e_200213
plain-language theorem explainer
For the six-index slot (2,0,0,2,1,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one of the 256 kernel decides that assemble the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}2,\,b{=}0,\,c{=}0,\,d{=}2,\,i{=}1,\,j{=}3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit integer kernel: $N(2,0,0,2,1,3)=8\,Z(2,0,0,2,1,3)$.
background
In the Regge exact-midpoint 4D analysis, the numerator m2Num is defined by folding a fixed coupling list: start at 0 and add each contribution contrib t a b c d i j. The companion explicitZ is a closed-form integer table on six Fin 4 indices (sample entries include 4 on diagonal-like pairs and -2 on mixed pairs).
This module is chunk 8 of the 256 kernel decides that certify m2Num = 8·explicitZ pointwise. The local setting is pure finite enumeration over the 4D index cube; no continuum limit or dynamical hypothesis is in play here.
Upstream, only the two definitions are needed: the fold that builds the numerator and the pattern-matched integer table that supplies the right-hand side.
proof idea
One-line kernel proof: by decide. Lean evaluates both sides at the concrete six-tuple (2,0,0,2,1,3) and checks integer equality. No lemmas are invoked beyond the definitions of the fold and the explicit table.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for all six Fin 4 indices by exhaustive fin_cases. Each chunk theorem such as this one discharges one concrete cell so the global forall can be built without a single monolithic decide.
In the gravity stack this identity is bookkeeping for the Regge midpoint TT kernel in 4D: it guarantees the folded coupling numerator is exactly eight times the closed-form Z table, a normalization step before continuum or continuum-limit arguments. It does not itself touch the forcing chain (T0–T8) or the Recognition Composition Law; it is infrastructure inside the discrete gravity analysis.
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