Gaussianization Flow Law

This paper studies the evolution of non-Gaussian phase fluctuations in stochastic and quantum systems. We construct an observable based on second and fourth cumulants and show that it approaches a constant value under broad physical conditions.The main result is the emergence of a universal invariant:Y(t) = σ⁴(t) / (κ(t) · t)We show that this quantity converges to a constant λ under short-memory, finite-cumulant dynamics. This connects stochastic processes, decoherence physics, and renormalization flow into a unified structure. In many physical systems, noise is commonly approximated as Gaussian. This assumption simplifies analysis but hides higher-order statistical structure.However, in real quantum systems (e.g., superconducting qubits, optical interferometers, spin environments), deviations from Gaussianity are observed due to:environmental memory effectsrare event contributionsinteraction-induced correlationsThe question addressed here is:Does non-Gaussianity decay in a universal way under time evolution?To answer this, we define a measurable invariant based on cumulants.