e_103011

In the 4D Regge midpoint analysis, the mass-squared numerator is the integer obtained by folding a fixed coupling list: each list entry contributes an integer term at a sextuple of indices in ${0,1,2,3}$, and the fold sums those contributions. The companion object is a sparse case-split table on the same sextuples, returning small integers (for example $4$ or $-2$ on the matched clauses shown upstream).

This module is chunk 4 of a kernel split whose sole job is to certify numerator $= 8\cdot Z$ at each concrete multi-index by computation. The ambient setting is the exact midpoint M2-TT identity in four dimensions; the chunks discharge the numerator side before assembly quantifies over all sextuples.

One-line computational proof. The tactic decide evaluates both sides as closed integer expressions: the left-hand side is the fold of the coupling contributions at $(1,0,3,0,1,1)$; the right-hand side is eight times the matched clause of the explicit kernel table at that same multi-index. Equality of the two integers closes the goal. No intermediate lemmas are invoked beyond the two definitions.

This cell is one of the pointwise facts consumed by the assembly theorem that states the numerator identity for every sextuple in ${0,1,2,3}^6$ by exhausting cases. That global equality is the numerator half of the exact midpoint Regge M2-TT identity in the 4D gravity analysis inside Recognition Science. The constant factor eight is the discrete normalization that matches the folded coupling sum to the explicit kernel table. Chunkwise decides of this form are what let the assembly close without leaving arithmetic obligations open.