e_200031

This module is chunk 8 of a 256-cell kernel certification that the folded numerator coupling equals eight times an explicit integer table on every 4D multi-index. The ambient setting is the exact midpoint M2 TT identity for Regge gravity in four dimensions.

The numerator side m2Num is defined by folding a fixed coupling list: it sums contrib t a b c d i j over every term t in couplingZList, yielding an Int. The right-hand side is explicitZ, a total function Fin 4^6 → Int given by a finite pattern match (e.g. diagonal blocks map to 4, certain off-diagonal pairs to -2, and all unmatched tuples implicitly to the remaining cases in the full definition).

The six arguments live in Fin 4, so there are exactly $4^6 = 4096$ conceivable tuples; the certification strategy reduces to a much smaller effective kernel and discharges it by exhaustive decide cells, of which this declaration is one.

One-line computational proof: by decide. Both sides are closed integer expressions once the six Fin 4 indices are fixed to the concrete values 2,0,0,0,3,1. Lean reduces m2Num through the fold over couplingZList and explicitZ through its pattern match, then checks integer equality by the kernel decision procedure. No lemmas are invoked beyond the definitions of m2Num and explicitZ.

Parent theorem m2Num_eq_eight_explicitZ assembles the universal identity $\forall a b c d i j,, m_2^{\mathrm{num}}=8,Z$ by nested fin_cases on all six indices; each leaf is one of these chunk theorems. Without the pointwise cells the assemble step has nothing to close on.

In the Recognition gravity stack this identity is the algebraic core of the exact midpoint M2 TT relation in 4D Regge calculus: it converts a summed coupling expansion into a sparse explicit kernel, which is what downstream curvature and continuum-limit arguments consume. It sits inside the Gravity analysis layer rather than the T0–T8 forcing chain, but it is the kind of fully discharged integer identity the framework demands before geometric claims are treated as proved.