e_020123

This module is chunk 2 of a 256-way kernel certification that the four-dimensional Regge midpoint $M_2$ numerator agrees with a closed integer table. Indices run over $\mathrm{Fin},4$, i.e. the discrete 4-label set used for simplex edge/face bookkeeping in the 4D Regge complex.

The numerator $N=\texttt{m2Num}$ is defined by folding a fixed coupling list: it accumulates every local contribution $\mathrm{contrib}(t;a,b,c,d,i,j)$ and returns an integer. The comparison table $Z=\texttt{explicitZ}$ is a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by finitely many pattern-matched constants (e.g. $4$, $-2$, and the remaining default).

The local claim is only the single cell $(0,2,0,1,2,3)$ of the identity $N=8Z$. Upstream, both $N$ and $Z$ live in the kernel-certificate module; downstream assembly will quantify over all six indices.

One-line computational proof: decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are substituted, because $N$ is a finite fold of integer contributions and $Z$ is a closed pattern match. Lean’s kernel decision procedure checks equality of those integers; no algebraic lemma is invoked.

The parent theorem is the universal assembly statement: for all $a,b,c,d,i,j:\mathrm{Fin},4$, $N(a,b,c,d,i,j)=8,Z(a,b,c,d,i,j)$. That proof runs nested fin_cases on the six indices and discharges each leaf by a chunk theorem of this form. Without the full 256-cell cover, the midpoint $M_2$ TT-identity certificate in 4D Regge analysis does not close.

In the broader gravity stack this is bookkeeping, not a new physical law: it certifies that the discrete curvature/coupling numerator used in the Regge midpoint scheme matches its explicit integer normal form, so later continuum or continuum-limit arguments can quote a single closed table instead of a fold. It does not itself touch the T0–T8 forcing chain, $\phi$-ladder masses, or the RCL; it is infrastructure under the discrete gravity side of the monolith.