versus a closed-form kernel table, both indexed by six coordinates in $\mathrm{Fin},4$.
The numerator sums contrib terms by folding a fixed coupling list over those six indices. The comparison target is a piecewise integer map on the same indices (sample clauses return $4$ or $-2$ on selected patterns). Both objects live in the kernel-certificate module imported here.
This file is chunk 7 of the 256 kernel decides that discharge numerator $= 8\cdot Z$ case by case. The local setting is pure decidable integer arithmetic; no analysis or continuum estimates enter.
proof idea
One-line wrapper proved by decide. With the six $\mathrm{Fin},4$ arguments fixed to $1,3,2,0,1,2$, the left side is a fully concrete fold of contributions over the coupling list, and the right side is eight times the matching clause of the explicit integer table. Both reduce to numerals; equality is checked by the kernel.
why it matters
Feeds the assembled theorem that asserts the numerator equals eight times the explicit kernel for every sextuple in $\mathrm{Fin},4$, proved by nested fin_cases over all coordinates. That global identity is the algebraic core of the Regge exact-midpoint M2TT check in four dimensions: once every kernel cell matches, the discrete identity is certified arithmetic rather than an open expansion.
It sits in the Gravity analysis path that supports discrete-to-continuum consistency for the geometric sector of Recognition Science (four-dimensional complex with the forced $D=3$ spatial slice from the forcing chain). Chunk 7 is scaffolding closure by exhaustion, not a new physical hypothesis.
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