e_013213

This module is chunk 1 of a 256-case kernel certification that the discrete numerator of a 4D Regge midpoint mass-squared / TT identity equals eight times a tabulated integer kernel. Indices run over $\mathrm{Fin},4$, matching the four spacetime directions of the discrete geometry.

The numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing a local contribution at each term. The comparison object $\mathrm{explicitZ}$ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by an exhaustive pattern match (typical nonzero values $\pm 2,,4$). The claim is pure integer arithmetic at one fixed six-tuple.

The ambient goal is the pointwise identity $\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ on all of $\mathrm{Fin},4^6$, assembled later by exhaustive case split.

One-line proof by decide. Both sides reduce to concrete integers: the left via the fold definition of the numerator, the right via the pattern match for the explicit kernel at $(0,1,3,2,1,3)$, then the equality $n=8m$ is discharged by the kernel decision procedure. No lemmas beyond the two definitions are invoked.

Feeds the assembler theorem $\mathrm{m2Num_eq_eight_explicitZ}$, which states the identity for every six-tuple by nested fin_cases and invokes each chunk cell such as this one. That global equality is the certified algebraic core of the Regge exact midpoint M2–TT identity in 4D discrete gravity analysis inside the monolith.

In the Recognition Science gravity stack this is bookkeeping, not a new physical law: it locks the numerator against the closed-form kernel so later continuum or continuum-limit arguments can quote a proved discrete identity rather than an unchecked table. It does not itself touch the forcing chain (T0–T8), $\phi$, or the eight-tick octave; those enter only if the Regge identity is wired into a larger RS gravity derivation.