e_330010
plain-language theorem explainer
For the six-index slot (3,3,0,0,1,0) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of the 256 kernel point checks. The proof is a pure decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(3,3,0,0,1,0)$ equals $8$ times the explicit kernel integer at the same indices: $N(3,3,0,0,1,0)=8\,Z(3,3,0,0,1,0)$.
background
This module is chunk 15 of a 256-point kernel certification that the folded numerator m2Num agrees with eight times a closed-form table explicitZ on every 6-tuple in $(\mathrm{Fin},4)^6$. The local setting is exact midpoint analysis for a 4D Regge-type M2TT identity in the gravity stack.
m2Num a b c d i j is defined by folding a fixed coupling list: start at 0 and add each contribution contrib t a b c d i j. explicitZ is an integer-valued pattern match on the six Fin-4 indices (sample clauses return $4$, $-2$, etc.). The identity under check is pointwise $N=8Z$ on that discrete cube.
Upstream, only those two definitions are needed; no analytic lemma is invoked. Downstream assembly will case-split all six indices and discharge each cell by a chunk theorem of this form.
proof idea
One-line kernel decide. Both sides reduce to concrete Int values once the six Fin-4 arguments are substituted: the fold that defines the numerator becomes a finite sum of integer contributions, and explicitZ hits a single match clause (or the default). decide compares the two closed integers and closes the goal. No lemmas beyond the definitions are applied.
why it matters
Feeds the parent universal statement m2Num_eq_eight_explicitZ, which asserts $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases on all six indices. Each chunk theorem such as this one is a single cell in that 256-case tree (module doc: "256 kernel decides").
In the Recognition gravity analysis path, equating the folded coupling numerator to an explicit sparse kernel is bookkeeping that lets later identities treat M2TT midpoint algebra in closed form rather than as an opaque fold. It does not itself invoke the forcing chain (T0–T8), RCL, or $\varphi$-ladder mass formulae; it is infrastructure inside the 4D Regge midpoint certification.
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