e_230331
plain-language theorem explainer
Pointwise identity: the folded Regge numerator m2Num at Fin-4 indices (2,3,0,3,3,1) equals eight times the explicit kernel value explicitZ at those indices. Gravity analysts cite it as one cell of the 256-case kernel that assembles the global m2Num = 8·explicitZ statement. The proof is a single kernel decide on concrete integers.
Claim. For the index sextuple $(a,b,c,d,i,j)=(2,3,0,3,3,1)$ with each entry in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(2,3,0,3,3,1)$ equals $8$ times the explicit integer kernel value $Z(2,3,0,3,3,1)$.
background
In the 4D Regge exact-midpoint analysis, the TT-sector numerator is packaged as an integer-valued map on six Fin-4 indices. The definition m2Num folds a fixed coupling list, accumulating a local contribution at each tuple; explicitZ is the matching closed-form table (sparse pattern-match on those indices, returning small integers such as $\pm 2,4$).
This module is chunk 11 of the 256 kernel decides that discharge m2Num = 8·explicitZ pointwise. The local setting is pure finite enumeration: every sextuple in $(\mathrm{Fin},4)^6$ is a separate decide goal. Upstream, only the two kernel definitions are needed; no analytic continuum limit enters.
proof idea
One-line computational proof: by decide. Lean evaluates both sides at the concrete indices $(2,3,0,3,3,1)$—the fold that defines m2Num and the pattern-match that defines explicitZ—and checks integer equality with the factor 8. No lemmas beyond the two definitions are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ by exhausting all Fin-4 cases. That global identity is the algebraic certificate that the midpoint Regge TT numerator collapses to the explicit kernel used downstream in the 4D gravity analysis. Within Recognition Science this is scaffolding for the discrete curvature side of the forcing chain (spatial $D=3$ already fixed at T8); it does not itself touch J-uniqueness, $\phi$, or the eight-tick octave, but it locks the integer combinatorics those continuum claims rest on.
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