Core Spectral Phase Transition

We prove that a single dimensionless parameter — the spectral decay rate ρ of a Markov generator — governs a universal phase transition in the compressibility of stochastic systems. For ρ > 1, any Lipschitz functional can be computed to accuracy ε using N = Θ(log(1/ε)/log ρ) spectral modes, independent of the system's dimension. For ρ < 1, the required complexity grows polynomially or exponentially in dimension. The critical point ρ = 1 manifests as six physically distinct but mathematically identical phenomena: the signal-noise boundary in financial markets, the pattern-noise threshold in machine learning, the quantum-classical boundary in decoherence, the Gaussian–non-Gaussian threshold in astrodynamics, the order-disorder transition in statistical physics, and the laminar-turbulent transition in fluid mechanics. We verify the core theorem in Lean 4 (machine-checked, zero axioms) and validate numerically across all six domains with a single code base. The result establishes spectral decay rate as a universal order parameter for the complexity of physical systems.

Keywords: spectral methods, phase transition, universality, compressibility, Markov generators, spectral gap