e_011023

This module is one chunk of the 4D Regge exact-midpoint M2–TT identity certification. The local goal, stated in the module header, is to prove $N=8Z$ on a block of the $4^6$ index space by kernel decides.

The numerator $N=\mathrm{m2Num}$ 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 closed form $Z=\mathrm{explicitZ}$ is an integer-valued pattern match on the six $\mathrm{Fin},4$ indices (sample clauses include $Z(0,0,1,1,2,2)=4$ and $Z(0,0,1,2,1,2)=-2$).

Both definitions live in the kernel certificate module imported here. Downstream assembly will range over all index sextuples by fin_cases and invoke one such equality per cell.

One-line proof by decide. Both sides reduce to concrete integers once the six indices are fixed: the fold that defines the numerator evaluates fully, and the pattern match for the explicit kernel returns a fixed Int. Lean’s decision procedure checks equality of those integers. No lemmas beyond the two definitions are required.

Feeds the parent theorem m2Num_eq_eight_explicitZ, which asserts $\forall a,b,c,d,i,j,, N=8Z$ by exhaustive fin_cases on $\mathrm{Fin},4$. Each chunk theorem such as this one discharges one cell of that case tree.

In the Recognition gravity stack this identity is part of the exact midpoint analysis of the Regge M2–TT sector in four dimensions. It is bookkeeping infrastructure rather than a forcing-chain landmark (T5–T8), but it underwrites the certified algebraic reduction used when matching discrete curvature couplings to continuum limits. Closing all 256 decides in the chunk completes one assembly block toward the global equality.