e_203322
plain-language theorem explainer
For the six Fin-4 indices (2,0,3,3,2,2), the summed numerator m2Num equals eight times the closed-form kernel explicitZ. Gravity analysts cite it as one atomic case in the 4D Regge midpoint M2TT numerator identity. The proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,0,3,3,2,2)$ in $\mathrm{Fin}\,4$, the folded 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 sits in the 4D Regge-calculus analysis of the exact midpoint M2TT identity. The numerator m2Num is the integer obtained by folding a fixed coupling list: each term contributes via a local contrib at six Fin-4 indices, then summing. The companion explicitZ is a pattern-matched closed form on those same six indices (sample values include $4$, $-2$, and so on).
The local claim of the file is the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z$ on a 256-case chunk of the $4^6$ index space. Upstream, both sides are pure definitions in the kernel-cert module; no analytic hypotheses remain once the indices are fixed.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values for the fixed sextuple $(2,0,3,3,2,2)$, so the kernel discharges equality without lemmas or rewriting.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the full universal identity $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8\cdot Z$ by exhaustive fin_cases on all six indices. Each chunk theorem such as this one supplies one decided leaf so the assembler can close without a monolithic kernel call. In the gravity stack this certifies that the summed Regge midpoint numerator matches the explicit eightfold kernel, a numerical backbone step for the 4D M2TT identity used downstream in curvature and mass-ladder checks.
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