e_331002

In the 4D Regge exact-midpoint analysis, two integer-valued kernels on $(\mathbb{F}_4)^6$ are compared. The folded numerator sums coupling contributions over a fixed list:

$$m_2^{\mathrm{num}}(a,b,c,d,i,j)=\sum_{t\in\mathrm{couplingZList}}\mathrm{contrib}(t;a,b,c,d,i,j).$$

The explicit kernel $Z$ is a closed-form case table on the same six indices (sample entries include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$).

This module is chunk 15 of the 256-point kernel certification that $m_2^{\mathrm{num}}=8Z$ everywhere. The local setting is pure finite enumeration: every coordinate is a Fin 4, so each sextuple is a concrete integer computation.

One-line computational discharge: by decide. Both sides reduce to concrete Int values for the fixed indices $(3,3,1,0,0,2)$, and Lean’s decision procedure checks equality. No algebraic lemmas are invoked; the proof relies only on the definitions of the folded numerator and the explicit kernel table.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states

$$\forall a,b,c,d,i,j:\mathbb{F}_4,\quad m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$$

and proves it by exhaustive fin_cases on all six indices. Each chunk theorem such as this one closes one branch of that case split. The identity is part of the Gravity-domain certification that the midpoint M2TT kernel in 4D Regge calculus matches its explicit closed form (scaled by eight), a computational step toward the continuum limit used in the RS gravity sector. It does not itself invoke the forcing chain (T0–T8) or the Recognition Composition Law; it is pure kernel arithmetic supporting that larger gravity analysis.