e_002102

In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list and summing a local contribution at each multi-index. The comparison target $Z_{\mathrm{expl}}$ is an explicit case table of small integers (entries such as $4$, $-2$, and so on).

The module is chunk 0 of a 256-way split of the kernel: each chunk discharges a block of concrete index tuples by decision procedure. The local claim is the single equality at $(0,0,2,1,0,2)$. Upstream, both sides are pure defs; no analytic lemma is required beyond evaluating those defs at fixed indices.

One-line computational proof: decide evaluates both sides at the concrete Fin-4 sextuple $(0,0,2,1,0,2)$ and checks integer equality. The left side reduces by unfolding the fold over the coupling list; the right side multiplies the matching explicitZ clause by eight. No lemmas beyond kernel evaluation are invoked.

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 exhaustive fin_cases. That global equality is the certified numerator half of the Regge exact midpoint M2 TT identity in 4D: it shows the folded coupling numerator is exactly eight times the hand-tabulated kernel, so the continuum or discrete TT check can quote a single closed form rather than a fold.

Within Recognition gravity analysis this is bookkeeping infrastructure, not a forcing-chain step (T0–T8). It closes one concrete cell of the 256-cell decide grid that makes the 4D kernel identity machine-checkable end to end.