e_102323
plain-language theorem explainer
For the Fin-4 index tuple (1,0,2,3,2,3), the folded M2 numerator equals eight times the explicit Z coupling. Gravity analysts checking the 4D Regge exact-midpoint M2 TT identity cite this as one atomic kernel equality among the chunked 256 decides. Proof is a single decide discharging the concrete integer equality.
Claim. For indices $a=1$, $b=0$, $c=2$, $d=3$, $i=2$, $j=3$ in $\mathrm{Fin}\,4$, the M2 numerator equals $8$ times the explicit integer coupling $Z$ at those indices: $m_2(1,0,2,3,2,3)=8\,Z(1,0,2,3,2,3)$.
background
In the 4D Regge midpoint stack, the M2 numerator is an integer obtained by folding a fixed coupling list: each list entry contributes a term depending on six Fin-4 indices, and the fold sums those contributions from zero. The companion map explicitZ is a closed-form pattern table on the same six indices, returning small integers (4, -2, and similar cases).
The module certifies the pointwise claim numerator = 8 · explicitZ on chunk 4 of the index space, via 256 kernel-level decides. Upstream definitions live in the kernel certificate module; this file only specializes them at concrete tuples.
proof idea
One-line computational wrapper: decide evaluates both sides at the literal indices 1,0,2,3,2,3 and accepts the integer equality. No named lemmas are applied beyond unfolding the fold-definition of the numerator and the case table for explicitZ.
why it matters
Feeds the assembly theorem that states the identity for every six-tuple in Fin 4 by exhaustive fin_cases, with this atom covering the (1,0,2,3,2,3) branch. That universal equality is the algebraic backbone of the Regge exact-midpoint M2 TT identity certification in four dimensions. In the Recognition gravity analysis it clears pure index bookkeeping needed before continuum or phenomenological claims; it is not itself a forcing-chain step (T0–T8).
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