e_312033
plain-language theorem explainer
For the six Fin-4 indices (3,1,2,0,3,3), the folded numerator m2Num equals eight times the closed-form table entry explicitZ. Gravity analysts cite it as one atomic case in the 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on the concrete integers.
Claim. With indices $(a,b,c,d,i,j)=(3,1,2,0,3,3)\in(\mathrm{Fin}\,4)^6$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(3,1,2,0,3,3)=8\,Z(3,1,2,0,3,3)$, where $Z$ is the explicit integer table for the midpoint kernel.
background
In the 4D Regge midpoint analysis, the M2–TT identity is checked by comparing a folded numerator against a closed form. The numerator m2Num a b c d i j is the fold of contrib over couplingZList, yielding an integer for each six-tuple of Fin 4 indices. The closed form explicitZ is a pattern-matched integer table on the same six indices (sample clauses give values such as 4, −2, …).
This module is chunk 13 of the 256-decide kernel split: each declaration pins one concrete six-tuple so the global identity can be assembled by exhaustive fin_cases. The local claim is only the equality at (3,1,2,0,3,3).
proof idea
One-line computational proof: by decide. Lean reduces both sides of m2Num 3 1 2 0 3 3 = 8 * explicitZ 3 1 2 0 3 3 to concrete integers (fold of the coupling list versus the table clause) and closes the goal in the kernel. No lemmas are invoked beyond the definitions of m2Num and explicitZ.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z$ and discharges the universal claim by fin_cases on all six indices. Each chunk theorem such as this one supplies one residual goal in that case split (or an equivalent decide path).
In the Recognition gravity stack this is bookkeeping for the exact midpoint M2–TT identity in 4D Regge calculus, not a forcing-chain (T0–T8) step. It closes a finite computational obligation so the continuum or continuum-limit arguments can treat the numerator/closed-form match as settled on the discrete kernel.
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