e_200020
plain-language theorem explainer
For the multi-index (2,0,0,0,2,0) on Fin 4, the folded Regge coupling numerator equals eight times the explicit integer kernel entry. Gravity analysts certifying the 4D midpoint M2TT identity cite this as one finite-case cell among the chunked kernel decides. Proof is a single kernel decision after unfolding both sides.
Claim. For $a=2$, $b=0$, $c=0$, $d=0$, $i=2$, $j=0$ with each index in $\{0,1,2,3\}$, the summed coupling numerator equals eight times the explicit kernel value: $m_2^{\mathrm{num}}(2,0,0,0,2,0)=8\,Z_{\mathrm{expl}}(2,0,0,0,2,0)$.
background
In the Regge-exact midpoint analysis for the 4D M2TT identity, two integer kernels on six indices in $\mathrm{Fin},4$ are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at the given multi-index. Its companion $Z_{\mathrm{expl}}$ is a sparse, pattern-matched table of small integers (entries such as $4$, $-2$, and so on).
This module is chunk 8 of the case-split certification that $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ holds on every sextuple. The surrounding kernel-cert file supplies both definitions; the assemble file recombines the cells into a single universal statement.
proof idea
One-line proof by decide. After unfolding the fold that defines the numerator and the match that defines the explicit kernel at $(2,0,0,0,2,0)$, both sides reduce to concrete integers; the kernel closes the equality.
why it matters
Feeds the universal identity m2Num_eq_eight_explicitZ, which introduces six $\mathrm{Fin},4$ variables and dispatches every cell by fin_cases. That assembled equality is the algebraic backbone of the Regge-exact midpoint M2TT certificate in the gravity analysis stack: once numerator and explicit kernel are locked at factor eight, downstream curvature and defect identities can quote a single clean relation rather than a raw fold. The chunk exists only to keep each decide goal tiny and deterministic.
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