e_221001

In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ accumulates every local contribution from a fixed coupling list by a left fold starting at zero. The companion $Z_{\mathrm{ex}}$ is an explicit case table of small integers (entries such as $4$, $-2$, and so on) that is meant to be the closed form of that sum, up to a universal factor of eight.

This module is chunk 10 of a 256-way split of the kernel: each declaration pins one concrete six-tuple so the global identity can be reassembled by exhaustive fin-cases. The local setting is pure integer arithmetic on the discrete index cube; no continuum limit is taken here.

One-line proof by decide. Lean reduces both sides at the concrete indices $(2,2,1,0,0,1)$: the fold that defines the numerator and the matching clause of the explicit kernel, then checks the integer equality $n=8z$ in the kernel.

The parent theorem m2Num_eq_eight_explicitZ states the identity for every six-tuple and proves it by nested fin-cases; each case lands on a chunk lemma of this form. Without the cell-by-cell certificates the assemble step cannot close. In the broader gravity stack this identity is bookkeeping for the exact midpoint M2TT kernel in four dimensions, a discrete curvature ingredient used when matching Regge-type actions to continuum limits. It does not itself invoke the RS forcing chain (T5–T8) or the J-cost, but it sits inside the gravity analysis that those foundations eventually constrain.