e_012203
plain-language theorem explainer
For the Fin-4 multi-index (0,1,2,2,0,3), the discrete Regge midpoint numerator equals eight times the explicit integer kernel at those indices. Analysts certifying the 4D M2-TT midpoint identity cite each such cell when assembling the full table. The proof is a single kernel `decide` on concrete integer arithmetic.
Claim. At multi-index $(0,1,2,2,0,3)$ on $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel value: $m_2^{\mathrm{num}}(0,1,2,2,0,3)=8\,Z_{\mathrm{expl}}(0,1,2,2,0,3)$.
background
In the 4D Regge midpoint M2-TT analysis, the numerator is an integer obtained by folding a contribution map over a fixed coupling list, for six indices each ranging in $\mathrm{Fin},4$. The explicit kernel is a sparse closed-form table on the same domain (pattern-matched integers such as $4$ or $-2$ on selected index patterns, zero elsewhere).
This module is chunk 1 of the kernel certification that every cell obeys numerator $= 8\cdot$ explicit kernel. The local setting is pure finite-domain integer equality: $4^6$ cells, discharged in decide-sized batches. Upstream definitions supply only the fold and the table; no continuum geometry is invoked here.
proof idea
One-line computational close: decide evaluates both sides as concrete Int values (the fold of contributions versus the pattern match on the explicit table) and accepts the equality. No intermediate lemmas are applied beyond unfolding those two definitions.
why it matters
This cell is one input to the universal assembly theorem stating that for all six $\mathrm{Fin},4$ indices the numerator equals eight times the explicit kernel. That parent result is proved by exhaustive fin_cases and is the algebraic backbone of the exact midpoint M2-TT identity in the 4D Regge gravity analysis. The constant factor $8$ links the folded coupling sum to the sparse kernel used in later curvature bookkeeping inside the Recognition gravity stack. It does not itself touch the forcing chain (T0–T8) or the J-cost; it is pure discrete-identity scaffolding for the gravity side.
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