e_031012
plain-language theorem explainer
For the six-index slot (0,3,1,0,1,2) on Fin 4, the folded numerator m2Num equals eight times the closed-form table explicitZ. Gravity analysts cite it when assembling the full 4D Regge midpoint M2–TT identity. The proof is a single kernel decide on concrete integers.
Claim. With indices in $\mathrm{Fin}\,4$, the folded coupling numerator satisfies $m_2^{\mathrm{num}}(0,3,1,0,1,2)=8\,Z_{\mathrm{explicit}}(0,3,1,0,1,2)$.
background
In the 4D Regge midpoint analysis, two integer-valued kernels on six Fin-4 indices are compared. The numerator m2Num folds a fixed coupling list, summing a local contribution at each triple; explicitZ is the matching closed-form lookup table (sparse integer values such as 4, -2, and defaults).
The module is chunk 3 of a 256-way case split that certifies pointwise equality m2Num = 8·explicitZ. Each chunk theorem pins one concrete multi-index so the assembler can discharge the universal statement by fin_cases without re-running the fold.
Upstream, m2Num and explicitZ are pure definitions in the kernel certificate module; no analytic hypotheses are carried.
proof idea
One-line proof by decide. Lean reduces both sides on the concrete Fin-4 tuple (0,3,1,0,1,2) to integers and checks equality in the kernel. No lemmas beyond the definitions of m2Num and explicitZ are invoked.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states ∀ a b c d i j, m2Num a b c d i j = 8 * explicitZ a b c d i j and proves it by exhaustive fin_cases, routing each cell to a chunk theorem such as this one.
That identity is the algebraic core of the Regge-exact midpoint M2–TT certificate in 4D gravity analysis inside the monolith. It sits downstream of the discrete curvature bookkeeping and upstream of continuum-limit comparisons; it does not itself invoke the T0–T8 forcing chain, but it stabilizes the discrete side of the gravity sector those landmarks eventually constrain.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.