e_023023

In the 4D Regge midpoint analysis, two integer-valued maps on six Fin 4 indices are compared. The numerator side $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: each term contributes an integer via a local contrib rule, summed from zero. The comparison side is an explicit pattern-matched kernel $Z$ that returns small integers (e.g. $4$, $-2$) on selected index patterns and is the closed form one wants to certify.

The module is chunk 2 of a 256-way split of the kernel: each chunk theorem pins one concrete six-tuple so the global identity can be reassembled by exhaustive fin_cases. Local setting: prove $m_2^{\mathrm{num}}=8\cdot Z$ pointwise on that grid.

One-line computational certificate: decide evaluates both sides at the concrete indices $(0,2,3,0,2,3)$ and checks integer equality. No algebraic lemmas are invoked; the kernel reduces the fold defining $m_2^{\mathrm{num}}$ and the pattern match defining $Z$, then compares $8Z$.

Parent theorem m2Num_eq_eight_explicitZ assembles the full $\forall$-statement by casing all six Fin 4 indices and dispatching to these atomic equalities. Without each chunk (including this one), the midpoint M2TT numerator identity in 4D Regge analysis does not close in Lean.

In the Recognition gravity stack this is bookkeeping infrastructure, not a new physical law: it locks the discrete curvature/coupling algebra that later feeds continuum or continuum-limit claims. It does not itself invoke the forcing chain (T5–T8), RCL, or $\phi$-ladder mass formulae; it only certifies one cell of the 4D kernel table those layers may rely on.