e_122222
plain-language theorem explainer
Pointwise identity: the folded Regge coupling numerator at multi-index (1,2,2,2,2,2) equals eight times the explicit integer table at that same sextuple. Gravity analysts cite it as one of the 256 kernel cells that assemble the global m2Num = 8·explicitZ statement. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,2,2,2,2,2)$ in $(\mathbb{F}_4)^6$, the folded coupling numerator equals eight times the closed-form integer table: $N(1,2,2,2,2,2)=8\,Z(1,2,2,2,2,2)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ 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 table $Z$ is an explicit pattern-matched integer function on the same domain (typical values $\pm 2,,4$, and zero off the matched patterns).
The module is chunk 6 of a 256-cell kernel that discharges $N=8Z$ one multi-index at a time. The ambient claim is the exact midpoint M2 TT identity in four dimensions: after all cells are checked, the folded numerator is identically eight times the closed form.
Upstream, $N$ is the fold of couplingZList contributions and $Z$ is the hand-written case table; neither depends on this cell.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete Int values at the fixed sextuple $(1,2,2,2,2,2)$, so the equality is a closed integer computation with no lemmas beyond evaluation of the fold and the pattern table.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j:,N=8Z$ and is proved by exhausting all $\mathrm{Fin},4$ cases. Each chunk cell such as this one supplies one of those 256 ground instances.
In the Recognition gravity stack this identity is bookkeeping for the exact midpoint M2 TT relation in 4D Regge calculus: it certifies that the folded coupling numerator matches the closed-form table used downstream in curvature and mass-ladder comparisons. It does not itself invoke the forcing chain (T5–T8) or the Recognition Composition Law; it is pure discrete linear-algebra certification inside the gravity analysis layer.
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