e_020103

In the 4D Regge exact-midpoint analysis, two integer-valued maps on six $\mathbb{F}_4$ indices are compared. The M2 numerator is the fold of a fixed coupling list: each term contributes an integer depending on the six indices, summed from zero. The explicit $Z$ map is a closed lookup table of small integers (entries such as $4$, $-2$, and so on) on the same domain.

The local module is chunk 2 of a 256-cell kernel: each cell asserts numerator $= 8\cdot Z$ at one concrete multi-index. The parent assembly theorem then ranges over all six indices by exhaustive fin_cases and invokes these cells.

Upstream, explicitZ and m2Num are pure definitions (table vs fold); no analytic lemma is required beyond decidable integer arithmetic at fixed points.

One-line computational proof: by decide. Both sides reduce to concrete Int values at the fixed six-tuple $(0,2,0,1,0,3)$, and Lean’s kernel decision procedure checks equality. No lemmas beyond the definitions of the numerator fold and the explicit table are applied.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in $(\mathbb{F}_4)^6$ by casing on each coordinate and discharging each cell. That global equality is the certified algebraic core of the Regge exact-midpoint M2–TT identity in 4D gravity analysis inside the monolith.

Within Recognition Science gravity work, such kernel certificates keep the midpoint curvature bookkeeping exact (no floating residual) before continuum or phenomenological layers. This cell is scaffolding only in the weak sense that it is one tile of a finite cover; the claim itself is fully decided.