e_203022
plain-language theorem explainer
For the six-index slot (2,0,3,0,2,2) on Fin 4, the folded numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of the 256 kernel decides. The proof is a single native decide on concrete integers.
Claim. For indices $a{=}2$, $b{=}0$, $c{=}3$, $d{=}0$, $i{=}2$, $j{=}2$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The folded numerator m2Num sums contrib terms over couplingZList. The companion explicitZ is a pattern-matched table of small integers (entries such as 4, -2, and so on) that is meant to be the closed form of that sum, up to a universal factor of 8.
This module is chunk 8 of the 256 kernel decides that discharge the pointwise identity m2Num = 8·explicitZ. The local setting is pure finite enumeration: every index lives in Fin 4, so each concrete sextuple is a closed integer computation.
Upstream, m2Num is the foldl of contrib over couplingZList, and explicitZ is the case table that the identity claims to match.
proof idea
One-line computational proof: by decide. Lean evaluates both sides at the concrete Fin-4 sextuple (2,0,3,0,2,2) and checks integer equality. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Parent theorem m2Num_eq_eight_explicitZ in the assemble module states the universal claim ∀(a b c d i j : Fin 4), m2Num = 8·explicitZ, proved by nested fin_cases. Each fin_cases leaf is one of these chunk theorems; e_203022 is the leaf for (2,0,3,0,2,2).
In the broader gravity stack this identity is the algebraic certificate that the midpoint M2TT numerator collapses to the explicit kernel used in the Regge exact-midpoint analysis. It is bookkeeping, not a new physical law, but without the full 256-leaf cover the assemble theorem does not close.
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