e_213101

This module is chunk 9 of a 256-case kernel certification that the folded numerator coupling equals eight times an explicit integer table on all six-tuples in $\mathrm{Fin},4$. The setting is the 4D Regge exact-midpoint M2TT identity analysis in the Gravity domain.

The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term evaluated at the six indices. The explicit kernel $Z$ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by a finite pattern of integer cases (typical nonzero values $\pm 4,\pm 2$). The claim is the pointwise identity of these two integers at one fixed multi-index.

One-line computational proof: decide evaluates both sides as concrete Int values (the fold of contributions versus the pattern-matched table entry scaled by 8) and closes the equality by kernel decision. No lemmas are invoked beyond the definitions of the numerator fold and the explicit table.

This is one atomic cell in the exhaustive certification that $m_2^{\mathrm{num}}=8Z$ on all of $\mathrm{Fin},4^6$. Downstream, m2Num_eq_eight_explicitZ assembles the full universal statement by casing all six indices; each chunk such as this one discharges a block of those cases. In the broader Recognition gravity stack, the identity is infrastructure for the Regge midpoint M2TT analysis rather than a forcing-chain landmark (T0–T8). It closes a pure computational obligation: every multi-index must match before the analytic identity can be quoted.