e_020111

This module is one chunk of a 256-case kernel certification that the 4D Regge midpoint numerator equals eight times a sparse explicit integer table. Indices run over $\mathrm{Fin},4$, labelling discrete directions in the 4-simplex edge data.

The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each coupling triple at the six indices. The companion table $\mathrm{explicitZ}$ is a pattern-matched integer function on the same six $\mathrm{Fin},4$ arguments; most slots are zero, with a few nonzero values such as $\pm 2$ and $4$ on diagonal-type patterns.

The local claim is the equality at one concrete hex-tuple inside chunk 2 of that case split. Upstream, only the two definitions (the fold and the table) are needed; no analytic lemma is required beyond decidable integer arithmetic.

One-line decide proof. Both sides reduce to concrete Int values: the left by evaluating the fold of contrib over couplingZList at $(0,2,0,1,1,1)$, the right by looking up explicitZ at the same indices and multiplying by $8$. Lean’s kernel decision procedure discharges the resulting integer equality.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all six $\mathrm{Fin},4$ indices by exhaustive fin_cases. That global equality is the algebraic core of the exact midpoint M2TT identity in the 4D Regge analysis stack: it replaces a folded coupling sum by a sparse closed form, enabling exact (not approximate) curvature and mass-term identities downstream.

Within Recognition Science gravity work, such kernel certificates keep discrete geometric identities fully machine-checked rather than numerically sampled. This particular hex-tuple is one of the 256 atomic decides; alone it is routine, but the assemble step depends on every chunk succeeding.