e_232202

In the 4D Regge midpoint analysis, two integer-valued kernels on six $\mathbb{F}4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: it sums a contribution term over every entry of couplingZList. The companion table $Z{\mathrm{explicit}}$ is a closed-form case-split on the same six indices, returning small integers such as $4$, $-2$, and so on.

The local module is chunk 11 of a 256-cell kernel certification: each cell checks $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ at one concrete multi-index. The factor eight is the global normalization that the assemble theorem will lift from pointwise decides to a universal statement.

One-line computational proof: decide evaluates both sides at the fixed indices $(2,3,2,2,0,2)$. The left side reduces by unfolding the fold over couplingZList; the right side multiplies the matching explicitZ clause by eight. Both land on the same Int, so the equality is discharged by the kernel decision procedure. No lemmas beyond the two definitions are invoked.

This cell is one of the 256 pointwise facts consumed by m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ and proves it by exhaustive fin_cases on all six indices. That universal identity is the algebraic core of the Regge exact-midpoint M2 TT certification in 4D gravity analysis. Without the chunk decides, the assemble theorem cannot close. The result is pure discrete kernel bookkeeping; it does not itself invoke Recognition forcing steps T0–T8, but it sits inside the gravity side of the monolith that those steps underwrite.