e_310230

In the 4D Regge midpoint analysis, the TT-sector mass-squared numerator is assembled by folding a fixed coupling list: $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ sums integer contributions over that list. The companion table $Z_{\mathrm{explicit}}$ is a sparse, pattern-matched integer kernel on six Fin-4 indices (typical nonzero values $\pm 2,,4$).

The local module is one of several decide-chunks that certify $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ on a block of the $4^6$ index space. The module header states the goal directly: "m2Num = 8·explicitZ, chunk 13 (256 kernel decides)." Upstream, both $m_2^{\mathrm{num}}$ and $Z_{\mathrm{explicit}}$ are pure definitions in the kernel-certificate file; no analytic lemma is required beyond evaluating those defs at a concrete point.

One-line computational proof: by decide. Lean reduces both sides at the fixed indices $(3,1,0,2,3,0)$ to concrete integers (the fold for $m_2^{\mathrm{num}}$ versus the pattern match for $Z_{\mathrm{explicit}}$) and checks equality in the kernel. No lemmas are invoked beyond the two definitions.

This pointwise certificate is one tile in the exhaustive cover consumed by m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ and discharges the claim by fin_cases over all six indices. That universal identity is the algebraic backbone of the Regge-exact midpoint M2 TT identity in 4D gravity analysis inside the Recognition Science stack. It does not itself touch the forcing chain (T0–T8) or the Recognition Composition Law; it is pure discrete kernel bookkeeping needed before continuum or continuum-limit claims can be stated cleanly.