e_311322

This module is chunk 13 of a case-split campaign proving that the 4D Regge midpoint numerator equals eight times an explicit integer kernel on every multi-index in $(\mathrm{Fin},4)^6$. The local slogan is "$m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{explicit}}$, chunk 13 (256 kernel decides)."

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: it sums a contribution functional over that list and returns an integer. The comparison object $Z_{\mathrm{explicit}}$ is a total function $(\mathrm{Fin},4)^6\to\mathbb{Z}$ given by an exhaustive pattern match (sample clauses include $(0,0,1,1,2,2)\mapsto 4$ and $(0,0,1,2,1,2)\mapsto -2$).

Both sides are pure integer expressions once the six Fin-4 indices are fixed, so each cell is a closed arithmetic identity.

One-line proof by decide. After the six concrete Fin-4 literals are substituted, both the fold that defines the numerator and the pattern match that defines the explicit kernel reduce to ground integers; the kernel decision procedure checks equality. No lemmas beyond the two definitions are invoked.

The parent theorem is the universal statement that $m_2^{\mathrm{num}}(a,b,c,d,i,j)=8,Z_{\mathrm{explicit}}(a,b,c,d,i,j)$ for all six indices in Fin 4. That assembler runs nested fin_cases over the six coordinates and discharges each leaf by a chunk theorem of this form.

In the broader Gravity analysis stack, the identity is the algebraic core of the exact 4D Regge midpoint M2TT certificate: once every cell matches, the midpoint discrete curvature expression collapses to the explicit kernel used in downstream curvature and continuum-limit arguments. The result is pure discrete GR bookkeeping inside Recognition Science gravity; it does not itself invoke the T0–T8 forcing chain or the J-cost, but it supplies a verified numerator identity those continuum comparisons rely on.