Part IX: Spectral Derivation of Fundamental Physical Constants

This paper presents a rigorous derivation of fundamental physical constants—Planck's constant (ℏ), the speed of light (c), Newton's gravitational constant (G), and the fine-structure constant (α)—directly from the intrinsic spectral geometry of spacetime deformation. Extending the spectral geometric framework developed in prior works, we define a self-adjoint deformation operator Ĉ, whose discrete spectrum encodes the fundamental structure of space and time. We demonstrate explicitly that these constants are not arbitrary parameters or empirical postulates, but universal invariants emerging naturally from the spectral hierarchy of geometric excitations. By analyzing spectral asymptotics, quantization conditions, and intrinsic scaling laws, we establish precise spectral formulas connecting geometry and observed physical laws. This result provides a unified geometric explanation of physical constants, linking quantum mechanics, relativity, and gravity through a single mathematical principle—spectral deformation geometry.