Macroscopic Parity Coherence in the Collatz Ensemble: A Haar–Wavelet Experimental Analysis

This paper presents a large-scale experimental investigation of structural coherence in the Collatz parity ensemble. Using Haar wavelet analysis applied across starting values rather than along individual trajectories, we detect a persistent macroscopic spectral slope in fixed-time parity columns. After residue-class debiasing (mod 210), the parity ensemble exhibits a stable power-law spectral decay across spatial frequency, with slope approximately −0.55 to −0.60 for sufficiently large time indices (t ≥ 40). Control sequences (iid Bernoulli and structured random mimics) do not reproduce this behavior. Furthermore, dyadic shuffling within 2-adic residue classes collapses the spectral slope, indicating that the coherence depends on cross-class interleaving rather than local 2-adic constancy. These results demonstrate that the Collatz map generates long-range spatial organization in the parity ensemble that is not reducible to trivial modular structure or stochastic effects. The observed spectral exponent appears robust across sampling regimes and ensemble sizes up to N ≈ 400,000. The work isolates a macroscopic structural invariant in Collatz dynamics and provides quantitative diagnostics for distinguishing deterministic arithmetic structure from stochastic controls.