e_311233

This module is chunk 13 of a 256-cell kernel certification that the Regge exact-midpoint numerator equals eight times an explicit integer table on $(\mathrm{Fin},4)^6$. The ambient setting is 4D discrete gravity analysis: six indices label pairs of edge/face directions in the midpoint TT sector.

The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add a contribution term for each coupling entry at the six indices. The comparison table $Z_{\mathrm{explicit}}$ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by exhaustive pattern match (typical nonzero values are $\pm 2,,4$).

The parent goal is the pointwise identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ for every six-tuple. Each chunk theorem discharges one concrete cell so the assembler can finish by finite case split.

One-line kernel proof: decide. Both sides reduce to closed integer expressions once the six Fin 4 arguments are literals (left via the fold over couplingZList and contrib; right via the pattern match in explicitZ), and the kernel checks equality of the resulting Int values.

Feeds the assembler theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ and proves it by six nested fin_cases sweeps. That global identity is the algebraic core of the Regge exact-midpoint M2 TT certification in 4D: it replaces a folded coupling sum by a sparse explicit table, making downstream curvature and mass-window arguments computable.

Within Recognition gravity analysis this is pure discrete kernel bookkeeping rather than a forcing-chain step (T0–T8), but it is required infrastructure for any claim that the midpoint TT sector matches the predicted eightfold normalization on the discrete lattice. Chunk 13 is one of many parallel cells; none is interesting alone, yet the assembler needs every cell closed with zero sorry.