The Refined Spectral–Geometric Regulator: An Extension to Quantum Mechanics

The author presents a refined spectral–geometric formulation of the additive–multiplicative (AM) regulator and demonstrates its natural extension to the core structures of quantum mechanics. Building on a seven-paper program on scale-regulated accumulation, this work shows that Planck’s quantization, Heisenberg’s operator formalism, and Schrödinger’s wave mechanics arise as constrained spectral limits of a single regulated accumulation principle. By integrating Fourier analysis into the spectral–geometric framework, the author attempts to clarify the deep connection between mode decomposition, regulation, and physical finiteness. The resulting formulation introduces no new postulates, preserves the validity of standard quantum theory within its domain, and explains both its success and its breakdown at extreme scales. Quantum behavior emerges here not as a fundamental mystery, but as the mathematically necessary response of accumulation to spectral constraint.