e_311100

In the 4D Regge exact-midpoint analysis, two integer-valued kernels 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 contrib map, summed from zero. The companion $Z_{\mathrm{explicit}}$ is a sparse case table of small integers (entries such as $4$, $-2$, and defaults), intended as the closed form of that sum divided by eight.

The ambient module is chunk 13 of a 256-cell decide sweep that checks $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ pointwise. The present cell fixes the multi-index $(3,1,1,1,0,0)$. Upstream, both maps live in the KernelCert module; the assemble theorem later quantifies over all six indices by fin_cases.

One-line kernel proof: by decide. Lean reduces both sides of the integer equality for the concrete Fin-4 sextuple $(3,1,1,1,0,0)$ and accepts the match. No lemmas are invoked beyond the definitions of m2Num (fold of contrib over couplingZList) and explicitZ (pattern-match table).

This cell is one brick in the pointwise identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ on $(\mathrm{Fin},4)^6$. Downstream, m2Num_eq_eight_explicitZ assembles every cell by nested fin_cases on $a,b,c,d,i,j$, quoting each chunk theorem such as this one. That global equality is the algebraic certificate that the Regge midpoint M2TT numerator collapses to the explicit sparse kernel used in the 4D gravity analysis. It does not itself touch the RS forcing chain (T5–T8) or the J-cost; it is infrastructure inside the discrete gravity side of the monolith.