e_030132

The module is one chunk of the 4D Regge exact-midpoint $M_2$ TT-identity certification. The local claim is the pointwise identity $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$ on a block of the $4^6$ index space (chunk 3; 256 kernel decides).

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is the fold of a contribution map over a fixed coupling list: it accumulates integer weights for each coupling triple at the six Fin-4 slots. The explicit kernel $Z_{\mathrm{expl}}$ is a closed integer lookup on the same six indices (sample values include $4$, $-2$, and so on for the tabulated patterns).

Upstream, both maps live in the kernel-certificate module; this chunk only evaluates one concrete sextuple.

One-line computational proof: decide evaluates both sides at the literal indices $(0,3,0,1,3,2)$ and checks integer equality. No lemmas are invoked beyond the reducible definitions of the numerator fold and the explicit kernel table.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every $a,b,c,d,i,j:\mathrm{Fin},4$ by exhaustive fin_cases. Each chunk theorem closes one cell so the universal midpoint identity is a pure case split over certified kernel facts rather than a symbolic expansion.

In the Recognition gravity stack this is bookkeeping for the exact 4D Regge midpoint $M_2$ TT relation: once every cell matches $8Z_{\mathrm{expl}}$, the continuum-facing mass-squared kernel is pinned to the explicit integer table used downstream in curvature and coupling identities. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete kernel arithmetic inside the gravity analysis layer.