Proof of Conjecture 3: The Boundary Term δX = 2/π from the Equatorial Membrane Mode

UCM Paper 40 (Theoretical). Proves Conjecture 3 of Paper 34: the boundary correction δX = 2/π to the membrane effective action arises from the Y₁1 azimuthal mode on the equatorial great circle, forced by the symmetry breaking SO (3) →SO (2) of the substrate flow. The proof is a geometric identity: (1/2π) ∫₀²π|sin φ|dφ = 2/π, requiring no free parameters. Combined with the bulk term X^ (0) = 2π³ (Paper 34, Zamolodchikov c-theorem, cₘem=3), the complete parameter-free prediction is E_Λ = π·EP·exp (−2π³ − 2/π) = 23. 75 eV, in agreement with the SPARC rotation-curve calibration (23. 9 eV) to 0. 63%. The UCM now derives both Ω_Λ = 0. 6834 and E_Λ = 23. 75 eV from the same substrate geometry without any free parameters. UCM preprint series on Zenodo, ORCID 0009-0005-5088-4339.