e_200322

In the 4D Regge exact-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: start at 0 and add each contribution term at the given indices. The comparison target $Z_{\mathrm{explicit}}$ is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and defaults).

The module is chunk 8 of a 256-way split of the kernel: each chunk theorem pins $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ at one concrete multi-index. The factor 8 is the global normalization relating the folded sum to the closed form.

Upstream, only the two definitions are needed; no analytic lemma is invoked. The ambient goal is the pointwise identity used in the M2 TT (midpoint, two-tensor) sector of the discrete gravity kernel.

One-line computational proof: by decide. Lean evaluates both sides as concrete integers (the fold for $m_2^{\mathrm{num}}$ at $(2,0,0,3,2,2)$ and eight times the matched $Z_{\mathrm{explicit}}$ entry) and checks equality in the kernel. No rewriting, induction, or named lemma beyond the two definitions is required.

Parent assembly theorem m2Num_eq_eight_explicitZ states the identity for every $a,b,c,d,i,j:\mathrm{Fin},4$ and proves it by exhaustive fin_cases. Each chunk such as this one supplies a single case, so the 4-gravity kernel identity is discharged as a finite conjunction of decide-able equalities rather than a symbolic closed-form argument.

In the Recognition gravity stack this underwrites the exact midpoint evaluation of the M2 TT block: once numerator and explicit table agree everywhere (up to the factor 8), downstream curvature and continuum-limit comparisons can quote a fully certified discrete identity. It is bookkeeping inside the gravity analysis, not a forcing-chain (T0–T8) step, but it removes a large finite proof obligation from the Regge side of the ledger.