e_330133
plain-language theorem explainer
For the six Fin-4 indices (3,3,0,1,3,3), the folded numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of the 256 kernel cells. The proof is a single decide on concrete integers.
Claim. With indices in $\{0,1,2,3\}$, the folded coupling numerator at $(3,3,0,1,3,3)$ equals eight times the tabulated explicit kernel entry at the same indices: $N(3,3,0,1,3,3)=8\,Z(3,3,0,1,3,3)$.
background
This module is chunk 15 of a 256-cell kernel certification that the folded numerator m2Num equals eight times a closed-form table explicitZ on every 6-tuple of Fin-4 indices. The setting is the 4D Regge midpoint analysis of the 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 explicit integer-valued pattern match on the six Fin-4 arguments (sample clauses give 4, 4, -2, ...). The identity m2Num = 8 · explicitZ is the algebraic content being certified cell by cell.
Upstream, both definitions live in the KernelCert module; this chunk only discharges one concrete sextuple.
proof idea
One-line proof by decide. Both sides reduce to concrete Int values once the six Fin-4 literals are substituted into the fold definition of m2Num and the pattern match of explicitZ, so the kernel decision procedure closes the equality with no further lemmas.
why it matters
Feeds the parent theorem m2Num_eq_eight_explicitZ, which states the identity for all Fin-4 sextuples and proves it by exhaustive fin_cases on each coordinate, dispatching each cell to a chunk theorem of this form. Without the 256 cell equalities, the assemble step cannot finish.
In the Recognition gravity analysis this certifies that the midpoint Regge M2 numerator is exactly eight times the explicit kernel table, a bookkeeping identity needed before continuum or continuum-limit claims. It is pure discrete algebra on Fin 4; it does not itself invoke the forcing chain (T0–T8), RCL, or the phi ladder, but it sits inside the gravity side of the monolith that those landmarks ultimately constrain.
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