e_013100

In the 4D Regge midpoint analysis, the M2–TT identity is checked by comparing a folded numerator against a closed-form integer kernel. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at 0 and add each term's contribution at the six indices. The comparison target is $\mathrm{explicitZ}$, a total function $\mathrm{Fin},4^6\to\mathbb{Z}$ given by a finite pattern of integer values (e.g. $4$, $-2$, and other sparse entries).

This module is chunk 1 of that case split: it discharges individual hexuples by kernel decision so the assembler can recombine them. The local claim is exactly one such hexuple equality $m_2^{\mathrm{num}}=8\cdot Z$.

One-line proof by decide. Both sides reduce to concrete integers: the left via the fold definition of $m_2^{\mathrm{num}}$ over couplingZList, the right via the pattern-matching definition of explicitZ at $(0,1,3,1,0,0)$. Lean’s decision procedure checks the integer equality; no lemmas beyond the two defs are required.

The parent theorem m2Num_eq_eight_explicitZ asserts the identity for every hexuple in $\mathrm{Fin},4^6$ by exhaustive fin_cases. Each atomic e_* lemma such as this one supplies one branch of that case tree. Closing the full identity certifies that the folded Regge coupling numerator is exactly eight times the explicit Z kernel, which is the algebraic content of the midpoint M2–TT check in this 4D gravity analysis stack. It is bookkeeping infrastructure rather than a new physical law, but without the chunk lemmas the assembler cannot finish.