e_100332

In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: it sums a local contribution at each table entry for the given multi-index. The companion map $Z_{\mathrm{ex}}$ is an explicit piecewise integer kernel (sample values include $4$, $-2$, and so on on diagonal and off-diagonal patterns).

The module is chunk 4 of a 256-case decide grid whose sole job is to check $m_2^{\mathrm{num}}=8,Z_{\mathrm{ex}}$ pointwise. The local setting is pure finite enumeration: no continuum limit or curvature hypothesis is in play inside the chunk.

One-line computational proof: decide evaluates both sides at the concrete six-tuple $(1,0,0,3,3,2)$ and checks integer equality. No lemmas beyond the definitions of $m_2^{\mathrm{num}}$ (fold of contributions) and $Z_{\mathrm{ex}}$ (pattern match) are required; the kernel reduces the closed terms directly.

This atom is consumed by the assembler m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in $(\mathrm{Fin},4)^6$ and discharges the universal quantifier by nested fin_cases. Each chunk theorem such as this one supplies one leaf of that case tree. In the broader gravity stack the identity certifies that the midpoint M2–TT numerator factors as eight times a sparse explicit kernel, a bookkeeping step toward exact Regge identities in four dimensions. It does not itself touch the RS forcing chain (T0–T8) or the J-cost; it is infrastructure inside the discrete gravity analysis layer.