Seeking Structural Patterns in Time Series with Diophantine Approximations

We introduce the Diophantine Dynamical Boundary Method (DDBM), a lattice‑based statistical procedure for distinguishing deterministic chaotic dynamics from periodic behavior and stochastic noise in univariate time series. DDBM applies rank‑normalization, two‑level residualization, lattice quantization, and cubic Diophantine phase extraction, then tests for phase non‑uniformity via Kolmogorov–Smirnov statistics with multiple‑testing correction. The method operates directly on scalar observables without phase‑space reconstruction, hyperparameter tuning, or training data, and remains effective for moderate sample sizes.