e_013323
plain-language theorem explainer
For the six-index tuple (0,1,3,3,2,3) on Fin 4, the folded numerator m2Num equals eight times the tabulated kernel value explicitZ. Gravity analysts certifying the exact midpoint M2TT identity in 4D Regge calculus cite this as one atomic kernel decide. The proof is a single decide closing the integer equality by computation.
Claim. For indices $(0,1,3,3,2,3)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(0,1,3,3,2,3)=8\,Z_{\mathrm{explicit}}(0,1,3,3,2,3)$.
background
In the 4D Regge exact-midpoint analysis, two integer kernels on six Fin 4 indices are compared pointwise. The numerator m2Num folds a fixed coupling list, summing each contribution at the given multi-index. The comparison target explicitZ is a sparse pattern-matched table of small integers (entries such as 4, -2, and so on).
This module is chunk 1 of a 256-case decide campaign whose goal is the identity m2Num = 8 · explicitZ everywhere. That identity is the algebraic content of the M2TT midpoint kernel certificate from the imported KernelCert module.
proof idea
One-line computational proof: by decide. Lean reduces both sides at the concrete multi-index (0,1,3,3,2,3): the fold that defines m2Num, and eight times the matching explicitZ clause. Both evaluate to the same Int, so the equality closes. No intermediate lemmas are applied beyond the two definitions.
why it matters
This lemma is one atomic case feeding m2Num_eq_eight_explicitZ, which asserts the same identity for every six-tuple in (Fin 4)^6 by exhaustive case split. That global equality is the kernel certificate under the exact midpoint M2TT identity in 4D Regge gravity analysis. In the Recognition gravity layer it locks discrete curvature bookkeeping used by continuum limits; it does not itself invoke the T0–T8 forcing chain or the J-cost.
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