e_023331

This module is chunk 2 of a 256-cell kernel certification that the 4D Regge midpoint numerator matches an explicit closed form. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the fold of a fixed coupling list: each term contributes an integer weight depending on the six Fin-4 indices, summed from zero.

The comparison target $Z_{\mathrm{explicit}}$ is a piecewise integer table on the same six indices (sample values include $4$, $-2$, and other small constants on the listed patterns). The local claim is the scalar identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ at one concrete multi-index inside that table.

Upstream, both sides are pure definitions in the kernel-cert module; no analytic lemma is required beyond evaluating those definitions at fixed Fin-4 data.

One-line computational proof: decide evaluates both integer sides at the literal indices $(0,2,3,3,3,1)$ and checks equality. No rewrite lemmas or induction; the kernel reduces the fold defining the numerator and the matching clause of the explicit table to concrete Int values and confirms $L=8R$.

Parent theorem m2Num_eq_eight_explicitZ assembles every Fin-4 cell by nested fin_cases, discharging each cell with a chunk lemma of this form. Without the cell-wise equalities, the universal statement $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ on all six indices would not close.

In the gravity stack this identity is the algebraic backbone of the exact midpoint M2TT relation in 4D Regge analysis: it replaces a summed coupling expression by a sparse explicit kernel, which is what downstream curvature and continuum-limit arguments consume. It is bookkeeping inside the Recognition gravity layer rather than a T0–T8 forcing step, but it is load-bearing for any claim that the discrete midpoint action matches the certified closed form.