The Geometric Ladder: Fundamental Constants as Eigenvalues of a Circle

Using only the intrinsic geometry of the circle S1 and the standard spectral theory of flat tori T n = (S1)n, we derive three unavoidable mathematical constraints: angular duality D = 180/π with reciprocal U = π/180, factorial escalation governed by 3! = 6 beyond dimension three, and transcendence-toll saturation with damping n!/π|n|−33|n|−3 for |n| ≥ 4. These constraints force a unique spectral ladder Tn on integer rungs n ∈ Z. Selected eigenvalues (or simple rational multiples fixed by the Basel regularisation ζ(2) = π2/6) reproduce the dimensionless combinations underlying the Planck constant h, the speed of light c, the fine-structure constant α, and the gravitational constant G to better than 10−10 relative precision against CODATA 2022, without adjustable parameters. The fundamental constants of physics are therefore exact eigenvalues of the circle.