e_112221

In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ 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 contribution function, summed from zero. The comparison target is an explicit piecewise kernel $Z$ that returns small integers (typically $\pm 2$ or $4$) on selected index patterns and is defined by exhaustive case analysis.

The module is chunk 5 of a 256-cell decide grid that discharges $m_2^{\mathrm{num}}=8Z$ pointwise. The global statement assembles every cell by nested fin_cases over the six indices. This cell fixes the concrete tuple $(1,1,2,2,2,1)$.

One-line decide proof. Both sides reduce to concrete Int values: the left by evaluating the fold that defines the numerator on the fixed indices, the right by evaluating the explicit kernel and multiplying by eight. Lean’s kernel decision procedure checks integer equality; no lemmas beyond the two definitions are invoked.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which asserts the identity for every six-tuple in $(\mathrm{Fin},4)^6$ by casing through all 4096 combinations and invoking the matching chunk cell. That global equality is the algebraic certificate that the midpoint Regge $M_2$ numerator is exactly eight times the closed-form kernel, a step in the discrete gravity analysis supporting Recognition Science continuum limits. It does not itself touch the forcing chain (T0–T8) or the Recognition Composition Law; it is pure integer bookkeeping inside the gravity layer.