e_021003

In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $N(a,b,c,d,i,j)$ is obtained by folding a fixed coupling list and summing each term's contribution at those indices. The closed form $Z$ is an explicit pattern-matched integer table (typical entries $\pm 2,,4$, and zero off the listed patterns).

The module is chunk 2 of a 256-cell decide grid that checks $N=8Z$ pointwise. The present cell fixes the concrete sextuple $(0,2,1,0,0,3)$. Upstream, $N$ is the fold over couplingZList and $Z$ is the tabulated kernel; neither is proved here, only evaluated.

One-line decide on a ground equality of integers. Lean reduces both sides: the fold defining the numerator at $(0,2,1,0,0,3)$ and the pattern match for the explicit kernel at the same indices, then checks $N=8Z$ by computation. No lemmas are invoked beyond kernel evaluation.

Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:\mathrm{Fin},4,; N=8Z$ by exhaustive fin_cases over the six indices. Each chunk theorem such as this one discharges one concrete cell so the universal identity is only a case split, not a fresh algebraic argument.

In the gravity stack this identity is the algebraic backbone of the exact midpoint M2TT relation in 4D Regge calculus: once $N=8Z$ holds everywhere, midpoint curvature contributions collapse to the tabulated kernel and can be matched to continuum limits. It is bookkeeping inside the Gravity domain, not a T0–T8 forcing step, but it is required scaffolding for any claim that the discrete midpoint action reproduces the intended continuum factor.