Information-Theoretic Limits of Time Distinguishability in Diffusion and Anomalous Diffusion

This record accompanies the paper: Information-Theoretic Limits of Time Distinguishability in Diffusion and Anomalous DiffusionAlexander YashinIndependent Researcher (Germany) --- Motivation In stochastic systems such as diffusion, time is typically treated as an external parameter rather than as an observable quantity. This work addresses a precise operational question: What is the minimal time difference that can be distinguished from data generated by stochastic dynamics? The answer is constrained not by metaphysical assumptions or new physics, but by information-theoretic limits inherent in statistical inference. --- Core idea Time is treated as an inferred parameter of probability distributions, not as a primitive variable. Temporal resolution is fundamentally limited by:- statistical sample size,- measurement noise,- the underlying dynamical law governing state evolution. This leads to sharp, estimator-independent bounds on temporal distinguishability. --- Main results Normal diffusion:For Brownian motion in d dimensions, Fisher information yields a universal lower bound on time resolution that scales linearly with time and inversely with the square root of the effective sample size. Hypothesis-testing equivalence:The same bound follows from a Kullback-Leibler divergence analysis, establishing local equivalence between estimation-theoretic limits (Cramer-Rao bound) and hypothesis-testing distinguishability. Photon-limited regime:In the experimentally relevant low-flux regime, a self-consistent analysis predicts a nontrivial scaling of the minimal distinguishable time with photon flux, proportional to flux to the power minus one third.This scaling is analytically derived and numerically verified. Anomalous diffusion:Subdiffusive and superdiffusive dynamics exhibit qualitatively distinct temporal resolution regimes governed by the anomalous exponent. All results are analytically derived, numerically verified, and experimentally falsifiable, without requiring any modification of known physical laws. --- Repository contents The repository provides:- LaTeX source of the paper,- reproducible numerical simulations,- scripts for Monte Carlo verification,- acceptance criteria and falsifiability protocols. --- Reproducibility All numerical claims are independently verified by Monte Carlo simulations.The repository is suitable for independent verification, citation, and extension.