e_033123

This module is chunk 3 of a 256-way kernel certification that the midpoint numerator of the 4D Regge M2TT identity coincides with eight times a closed-form table. Indices run over $\mathrm{Fin},4$, matching the four discrete directions of the 4-simplex edge data.

The numerator $\mathrm{m2Num}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib},t,a,b,c,d,i,j$. The comparison target $\mathrm{explicitZ}$ is an integer-valued pattern match on the six indices (sample clauses return $4$, $-2$, and so on). The local claim is only the equality at one concrete sextuple; the full $\forall$ statement is assembled downstream by exhaustive $\mathrm{fin_cases}$.

One-line proof by decide. Both sides reduce to concrete Int values: the left by evaluating the fold of contrib over couplingZList at $(0,3,3,1,2,3)$, the right by multiplying the matched explicitZ clause by $8$. No lemmas are invoked beyond kernel computation of those integers.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,;\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ by nested fin_cases over all six $\mathrm{Fin},4$ arguments. Each chunk equality such as this one discharges one leaf of that case tree.

In the Recognition gravity stack this identity is part of the exact midpoint analysis of the 4D Regge M2TT kernel: once numerator and explicit table agree everywhere, the continuum-limit and curvature-matching arguments can quote a fully certified algebraic identity rather than a numerical check. It is pure discrete linear algebra on the coupling table; it does not itself invoke the forcing chain (T5–T8) or the J-cost, but it underwrites the gravity side that those foundations later calibrate.