Dimensionality Reduction in the Collatz Process: Locating Extreme Anomalies via Sturmian Parameterization

Finding integers with anomalously long Collatz trajectories is a challenging optimization problem. For 5000-bit integers, exhaustive search is infeasible due to the exponential search space (2^5000). This paper proposes Sturmian word parameterization as a dimensionality reduction technique, compressing the search from 5000 bits to 30 directive coefficients. We employed an Island Model Genetic Algorithm with LightGBM surrogate filtering to optimize directive sequences. To ensure statistical validity, we generated a rigorous control group of 3,000 samples: 2,000 true random integers and 1,000 random Sturmian sequences. Our central finding refutes the hypothesis that Sturmian structure inherently produces anomalies. Both random integers and random Sturmian sequences exhibit identical baseline behavior (μ_γ ≈ 1.50). However, the reduced parameter space enables efficient evolutionary search. The algorithm identified a champion with anomaly score γ = 1.731, substantially exceeding the control maximum of 1.649 (empirical p < 0.0003). Feature importance analysis reveals that the second and third directive coefficients dominate prediction, accounting for combined gain exceeding 449. This suggests that local recurrence structure controls early trajectory dynamics more strongly than global statistics. The results demonstrate that Sturmian parameterization provides effective dimensionality reduction without distributional bias. This method located a 1-in-100,000 outlier with computational efficiency orders of magnitude superior to brute-force search. Complete directive sequences and reproducible implementation details are provided.