e_300331
plain-language theorem explainer
For the Fin-4 multi-index (3,0,0,3,3,1), the folded M2 numerator equals eight times the explicit Z kernel entry. Gravity analysts building the exact midpoint Regge M2-TT identity cite it as one decided cell in chunk 12 of the 256-kernel table. The proof is a single kernel decide on the two concrete integers.
Claim. For indices $a=3$, $b=0$, $c=0$, $d=3$, $i=3$, $j=1$ in $\{0,1,2,3\}$, the folded M2 numerator at $(a,b,c,d,i,j)$ equals $8$ times the explicit integer Z-kernel value at the same multi-index.
background
In the 4D Regge exact-midpoint analysis, the M2 numerator at a six-index is the fold of a contribution map over a fixed coupling list. The companion explicit Z object is a piecewise integer function on six Fin-4 arguments, listing the closed-form values the identity claims to match up to a universal factor of eight.
This module is chunk 12 of those kernel decides: each declaration pins one concrete cell of the table. Upstream, both the folded numerator and the explicit Z table live in the kernel-certificate module; the numerator sums contributions, while explicit Z pattern-matches on the six indices.
proof idea
One-line kernel decide. With all six Fin-4 arguments fixed, both the folded numerator and eight times the explicit Z entry reduce to concrete integers, so decide closes the equality with no further lemmas.
why it matters
The cell is consumed by the universal assembly theorem that states the folded M2 numerator equals eight times explicit Z for every six-tuple in Fin 4. That assembly runs by exhaustive fin_cases and is the algebraic backbone of the exact midpoint M2-TT identity in the 4D Regge gravity analysis. The factor-of-eight match is a discrete computational certificate inside the gravity stack, not a step of the T0-T8 forcing chain.
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