e_020110

In the 4D Regge midpoint analysis, two integer-valued kernels on six indices in $\mathrm{Fin},4$ are compared. The folded numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the sum of local contributions obtained by folding a fixed coupling list. The explicit table $Z_{\mathrm{expl}}$ is a closed-form pattern-matched integer function on the same six indices (nonzero only on a sparse set of index patterns, with values such as $\pm 2,\pm 4$).

This module is chunk 2 of the 256-point kernel certification that $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ holds at every sextuple. The local setting is pure finite enumeration: each chunk theorem pins one concrete multi-index so the global assembly can exhaust $\mathrm{Fin},4^6$ without re-deriving the fold.

One-line computational proof: decide evaluates both sides at the fixed indices $(0,2,0,1,1,0)$. The left side runs the fold that defines the numerator; the right side looks up (or defaults) the explicit table and multiplies by 8. Equality of the resulting integers is discharged by the kernel decision procedure. No lemmas beyond the two definitions are invoked.

Feeds the assembly theorem $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ for all six $\mathrm{Fin},4$ arguments, which case-splits on every index and consumes these pointwise chunk facts. That global identity is the algebraic core of the Regge exact-midpoint M2TT certification in 4D: it replaces a summed coupling expression by a sparse closed form, enabling exact (not approximate) midpoint identities in the discrete gravity sector.

Within Recognition Science gravity work, such kernel equalities underwrite curvature and mass-side discrete checks that sit downstream of the forcing chain's $D=3$ and eight-tick structure; they do not themselves invoke $J$, $\varphi$, or the RCL, but they keep the 4D Regge bookkeeping exact.