e_022010
plain-language theorem explainer
For the six-index slot (0,2,2,0,1,0) 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 pure kernel decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(0,2,2,0,1,0)$ equals eight times the explicit integer kernel at the same indices: $N(0,2,2,0,1,0)=8\,Z(0,2,2,0,1,0)$.
background
This module is chunk 2 of a 256-cell decide grid that checks the identity $N=8Z$ pointwise on $(\mathrm{Fin},4)^6$. The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$. The closed form $Z=\mathrm{explicitZ}$ is an integer-valued pattern match on the six indices (typical nonzero values $\pm 2,,4$).
The ambient setting is the exact midpoint analysis of the 4D Regge M2TT identity in the Gravity.Analysis stack. Each chunk theorem pins one concrete sextuple so the assembler can recombine them without re-running the fold.
proof idea
One-line decide on a fully concrete equality of integers. Both sides reduce by definition: the left-hand fold over couplingZList evaluates to a fixed Int, the right-hand side is 8 times the matching explicitZ clause (or zero if unmatched). No lemmas are invoked beyond kernel evaluation.
why it matters
Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases over the six $\mathrm{Fin},4$ indices and dispatch into these chunk cells. That global identity is the certified algebraic core of the Regge exact-midpoint M2TT numerator in 4D, tying the folded coupling expansion to the sparse explicit kernel used downstream in the gravity analysis. It is bookkeeping infrastructure rather than a new physical law, but without the pointwise cells the assembler cannot close.
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