e_312201
plain-language theorem explainer
For the six-index slot (3,1,2,2,0,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it when assembling the global identity m2Num = 8·explicitZ over the 4D Regge midpoint kernel. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,2,2,0,1)$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is one chunk of a 256-case kernel certification that the 4D Regge midpoint numerator equals eight times a closed-form integer table. The ambient setting is discrete gravity analysis: couplings on six Fin-4 indices are summed into a numerator, then matched against an explicit sparse table.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a contribution function over a fixed coupling list, starting from zero. The table $Z$ is an explicit pattern-match on six Fin-4 arguments returning small integers (e.g. $\pm 2,4$ on the listed support, and implicitly zero elsewhere in the full definition).
Chunk 13 discharges one concrete six-tuple so the assembler can recombine all cases into a single universal identity.
proof idea
One-line proof by decide. Both sides reduce to concrete integers: the left-hand side evaluates the fold of contributions at $(3,1,2,2,0,1)$, the right-hand side multiplies the table lookup explicitZ 3 1 2 2 0 1 by eight. Kernel decision closes the equality with no further lemmas.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ by exhaustive fin_cases over Fin 4. Each chunk such as this one supplies one decided instance so the universal statement does not re-decide the whole kernel inline.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint Regge analysis in 4D: it certifies that the folded coupling numerator is exactly eight copies of the sparse integer kernel, a prerequisite for later continuum or continuum-limit comparisons. It does not itself invoke the T0–T8 forcing chain, but sits inside the gravity-analysis layer that consumes those geometric constraints.
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