e_020102

This module sits inside the 4D Regge exact-midpoint analysis for the M2–TT identity. The local goal, stated in the module header, is to verify m2Num = 8·explicitZ on one chunk of the 256 kernel cells (all sextuples in Fin 4).

The numerator m2Num(a,b,c,d,i,j) is defined by folding a fixed coupling list: start at 0 and add contrib(t,a,b,c,d,i,j) for each table entry t. The comparison target explicitZ is a total function Fin 4^6 → ℤ given by an exhaustive pattern match (sample clauses include (0,0,1,1,2,2) ↦ 4 and (0,0,1,2,1,2) ↦ −2).

Chunk theorems such as this one pin individual cells so a later fin_cases sweep can reassemble the universal identity.

Both sides evaluate to concrete integers once the six Fin 4 arguments are fixed. The tactic decide runs the kernel decision procedure on the ground equality m2Num 0 2 0 1 0 2 = 8 * explicitZ 0 2 0 1 0 2 and closes the goal. No lemmas are invoked beyond the unfolding of the two defs.

The parent theorem m2Num_eq_eight_explicitZ asserts ∀(a b c d i j : Fin 4), m2Num a b c d i j = 8 * explicitZ a b c d i j. Its proof is a six-fold fin_cases that dispatches each cell to a chunk lemma of this form. Without the cell-wise checks, the assembly cannot finish.

In the broader gravity stack this identity is the algebraic core of the Regge exact-midpoint M2–TT certificate: it converts the folded coupling sum into the closed explicitZ table, which is what downstream curvature and mass-gap arguments consume. It is pure discrete bookkeeping, not a continuum limit claim, and does not itself invoke the RS forcing chain (T5–T8) or the Recognition Composition Law.