e_023232
plain-language theorem explainer
For the six-index cell (0,2,3,2,3,2) on Fin 4, the coupling-folded midpoint numerator equals eight times the tabulated explicit integer kernel. Gravity analysts certifying the 4D Regge exact-midpoint M₂TT identity cite this as one of 256 kernel cells. The proof is a single kernel decision on the concrete integer equality.
Claim. For indices $a=0$, $b=2$, $c=3$, $d=2$, $i=3$, $j=2$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel entry at those indices: $N(0,2,3,2,3,2)=8\,Z(0,2,3,2,3,2)$.
background
In the 4D Regge exact-midpoint analysis, two integer maps on six $\mathrm{Fin},4$ indices are compared. The numerator folds a fixed coupling list, summing a local contribution at each coupling triple. The explicit kernel is a sparse case table of small integers (entries such as $4$, $-2$, and so on).
This module is chunk 2 of a 256-cell partition of the $\mathrm{Fin},4^{6}$ domain. Each cell asserts the local identity numerator $= 8\cdot$ explicit kernel, so a later exhaustive case split can discharge the universal statement. The module doc states the goal directly: numerator equals eight times the explicit kernel, via 256 kernel decides.
proof idea
One-line proof by decide. With all six indices fixed, both sides reduce to concrete integers: the fold in the numerator becomes a finite sum of known contributions, and the explicit table returns a single Int. Lean’s decision procedure checks the equality; no lemmas beyond the two definitions are required.
why it matters
This cell is consumed by the assembly theorem that states the identity for every six-tuple of $\mathrm{Fin},4$ indices, proved by nested fin_cases on all six arguments. That global identity is part of the Regge exact-midpoint M₂TT certification stack in the gravity analysis layer. It is discrete geometric bookkeeping for the 4D midpoint kernel, not a step on the T0–T8 forcing chain, the Recognition Composition Law, or the $\varphi$-ladder mass formula. Its role is to close one of the 256 concrete obligations that make the assembled equality hold definitionally on the nose.
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