e_311220
plain-language theorem explainer
For the index sextuple (3,1,1,2,2,0), the midpoint Regge mass-squared numerator equals eight times the explicit kernel integer. Gravity analysts assembling the 4D midpoint M2TT identity cite this as one decided cell among the 256-kernel case split. The proof is a single computational `decide` on two concrete integers.
Claim. The integer-valued midpoint numerator $m_2^{\mathrm{num}}(3,1,1,2,2,0)$ equals $8$ times the explicit kernel value $Z(3,1,1,2,2,0)$, where both maps take six arguments in $\{0,1,2,3\}$.
background
In the 4D Regge midpoint M2TT analysis, two integer kernels on six Fin 4 indices are compared. The numerator m2Num is the fold of a coupling list: it sums a local contribution over every term in couplingZList. The comparison target explicitZ is a piecewise table of small integers (entries such as $4$, $-2$, and so on) on the same six-index domain.
The ambient module is chunk 13 of a 256-cell kernel certification whose sole job is to prove the pointwise identity numerator $= 8\cdot$ explicit table. Each cell is an independent equality of two concrete Int values. The present declaration is the cell at indices $(3,1,1,2,2,0)$.
proof idea
One-line computational proof: by decide. Both sides of the equality reduce to closed integer literals (the fold for the numerator and the matching clause of the explicit table), so Lean’s decision procedure discharges the goal with no lemmas and no case split inside this declaration.
why it matters
The parent theorem m2Num_eq_eight_explicitZ asserts the identity for every sextuple in Fin 4^6 by exhaustive fin_cases. This cell is one of the concrete equalities that fill that case tree (chunk 13 of the 256 kernel decides). Without the full set of cells, the midpoint M2TT numerator cannot be replaced by the closed explicit kernel, blocking later exact identities in the 4D Regge gravity analysis. The result is pure discrete linear algebra on the coupling table; it does not itself invoke the RS forcing chain, but it sits inside the gravity-side exactness pipeline that those landmarks ultimately constrain.
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