e_232103
plain-language theorem explainer
For the six Fin-4 indices (2,3,2,1,0,3), the folded coupling numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts cite it inside the exhaustive 4^6 case split that proves the Regge midpoint M2TT numerator identity in four dimensions. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the coupling numerator satisfies $m_2^{\mathrm{num}}(2,3,2,1,0,3)=8\,Z_{\mathrm{explicit}}(2,3,2,1,0,3)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each local contribution contrib t a b c d i j. The comparison target explicitZ is a sparse lookup table of small integers (entries such as $4$, $-2$, and defaults) that encodes the expected closed form.
This module is chunk 11 of the 256-kernel decide campaign whose sole job is to discharge $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ pointwise. The ambient claim is the exact midpoint M2TT identity for the discrete gravity kernel, reduced to finite arithmetic on index sextuples.
proof idea
One-line computational proof: by decide. Lean evaluates both sides at the concrete sextuple $(2,3,2,1,0,3)$ inside the kernel and confirms integer equality. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles every Fin-4 sextuple by nested fin_cases and needs each pointwise fact such as this one. Establishing the numerator identity is a concrete algebraic step toward the exact Regge midpoint M2TT relation used in the discrete gravity sector of Recognition Science. It does not itself touch the forcing chain (T0–T8) or the J-cost uniqueness, but it hardens the 4D curvature-kernel bookkeeping those continuum limits rely on.
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