e_200301
plain-language theorem explainer
For the six-index slot (2,0,0,3,0,1) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one atomic decide-cell in the 4D Regge midpoint M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(2,0,0,3,0,1)$ equals eight times the explicit integer kernel at the same indices: $N(2,0,0,3,0,1)=8\,Z(2,0,0,3,0,1)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2TT identity certification: the claim that the folded numerator m2Num equals eight times a sparse explicit integer table explicitZ on every sextuple of Fin-4 indices. The module header states the local goal as "m2Num = 8·explicitZ, chunk 8 (256 kernel decides)."
m2Num a b c d i j is defined by folding a fixed coupling list and summing per-term contributions at those indices. explicitZ is a pattern-matched integer table on six Fin-4 arguments (nonzero only on a short list of index patterns such as (0,0,1,1,2,2) ↦ 4 and several −2 off-diagonal patterns). The present declaration fixes one concrete sextuple, (2,0,0,3,0,1), inside that finite grid.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete integers once the six Fin-4 arguments are substituted into the fold definition of the numerator and the pattern table for the explicit kernel, so the equality is a pure integer decision with no further lemmas.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for all $a,b,c,d,i,j:\mathrm{Fin},4$ by exhaustive fin_cases and dispatches each cell to a chunk theorem of this form. That global equality is the certified numerator half of the Regge exact-midpoint M2TT identity in 4D gravity analysis inside the monolith. It is bookkeeping infrastructure rather than a forcing-chain landmark (T0–T8), but it closes a finite kernel obligation needed before continuum or continuum-limit gravity statements can quote an exact discrete identity.
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