e_031002

In the 4D Regge midpoint M2/TT analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: each term contributes an integer via a local contrib, and the fold starts at 0. The comparison target is an explicit piecewise kernel $Z$ on the same six indices, tabulated by pattern (e.g. diagonal blocks map to 4, certain off-diagonal swaps to $-2$).

This module is chunk 3 of a 256-way case split that certifies $m_2^{\mathrm{num}}=8Z$ pointwise. The local setting is pure finite enumeration: no continuum limit, no metric signature choice beyond the discrete kernel data already fixed upstream.

One-line computational proof: decide evaluates both sides on the concrete six-tuple $(0,3,1,0,0,2)$. The left side runs the fold that defines the numerator; the right side looks up (or reduces to) the matching clause of the explicit kernel and multiplies by 8. Equality of the resulting integers is discharged by the kernel decision procedure.

Parent theorem m2Num_eq_eight_explicitZ assembles every chunk into the universal statement $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8Z$, proved by exhaustive fin_cases on the six indices. Each chunk lemma such as this one supplies one concrete cell of that table. In the broader gravity stack, the identity is the algebraic certificate that the midpoint M2 numerator factors through the explicit Z kernel, a step toward exact discrete curvature bookkeeping in the Recognition Science Regge analysis. It does not itself touch continuum GR limits or the T0–T8 forcing chain; it is pure finite-kernel bookkeeping.