e_123323

This module is chunk 6 of a 256-cell kernel certification that the folded numerator of the 4D Regge midpoint M2TT identity equals eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$ (four spacetime directions).

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at those six indices. The explicit kernel $Z_{\mathrm{expl}}$ is a pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and so on).

The local claim is one concrete cell of the identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$, discharged by kernel decision rather than by symbolic algebra on the fold.

One-line proof by decide. Both sides reduce to concrete integers once the six Fin-4 indices are fixed: the left-hand side evaluates the fold of couplingZList contributions at $(1,2,3,3,2,3)$; the right-hand side looks up explicitZ at the same tuple and multiplies by 8. Lean’s decision procedure checks integer equality.

Parent theorem m2Num_eq_eight_explicitZ assembles the full universal statement $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ by exhaustive fin_cases on all six indices; each leaf is one of these chunk cells. Without the cell-wise equalities, the assemble step has nothing to close on.

In the gravity analysis stack this certifies that the midpoint M2TT numerator in 4D Regge calculus matches the explicit kernel used downstream, a bookkeeping identity rather than a new dynamical law. It sits inside the broader RS gravity pipeline that feeds curvature and mass-ladder work, but does not itself invoke $\phi$, the eight-tick octave, or the forcing chain T0–T8.