e_032201

This module is chunk 3 of a 256-cell kernel certification that the 4D Regge midpoint M2TT numerator equals eight times a sparse explicit integer table. Indices run over $\mathrm{Fin},4$, matching the four discrete directions of the midpoint stencil.

The numerator $N(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each contribution term for the six indices. The explicit table $Z$ is a pattern-matched integer function on the same six indices, nonzero only on a short list of symmetry orbits (e.g. $4$ on diagonal pairs, $-2$ on certain swaps).

The local claim is one concrete instance of $N=8Z$. The parent assembly theorem will range over all $4^6$ tuples by fin_cases and invoke these cells.

One-line proof by decide. Both sides reduce to concrete Int values: the left by evaluating the fold of contrib over couplingZList at the fixed indices $(0,3,2,2,0,1)$, the right by looking up explicitZ (or defaulting to 0 if unmatched) and multiplying by 8. Lean’s kernel decision procedure checks integer equality; no lemmas are invoked.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases over $\mathrm{Fin},4$. That identity is the algebraic core of the Regge exact midpoint M2TT certification in 4D gravity analysis: it collapses a folded coupling sum to a sparse closed form, enabling exact discrete curvature bookkeeping without floating-point residue.

Within Recognition Science gravity work, such kernel equalities underwrite discrete geometric identities used when matching continuum limits and ladder normalizations. This cell is pure bookkeeping; it does not itself invoke $\phi$, the eight-tick octave, or $D=3$, but it is infrastructure those continuum comparisons rely on once the 4D stencil is certified.