fXOR Ordering of Time–Frequency Data: A Phase-Based Alternative to Cartesian Representation

This work demonstrates a symbolic re-ordering method (fXOR) that transforms conventional time–frequency–amplitude data into a phase-accumulated space based on digital-root (mod-9) logic. Using identical data, a direct comparison is shown between Cartesian coordinates and fXOR ordering. While Cartesian representation preserves magnitude and locality, fXOR ordering preserves phase transitions, balance relationships, and long-range structural coherence. No filtering, smoothing, or learned models are applied. The work is intended as a methodological note establishing priority and providing a reproducible reference for symbolic phase-based data ordering.