e_211303
plain-language theorem explainer
For the six-index tuple (2,1,1,3,0,3) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite it as one cell of the 256-point Regge midpoint identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a{=}2,\,b{=}1,\,c{=}1,\,d{=}3,\,i{=}0,\,j{=}3$ in $\mathrm{Fin}\,4$, the folded coupling numerator equals eight times the explicit kernel integer: $N(2,1,1,3,0,3)=8\,Z(2,1,1,3,0,3)$.
background
In the Regge exact-midpoint analysis for 4D gravity, two integer-valued kernels on six Fin-4 indices are compared. The numerator $N=\mathrm{m2Num}$ is obtained by folding a fixed coupling list and summing local contributions at $(a,b,c,d,i,j)$. The closed form $Z=\mathrm{explicitZ}$ is a piecewise integer table on the same six indices (typical nonzero values $\pm 2,,4$).
The module is chunk 9 of a 256-cell decide grid that exhausts every index combination. The local claim is the scalar identity $N=8Z$ at one fixed tuple. Upstream definitions supply only the table and the fold; no analytic closed form is assumed beyond the tabulated cases.
proof idea
One-line kernel proof: by decide. Lean evaluates both sides as concrete Int values (the fold of couplingZList versus the matching clause of explicitZ) and discharges equality by computation. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the universal statement m2Num_eq_eight_explicitZ, which asserts $N=8Z$ for every six-tuple by fin_cases on all six indices and routes each cell to its chunk theorem. That identity is the algebraic core of the Regge exact-midpoint $M_2$ TT certificate in 4D. Within Recognition gravity it certifies that the discrete curvature numerator matches the explicit kernel used downstream in the midpoint mass and defect analysis. It is pure bookkeeping scaffolding: one of 256 identical decide cells, not a new physical law.
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