e_023030
plain-language theorem explainer
For the fixed index sextuple (0,2,3,0,3,0) on Fin 4, the folded numerator m2Num equals eight times the tabulated explicitZ value. Gravity analysts assembling the full Regge midpoint M2–TT identity cite this as one kernel cell. The proof is a single decidability discharge on concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the coupling numerator satisfies $m_2^{\mathrm{num}}(0,2,3,0,3,0)=8\,Z_{\mathrm{explicit}}(0,2,3,0,3,0)$.
background
This module is chunk 2 of a 256-cell kernel that checks the identity $m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ pointwise on $(\mathrm{Fin},4)^6$. The setting is the exact midpoint analysis of the 4D Regge M2–TT identity in the Gravity.Analysis stack.
$m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding couplingZList and summing the local contribution of each coupling term at those six indices. $Z_{\mathrm{explicit}}$ is the closed integer table for the same sextuple (pattern-matched constants such as $4$, $-2$, and so on). The claim is the numerical equality of those two integers at one concrete point.
proof idea
One-line kernel proof: decide evaluates both sides as concrete Int values (the fold for $m_2^{\mathrm{num}}$ and the table lookup for $Z_{\mathrm{explicit}}$) and closes the equality by computation. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,, m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ by exhaustive fin_cases on all six indices. Each chunk cell such as this one is a leaf of that case tree. In the broader RS gravity line, the identity certifies that the midpoint M2 numerator matches the explicit eightfold Z kernel used in the Regge exact-midpoint analysis; without the pointwise cells the global equality does not typecheck.
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