e_020330

This module is one chunk of a 256-case kernel certification that the Regge midpoint numerator $m_2^{\mathrm{num}}$ is identically eight times a sparse explicit integer table $Z$ on $(\mathrm{Fin},4)^6$. The local setting is pure integer arithmetic: no continuum limit, only exact evaluation of a finite fold.

Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing a contribution term at each tuple. The companion table $Z$ is a pattern-matched integer function on six $\mathrm{Fin},4$ indices (typical nonzero values $\pm 2,,4$). The identity $m_2^{\mathrm{num}}=8Z$ is the algebraic content being certified case by case.

The present declaration fixes one concrete six-tuple in that grid. Sibling lemmas cover the other tuples in the same chunk; the assemble theorem will recombine them.

One-line decide proof. Both sides reduce to concrete Int values once the six indices are substituted: the fold that defines $m_2^{\mathrm{num}}$ evaluates fully, and $Z$ matches its pattern (or defaults). Lean’s kernel decision procedure checks the resulting integer equality. No lemmas beyond the two definitions are invoked.

Feeds the parent assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all six indices by exhaustive fin_cases and dispatches each leaf to a chunk lemma of this form. That global equality is the certified algebraic core of the 4D Regge exact-midpoint M2 TT identity used in the gravity analysis stack.

In the Recognition framework this sits in the discrete-geometry layer that underwrites continuum gravitational identities, not in the T0–T8 forcing chain itself. It closes one of 256 kernel obligations so the assemble proof stays sorry-free. No open physics hypothesis is at stake here: pure finite-case arithmetic.