e_321301
plain-language theorem explainer
For the six Fin-4 indices (3,2,1,3,0,1), the midpoint Regge numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one atomic cell in the 4D M2TT identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}3,b{=}2,c{=}1,d{=}3,i{=}0,j{=}1$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $N(3,2,1,3,0,1)=8\,Z(3,2,1,3,0,1)$.
background
In the 4D Regge midpoint analysis, the numerator m2Num is the fold of a fixed coupling list: each term contributes an integer contrib at six face/edge indices drawn from Fin 4, and the fold starts at 0. The companion explicitZ is a sparse pattern-matched integer table on the same six indices (typical nonzero values are $\pm 2,\pm 4$).
The module is chunk 14 of a 256-cell kernel certification whose sole job is to discharge m2Num = 8 · explicitZ pointwise. The local setting is pure integer arithmetic on a finite index set; no continuum limit or curvature hypothesis is in play yet.
Upstream, both sides are pure definitions from the kernel certificate module: the left-hand side is the fold, the right-hand side is the lookup table.
proof idea
One-line decide proof. Both sides evaluate to concrete Int values at the fixed six-tuple (3,2,1,3,0,1), so the kernel closes the equality by computation with no lemmas or rewriting.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all six Fin 4 arguments by exhaustive fin_cases. That global equality is the algebraic backbone of the exact midpoint M2TT identity in 4D Regge gravity inside the Recognition stack.
The chunking (256 decides) keeps each certificate tiny and machine-checkable. Once assembled, the factor-of-eight relation lets later gravity lemmas replace the folded coupling sum by the sparse closed form explicitZ, which is the form used in continuum matching and curvature identities. No T0–T8 forcing step is touched directly; this is infrastructure under the gravity analysis layer.
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