Universal Decay Field Theory and the Unified Spectral Decay Theorem: A Cross-Disciplinary Framework for Multi-Scale Systems

We present a unified mathematical framework characterizing spectral decay rates in systems exhibiting inter-scale coupling. The Unified Spectral Decay Theorem establishes that the decay rate γ of inter-scale correlations is governed by three independent regularity constraints: geometric (spectral dimension), analytic (Fourier regularity), and arithmetic (number-theoretic). The minimum of these constraints determines system behavior. We extend this to Universal Decay Field Theory (UDFT), a Lagrangian formulation treating spectral decay as a dynamical field with universal coupling constant λ = φ ≈ 0.618. The framework unifies phenomena across algorithmic complexity, quantum phase transitions, and spectral geometry, generating quantitative cross-domain predictions.