e_020321

In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at 0 and add each contribution term at the given indices. The comparison target $Z_{\mathrm{explicit}}$ is a sparse pattern table on the same six indices (nonzero only on a short list of match arms, e.g. values $\pm 2,4$).

This module is chunk 2 of the 256-cell kernel certification that $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ pointwise. The local setting is pure finite enumeration: every cell is an equality of two concrete integers obtained from those defs.

One-line computational proof: decide evaluates both sides of the equality at the concrete indices $(0,2,0,3,2,1)$. The left side runs the fold that defines the numerator; the right side multiplies the table lookup for the explicit kernel by 8. No lemmas are invoked beyond the kernel's decision procedure on closed integer terms.

The parent theorem is the full pointwise identity $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$, assembled by exhaustive fin_cases on all six indices. Each chunk theorem such as this one discharges one cell so the assembler can finish without re-deciding the whole $4^6$ grid in one tactic block.

In the gravity stack this identity is the algebraic core of the Regge exact midpoint M2–TT certificate in 4D: once numerator and explicit kernel match up to the universal factor 8, downstream curvature and deficit bookkeeping can treat the closed form as authoritative. It does not itself touch the T0–T8 forcing chain or the Recognition Composition Law; it is infrastructure inside the discrete gravity analysis layer.