e_000203
plain-language theorem explainer
For the six-index slot (0,0,0,2,0,3) on Fin 4, the folded numerator m2Num equals eight times the tabulated kernel value explicitZ. Gravity analysts assembling the exact midpoint M2 TT identity cite this as one of the 256 kernel cells. The proof is a single decide on two concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(0,0,0,2,0,3)=8\,Z_{\mathrm{explicit}}(0,0,0,2,0,3)$.
background
This module is chunk 0 of a 256-cell kernel certification that the Regge midpoint numerator $m_2^{\mathrm{num}}$ equals eight times a closed-form integer table $Z_{\mathrm{explicit}}$ on every six-tuple in $(\mathrm{Fin},4)^6$.
Upstream, $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: it sums the local contribution of each coupling term at those indices. The companion table $Z_{\mathrm{explicit}}$ is a pattern-matched integer function on the same six indices (sample values include $4$, $-2$, and so on for the listed patterns).
The local goal is purely arithmetic: evaluate both sides at one concrete multi-index and check equality. The parent assembly later quantifies over all of $\mathrm{Fin},4$ by fin_cases.
proof idea
One-line kernel decision. Both sides reduce to concrete integers once the six Fin-4 arguments are fixed, so decide discharges the equality with no further lemmas. No algebraic rewriting is needed beyond evaluation of m2Num (the fold) and explicitZ (the table).
why it matters
Feeds the universal identity m2Num_eq_eight_explicitZ in the assemble module, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{explicit}}$ and proves it by exhaustive fin_cases on each coordinate, invoking one cell theorem per tuple. That identity is the certified numerator half of the exact midpoint M2 TT relation in 4D Regge analysis inside the Gravity domain of the monolith. It does not itself touch the T0–T8 forcing chain or the RCL; it is infrastructure for the discrete gravity side of Recognition Science.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.