e_211103
plain-language theorem explainer
One kernel case of the 4D Regge midpoint identity: the folded coupling numerator at index sextuple (2,1,1,1,0,3) equals eight times the explicit integer table entry. Gravity analysts cite it only as a brick in the exhaustive Fin-4 assembly. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(2,1,1,1,0,3)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint M2 TT analysis, two integer-valued maps on sextuples of Fin 4 indices are compared. The numerator m2Num folds a fixed coupling list, summing a local contribution at each term. The companion explicitZ is a closed-form integer table on the same domain (sample entries include $4$, $-2$, and so on for distinguished index patterns).
The module is chunk 9 of a 256-case kernel certification that m2Num = 8 · explicitZ pointwise. Each chunk theorem pins one concrete sextuple so the global identity can be assembled by exhaustive case split on Fin 4.
proof idea
One-line computational proof: decide evaluates both sides at the fixed indices $(2,1,1,1,0,3)$ and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit table.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhausting cases. That global equality is the certified algebraic core of the Regge-exact midpoint M2 TT identity in four dimensions inside the Gravity analysis stack. It is bookkeeping, not a new physical law: it closes one of the 256 kernel cells needed before continuum or continuum-limit claims can rest on a fully checked discrete identity.
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