e_000232

This module sits in the Gravity analysis of the exact Regge midpoint M2–TT identity in 4D. The local setting is the chunk-0 slice of the claim that the folded numerator equals eight times an explicit kernel, certified by 256 kernel decisions.

The numerator m2Num is the fold of a contribution function over a fixed coupling list: it accumulates integer contributions at a six-index multi-index in $(\mathrm{Fin},4)^6$. The companion explicitZ is a pattern-matched integer table on the same multi-indices (typical values $\pm 2,,4$ on the listed patterns).

The identity under audit is pointwise equality of these two integer-valued maps, scaled by eight on the explicit side. Upstream definitions supply only the table and the fold; no analytic closed form is assumed beyond what the kernel can evaluate.

One-line computational proof: decide evaluates both sides as concrete integers and checks equality. No lemmas are invoked beyond the definitions of the fold m2Num and the pattern table explicitZ at the fixed multi-index $(0,0,0,2,3,2)$.

This is one atomic case in the chunk-0 certification that $m_2^{\mathrm{num}}=8,Z$ on all of $(\mathrm{Fin},4)^6$. Downstream, m2Num_eq_eight_explicitZ assembles the universal statement by exhaustive fin_cases on the six indices; each concrete equality such as this one discharges a branch.

In the broader Recognition gravity stack, the midpoint M2–TT identity is part of the discrete curvature bookkeeping that must match continuum limits. The factor of eight and the explicit kernel table are the computational content of that match on the finite index set. The declaration does not itself touch the forcing chain (T0–T8) or the J-cost; it is pure integer certification inside the Regge analysis layer.