e_212230
plain-language theorem explainer
For the six-index tuple (2,1,2,2,3,0) in (Fin 4)^6, the folded numerator m2Num equals eight times the closed-form kernel explicitZ. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of the 256 kernel cells. The proof is a single kernel decide on integer equality.
Claim. For indices $a{=}2,\,b{=}1,\,c{=}2,\,d{=}2,\,i{=}3,\,j{=}0$ in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a coupling list: it sums contribution terms contrib over couplingZList at the given multi-index. The companion map explicitZ is a closed-form case table on $(\mathrm{Fin},4)^6$ returning small integers (e.g. $\pm 2,,4$).
The local module is chunk 9 of a 256-cell kernel certification that $m_2^{\mathrm{num}}=8\cdot Z$ pointwise. Each chunk theorem pins one concrete multi-index so the universal statement can be assembled by exhaustive fin_cases without re-running the fold at every site.
proof idea
One-line computational proof: by decide. Lean reduces both sides at the concrete Fin-4 tuple $(2,1,2,2,3,0)$ to integers and checks equality in the kernel. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
This cell is one of the 256 pointwise facts consumed by m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8\cdot Z$ and discharges the universal claim by nested fin_cases on all six indices. That identity is the algebraic backbone of the Regge exact midpoint M2TT certification in 4D gravity analysis inside the monolith. It does not itself touch the T0–T8 forcing chain or the Recognition Composition Law; it is pure discrete kernel bookkeeping for the curvature/mass-side midpoint identity.
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