e_010102
plain-language theorem explainer
For the six-index tuple (0,1,0,1,0,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts cite this as one atomic cell in the 4D Regge midpoint M2 identity. The proof is a single kernel decide on concrete integers.
Claim. For indices $a=0$, $b=1$, $c=0$, $d=1$, $i=0$, $j=2$ in $\mathrm{Fin}\,4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $Z(a,b,c,d,i,j)$.
background
This module is chunk 1 of a 256-cell kernel certification that the Regge midpoint numerator $m_2^{\mathrm{num}}$ is identically eight times a sparse explicit integer table $Z$ on $(\mathrm{Fin},4)^6$.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list and summing local contributions at those indices. The companion table $Z$ is a pattern-matched integer function on the same six indices (nonzero only on a short list of index patterns, e.g. values $\pm 2,4$).
The local claim is the single cell at $(0,1,0,1,0,2)$: both sides are concrete integers, so equality is a finite decision problem.
proof idea
One-line proof by decide. Lean reduces both the fold that defines $m_2^{\mathrm{num}}$ at the fixed indices and the pattern match for $Z$ at those same indices to integers, then checks equality in the kernel. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem $m_2^{\mathrm{num}}=8Z$ for all six Fin-4 indices, which exhausts the index space by nested fin_cases and dispatches each cell to a chunk decide such as this one. That global identity is part of the 4D Regge-exact midpoint analysis in the Gravity domain: it certifies that the discrete curvature/mass numerator matches an eightfold multiple of a sparse closed-form kernel, a bookkeeping step toward exact midpoint identities used in the RS gravity stack.
It does not itself touch the T0–T8 forcing chain or the Recognition Composition Law; it is pure finite combinatorial certification inside the gravity analysis layer.
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