e_112131

In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer via a local kernel, and the fold starts at zero. The companion map $Z$ is an explicit pattern-matched table on the same six indices (sample entries include $4$, $-2$, and so on).

The module is chunk 5 of a 256-case kernel split: each chunk theorem pins one concrete sextuple so that the global identity $m_2^{\mathrm{num}}=8Z$ can be reassembled by exhaustive fin_cases. The local setting is pure integer arithmetic on a finite index set; no continuum limit is taken here.

One-line computational certificate: decide evaluates both sides at the fixed indices $(1,1,2,1,3,1)$ and checks integer equality in the kernel. No lemmas are invoked beyond the definitions of $m_2^{\mathrm{num}}$ (fold of contributions) and $Z$ (explicit table).

Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$ and discharges the universal quantifier by six nested fin_cases over Fin 4. Each chunk theorem such as this one is a leaf of that case tree. In the broader gravity stack the identity is the algebraic backbone of the exact midpoint M2–TT comparison in 4D Regge calculus; closing every leaf removes scaffolding from that comparison. It does not itself touch the T0–T8 forcing chain or the Recognition Composition Law, but it is part of the discrete curvature bookkeeping those layers later consume.