e_103202
plain-language theorem explainer
For the six-index slot (1,0,3,2,0,2) on Fin 4, the Regge midpoint numerator m2Num equals eight times the explicit kernel value explicitZ. Gravity analysts assembling the full 4D m2Num = 8·Z identity cite this as one of the 256 kernel cases. The proof is a single decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j) = (1,0,3,2,0,2)$ with each index in $\{0,1,2,3\}$, the folded numerator coupling $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
In the 4D Regge exact-midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator $m_2^{\mathrm{num}}$ is obtained by folding a fixed coupling list: each term contributes via a local contrib and the accumulator starts at 0. The comparison target is explicitZ, a sparse case table of small integers (entries such as $4$, $-2$, and so on) that encodes the closed-form kernel.
The ambient module is chunk 4 of a 256-way case split whose sole purpose is to discharge $m_2^{\mathrm{num}} = 8\cdot Z$ pointwise. The parent certificate lives in the KernelCert import; this file only records individual decided equalities.
proof idea
One-line computational proof: decide evaluates both sides at the concrete six-tuple $(1,0,3,2,0,2)$ and checks integer equality. No lemmas are invoked beyond the reducible definitions of m2Num and explicitZ.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8\cdot Z$ and proves it by exhaustive fin_cases on all six indices. Each chunk theorem such as this one supplies one concrete cell of that $4^6=4096$-point (here organized as 256 kernel decides) table. The identity is infrastructure for the Regge midpoint $M_2$/$TT$ analysis in the Gravity domain; it does not itself touch the T0–T8 forcing chain or the Recognition Composition Law, but it hardens a discrete curvature kernel used downstream in RS gravity numerics.
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