e_211123

In the 4D Regge midpoint analysis, the M2 numerator is the integer obtained by folding a fixed coupling list: each term contributes via a local contrib at six Fin 4 indices, and the fold starts from zero. The companion object explicitZ is a sparse case table on the same six indices, returning small integers such as $4$, $-2$, and so on for the nonzero patterns.

The module is chunk 9 of a 256-way kernel split: every concrete six-tuple is checked separately so that the global identity $m_2^{\mathrm{num}}=8,Z$ can be assembled by exhaustive fin_cases. Local setting is pure integer equality at one point of that grid.

One-line kernel proof: decide evaluates both sides at the fixed indices $(2,1,1,1,2,3)$. The left side runs the fold that defines the M2 numerator; the right side multiplies the looked-up explicitZ entry by eight. No lemmas beyond definitional reduction are required.

Feeds the parent theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in Fin 4 and closes by casing on all six indices. Each chunk theorem such as this one discharges one residual goal in that assembly, so the global factor-of-eight relation between the folded numerator and the explicit Z table becomes a proved fact rather than a hypothesis.

In the broader gravity stack this is bookkeeping for the Regge midpoint M2–TT identity in 4D, not a new physical law. It does not itself invoke the forcing chain (T0–T8), RCL, or the phi ladder; it only certifies one integer cell needed upstream of continuum or continuum-limit arguments.