e_011122

This module is chunk 1 of a 256-cell kernel certification that the 4D Regge exact-midpoint M2TT numerator agrees with an explicit closed form, pointwise on $\mathrm{Fin},4^6$.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: each term contributes an integer via a local contrib map, and the fold starts at 0. The comparison target $Z_{\mathrm{expl}}$ is a pattern-matched integer table on the same six indices (sample values include $4$, $-2$, and other small integers on the diagonal and off-diagonal patterns).

The local claim is one concrete cell of the identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$. Downstream assembly will quantify over all six indices by exhaustive fin_cases.

One-line kernel proof: decide evaluates both sides as concrete integers (the fold for m2Num at (0,1,1,1,2,2) and eight times the matched explicitZ entry) and discharges equality by computation. No lemmas beyond the two definitions are invoked.

Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and proves it by six nested fin_cases, each leaf a chunk theorem of this form. That global identity is the certified algebraic core of the 4D Regge exact-midpoint M2TT analysis in the Gravity domain: it replaces a summed coupling expression by a sparse explicit integer kernel, enabling exact midpoint identities without residual symbolic sums. Within Recognition Science gravity work this is bookkeeping infrastructure, not a forcing-chain step (T0–T8), but it locks the discrete curvature/numerator side used in continuum-limit comparisons.