e_321003

This module is chunk 14 of the 256-case kernel certification that the Regge exact-midpoint M2 numerator equals eight times an explicit integer table on $(\mathrm{Fin},4)^6$. The numerator is defined by folding a fixed coupling list: each term contributes an integer depending on the six indices, and the fold starts at zero. The explicit table $Z$ is a pattern-matched integer function on the same six indices (sample values include $4$, $-2$, and so on for the listed patterns).

The local goal is purely algebraic identity checking: no continuum limit or variational argument is invoked here. Upstream, both the fold definition and the explicit table live in the kernel-certificate module; this chunk only evaluates one concrete multi-index.

One-line computational proof: decide evaluates both integer sides for the fixed indices $(3,2,1,0,0,3)$ and checks equality. No lemmas are rewritten; the kernel reduces the fold that defines the numerator and the pattern match that defines the explicit table to concrete Int values, then confirms they stand in the ratio $1:8$.

Parent theorem m2Num_eq_eight_explicitZ assembles all 256 index tuples by nested fin_cases and needs each cell equality such as this one. That universal identity is the algebraic backbone of the 4D Regge exact-midpoint M2TT certification in the Gravity analysis stack: once every cell matches, the folded numerator may be replaced by the closed-form kernel everywhere.

Within Recognition Science gravity work, this is bookkeeping rather than a forcing-chain step (not T0–T8), but it is required scaffolding for trustworthy discrete curvature identities before continuum or phenomenological claims. It closes one decide obligation inside the 256-kernel grid.