Ground Eigenvalue of the Twisted Laplacian on a Constant-Curvature Möbius Band in the Reduced Anti-Periodic Sector

We establish that the ground eigenvalue of the twisted Laplace-Beltrami operator on a Constant-Curvature Möbius band in the Reduced Anti-Periodic Sector, equals the surface scalar curvature exactly: λ₀ = R_Σ = 2/R² Two independent paths (direct Rayleigh quotient and Bochner lower bound) yield the same result, with geodesic boundaries ensuring vanishing flux. The curvature of S³ supplies a factor of 2 over the flat-strip value. To the authors' knowledge this specific model has not received prior treatment. The result grounds the topological derivation of the cosmological constant in Mode Identity Theory. v2 update: the title now reads Constant-Curvature Möbius Band rather than Totally Geodesic Möbius Band in S3. Section 4 describes the construction as the quotient of a spherical band on S2 ⊂ S3 by twisted boundary-edge identication, with the involution τ dened explicitly in Section 2. All references to an ambient R4 isometry have been removed. The spectral result and both proof paths are unchanged.