e_320301

In the 4D Regge midpoint analysis, the mass-squared numerator is assembled by folding a fixed coupling list: each term contributes an integer depending on six $\mathrm{Fin},4$ indices, and $m_2^{\mathrm{num}}$ is that fold starting from zero. The companion map $Z_{\mathrm{explicit}}$ is a closed-form integer table on the same six indices (nonzero only on a sparse set of patterns such as paired equal legs with values $\pm 2,4$).

This module is chunk 14 of the exhaustive certification that $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{explicit}}$ at every multi-index. The local setting is pure integer arithmetic on a finite domain: no continuum limit or variational argument is invoked inside the chunk.

Both sides evaluate to concrete integers once the six indices are fixed. The tactic decide runs the kernel decision procedure on the resulting closed integer equality and closes the goal. No lemmas beyond the definitions of the numerator fold and the explicit table are required.

The parent assembly theorem m2Num_eq_eight_explicitZ states the identity for all six indices and discharges it by nested fin_cases; each leaf is one of these chunk theorems. Establishing the factor-of-eight match between the folded coupling numerator and the explicit kernel is the algebraic core of the Regge exact midpoint M2–TT identity in 4D, which sits in the gravity side of the Recognition stack (discrete curvature bookkeeping compatible with the eight-tick and $D=3$ forcing landmarks). Without the pointwise cells, the universal statement cannot be certified computationally.