e_101212
plain-language theorem explainer
For the six-index slot (1,0,1,2,1,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the exact midpoint M2–TT identity cite this as one of the 256 kernel decides. The proof is a single decide on concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the folded coupling numerator at $(a,b,c,d,i,j)=(1,0,1,2,1,2)$ equals eight times the explicit integer kernel at those same indices: $m_2^{\mathrm{num}}(1,0,1,2,1,2)=8\,Z_{\mathrm{expl}}(1,0,1,2,1,2)$.
background
In the Regge exact-midpoint analysis for 4D gravity, two integer-valued six-index kernels appear. The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is obtained by folding a fixed coupling list and summing each term's contribution at those indices. The companion $Z_{\mathrm{expl}}$ is a sparse pattern-matched table of small integers (entries such as $4$, $-2$, and so on) that is meant to be the closed form of that fold, up to a universal factor of eight.
This module is chunk 4 of the 256-case decide grid that checks $m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ pointwise on $(\mathrm{Fin},4)^6$. The local setting is pure finite enumeration: no continuum limit and no curvature hypothesis beyond the discrete kernel definitions imported from the KernelCert module.
proof idea
One-line kernel decide. Both sides reduce to concrete Int values once the six Fin 4 arguments are substituted, so decide discharges the equality with no lemmas beyond the definitions of m2Num and explicitZ.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple by fin_cases on each index and routes each leaf to a chunk decide such as this one. That global equality is the algebraic backbone of the Regge exact-midpoint M2–TT identity in 4D: it replaces a folded coupling sum by an explicit sparse kernel times eight, clearing the numerator side of the discrete gravity certificate. Within Recognition Science gravity work this is bookkeeping infrastructure rather than a forcing-chain step (T0–T8), but without the pointwise decides the midpoint identity cannot be closed in Lean.
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