e_032000

In the 4D Regge midpoint analysis, two integer kernels on six Fin 4 indices are compared. The folded numerator $m_2^{\mathrm{num}}$ is the sum of all coupling contributions obtained by folding couplingZList with a local contrib term. The companion table $Z_{\mathrm{expl}}$ is an explicit case-split integer function on the same six indices (sample values include $4$, $-2$, and so on).

This module is chunk 3 of a 256-cell decide kernel whose sole job is to check $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ at concrete index tuples. The local setting is pure finite enumeration: no continuum limit and no variational argument, only integer equality at fixed multi-indices.

One-line computational proof: decide evaluates both sides at the concrete sextuple $(0,3,2,0,0,0)$ and closes the integer equality. No lemmas beyond the definitions of $m_2^{\mathrm{num}}$ and $Z_{\mathrm{expl}}$ are required; the fold and the case table reduce to numerals that Lean’s decision procedure compares.

Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j):(\mathrm{Fin},4)^6$ by exhausting all six indices with fin_cases. Each chunk cell such as this one discharges one concrete branch of that case tree.

In the broader Gravity track this certifies that the folded midpoint numerator is exactly eight times the closed-form kernel, a bookkeeping identity needed before continuum or continuum-limit statements about the Regge M2TT sector. It is scaffolding for the exact midpoint identity rather than a physical law by itself; the Recognition landmarks (T5–T8, RCL, $\varphi$-ladder) are not invoked here.