e_133003
plain-language theorem explainer
Pointwise identity: the folded Regge midpoint numerator at index sextuple (1,3,3,0,0,3) equals eight times the explicit integer table entry. Gravity analysts cite it as one cell of the 256-kernel certification that m2Num = 8·explicitZ on (Fin 4)^6. Proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,3,3,0,0,3)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit closed-form integer: $N(1,3,3,0,0,3)=8\,Z(1,3,3,0,0,3)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued maps on sextuples of Fin 4 indices are compared. The numerator m2Num is obtained by folding a fixed coupling list: it sums a local contribution at each coupling triple against the six free indices. The companion explicitZ is a sparse case table of small integers (entries such as 4, -2, and so on) meant to be the closed form of that sum divided by eight.
This module is chunk 7 of the 256-cell kernel certification announced in the module doc: every concrete sextuple must satisfy the scalar identity numerator = 8 · table. The present declaration treats only the cell (1,3,3,0,0,3).
proof idea
One-line kernel proof: decide evaluates both sides as concrete Int values (the fold that defines the numerator, and the matching clause or default of the explicit table) and discharges the resulting numeral equality. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in (Fin 4)^6 by exhaustive fin_cases. Each chunk cell such as this one is a certified brick in that cover; together they turn the sparse table explicitZ into a proven closed form for the folded Regge midpoint numerator. In the broader gravity stack this underwrites exact midpoint identities used in the discrete curvature analysis, rather than a continuum forcing step (T0–T8).
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