e_001020
plain-language theorem explainer
For the six Fin-4 indices (0,0,1,0,2,0), the midpoint Regge numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one cell of the 256-case kernel that assembles the global identity m2Num = 8·explicitZ. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}0,b{=}0,c{=}1,d{=}0,i{=}2,j{=}0$ in $\mathrm{Fin}\,4$, the summed midpoint coupling numerator equals eight times the explicit integer kernel: $N(0,0,1,0,2,0)=8\,Z(0,0,1,0,2,0)$.
background
In the 4D Regge midpoint analysis, the numerator m2Num accumulates integer contributions from a fixed coupling list over six Fin-4 indices (two edge pairs and a midpoint pair). The companion explicitZ is a closed-form integer table on the same index domain, with sparse nonzero entries such as $4$ on matched diagonal pairs and $-2$ on selected off-diagonal patterns.
This module is chunk 0 of the 256 kernel decides that discharge, index-by-index, the equality of the folded numerator to eight times that table. The local setting is pure finite arithmetic: no continuum limit and no metric hypotheses beyond the discrete index set.
Upstream, both sides are definitions in the kernel certificate module; the present theorem only evaluates them at one concrete sextuple.
proof idea
One-line decide proof. Lean reduces both m2Num 0 0 1 0 2 0 (the fold of contrib over couplingZList) and 8 * explicitZ 0 0 1 0 2 0 to concrete integers and checks equality by kernel computation. No lemmas are invoked beyond definitional unfolding.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states the identity for all six Fin-4 indices by exhaustive fin_cases and dispatches each cell to a chunk theorem of this form. That global identity is the algebraic backbone of the exact midpoint M2 TT identity in the 4D Regge analysis used on the gravity side of Recognition Science.
Within the framework it is bookkeeping, not a forcing-chain step: it certifies that the discrete curvature numerator matches the closed kernel before continuum or phenomenological claims are attached. Closing all 256 cells removes scaffolding from the midpoint identity path.
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