e_002002

In the 4D Regge midpoint M2TT analysis, two integer kernels on six indices in $\mathrm{Fin},4$ are compared pointwise. The numerator is defined by folding a fixed coupling list and summing a contribution at each tuple. The comparison table is a sparse case-split map into small integers (entries such as $4$ and $-2$ on selected patterns).

This module is chunk 0 of the 256-case certification that the folded numerator equals eight times the table everywhere. The local setting is pure finite enumeration: every index lives in a four-element type, so each cell is a concrete integer equality.

Upstream, the table and the fold are the only ingredients; no continuum geometry enters the cell check.

Both sides are closed integer terms once the six indices are fixed. The tactic decide evaluates the fold on the left and the table lookup on the right, then confirms the integers match. No named algebraic lemma is applied; the goal is discharged by the kernel decision procedure on Int equality.

The parent theorem asserts the numerator-table identity for every six-tuple in $\mathrm{Fin},4$ by exhaustive case split on each index. Each generated goal is one cell of this form; the present declaration closes the cell $(0,0,2,0,0,2)$. Completing all cells certifies that the folded coupling numerator is exactly eight times the closed-form kernel used in the Regge exact midpoint M2TT identity in four dimensions. That identity sits in the gravity analysis stack, where discrete curvature bookkeeping is checked before any continuum or phenomenological claim.