e_001003
plain-language theorem explainer
For the six Fin-4 indices (0,0,1,0,0,3), the folded numerator coupling m2Num equals eight times the explicit integer kernel explicitZ. Gravity analysts cite it as one cell of the 256-way case split that certifies the Regge exact-midpoint M2TT identity in 4D. The proof is a single kernel decide on concrete integers.
Claim. With indices $(a,b,c,d,i,j)=(0,0,1,0,0,3)$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel value: $m_2^{\mathrm{num}}(0,0,1,0,0,3)=8\,Z_{\mathrm{expl}}(0,0,1,0,0,3)$.
background
This module is chunk 0 of a 256-cell decide grid that checks the pointwise identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ on every 6-tuple of $\mathrm{Fin},4$ indices. The setting is the 4D Regge exact-midpoint analysis of the M2TT gravitational kernel.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the integer obtained by folding contrib over couplingZList from zero. The companion map $Z_{\mathrm{expl}}$ is an explicit piecewise integer function on the same six indices (e.g. value $4$ on $(0,0,1,1,2,2)$, $-2$ on several mixed pairs, and $0$ on unmatched patterns). The present cell evaluates both sides at $(0,0,1,0,0,3)$.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete Int values (the fold for $m_2^{\mathrm{num}}$ and the pattern match for $Z_{\mathrm{expl}}$), so the equality is discharged by the decidable equality instance on integers with no further lemmas.
why it matters
Parent theorem m2Num_eq_eight_explicitZ assembles all 256 index cells by nested fin_cases on $a,b,c,d,i,j$; this declaration is the cell for $(0,0,1,0,0,3)$. That universal equality is the algebraic backbone of the Regge exact-midpoint M2TT identity in 4D gravity analysis inside the Recognition Science stack. It does not itself invoke the forcing chain (T5–T8) or the RCL, but it supplies a certified discrete kernel identity used downstream in the gravity sector.
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