e_021233

This module is chunk 2 of a 256-cell kernel certification that the Regge midpoint numerator equals eight times a closed-form table. Indices run over $\mathrm{Fin},4$, matching the four spacetime directions in the 4D discrete gravity setting.

The numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: it sums $\mathrm{contrib},t,a,b,c,d,i,j$ over every term $t$ in $\mathrm{couplingZList}$. The companion table $\mathrm{explicitZ}$ is a pattern-matched integer function on six $\mathrm{Fin},4$ arguments (sample clauses return $4$, $-2$, etc.).

The local claim is one concrete instance of the global identity $\mathrm{m2Num}=8\cdot\mathrm{explicitZ}$ that the assemble module will reassemble by exhaustive $\mathrm{fin_cases}$.

One-line proof by decide. Both sides reduce to concrete integers: the left-hand side evaluates the fold of m2Num at the fixed indices $(0,2,1,2,3,3)$; the right-hand side multiplies the matching explicitZ clause by eight. Lean’s kernel decision procedure checks integer equality, with no lemmas or rewriting required.

Feeds the parent theorem m2Num_eq_eight_explicitZ, which states the identity for all six $\mathrm{Fin},4$ indices and is proved by nested fin_cases that discharge each cell (including this one). That global equality is the algebraic backbone of the Regge exact midpoint M2TT identity in 4D discrete gravity analysis inside Recognition Science.

In the broader framework this sits in the gravity domain supporting continuum limits and curvature bookkeeping on the discrete side; it does not itself invoke the forcing chain (T0–T8), the J-cost, or the phi ladder, but supplies a certified numerator identity those continuum arguments rely on when matching Regge data to continuum Einstein–Hilbert structure.