e_310331
plain-language theorem explainer
For the six-index tuple (3,1,0,3,3,1) in (Fin 4)^6, the folded numerator m2Num equals eight times the closed-form kernel explicitZ. Gravity analysts cite it as one atomic cell in the 4D Regge midpoint M2 identity. The proof is a single decide, discharging the equality by kernel evaluation.
Claim. For indices $a{=}3,\,b{=}1,\,c{=}0,\,d{=}3,\,i{=}3,\,j{=}1$ in $\mathbb{F}_4$, the folded coupling numerator equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(3,1,0,3,3,1)=8\,Z_{\mathrm{ex}}(3,1,0,3,3,1)$.
background
This module is chunk 13 of a 256-cell kernel certification that the 4D Regge exact-midpoint numerator equals eight times a closed integer table. The ambient setting is discrete gravity analysis: index sextuples range over Fin 4, matching the four spacetime directions of the midpoint identity.
The numerator m2Num is defined by folding a contribution map over a fixed coupling list, accumulating an integer. The companion explicitZ is a pattern-matched table Fin 4^6 → ℤ giving the expected closed values (e.g. ±2, 4 on selected support). The claim is the pointwise identity of these two integers at one concrete sextuple.
Upstream, both definitions live in the KernelCert module; this chunk only evaluates them at (3,1,0,3,3,1).
proof idea
One-line computational discharge: by decide. Lean reduces both sides (the fold defining the numerator, and the pattern match defining the explicit kernel) to concrete integers and checks equality. No lemmas are invoked beyond the decidable equality instance on ℤ.
why it matters
The parent theorem m2Num_eq_eight_explicitZ asserts the identity for every sextuple in (Fin 4)^6. Its proof is a nested fin_cases tree that bottoms out at atomic cells such as this one. Without the chunk equalities, the universal statement does not close.
In the Recognition gravity stack this certifies that the discrete midpoint M2 numerator is exactly eight times the tabulated kernel, a structural factor that keeps the 4D Regge analysis aligned with the forced D = 3 spatial plus time indexing. It is pure bookkeeping infrastructure, not a new physical law, but the assembly step depends on every cell.
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