e_313202

This module is one chunk of the 4D Regge exact-midpoint analysis: it certifies, case by case, that the discrete mass numerator built from the coupling list equals eight times a closed-form integer kernel. The ambient setting is discrete gravity on a 4-index lattice (each index in $\mathrm{Fin},4$), where midpoint TT identities reduce curvature/mass bookkeeping to finite integer sums.

The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each tuple $(a,b,c,d,i,j)$. The comparison object $\mathrm{explicitZ}$ is a pattern-matched integer table on the same six indices (typical nonzero entries are $\pm 2,,4$). The module goal is the pointwise identity $m_2^{\mathrm{num}}=8\cdot\mathrm{explicitZ}$ over the full $4^6$ grid, split into decide-sized chunks.

One-line kernel proof: decide evaluates both sides as concrete integers for the fixed indices $(3,1,3,2,0,2)$ and checks equality. No lemmas beyond the definitions of the numerator fold and the explicit kernel table are required; the checker closes the arithmetic.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for all six $\mathrm{Fin},4$ indices by exhaustive fin_cases and invokes each chunk equality such as this one. That global identity is the algebraic core of the Regge exact-midpoint M2–TT certification in 4D discrete gravity: it replaces a summed coupling expansion by an eightfold multiple of a sparse explicit kernel, making later curvature and mass identities purely combinatorial.

Within Recognition Science gravity work, this is infrastructure rather than a forcing-chain landmark (T0–T8). It supports exact discrete checks that continuum limits and effective Newtonian bookkeeping can later quote without residual summation error.