A Bounded, Memoryless, Cycle-Slip-Free, and Reliability-Weighted Angular Framework for Multi-Harmonic Signal Analysis

Comparing scale-parameter ratios in digital signal processing via unbounded logarithmic axes or periodic phase-based comparators introduces structural costs, including unbounded axis growth and cycle slips. We introduce GTFS, a bounded, nonperiodic angular projection mapping an unbounded ratio space onto a finite angular interval using a monotonically increasing, asymptotically saturating nonlinear function. Analytical properties, including monotonicity, boundedness, and octave-dependent angular contraction, are derived and verified across three closed-form realizations (arctangent, logistic, hyperbolic tangent). We examine four properties for multi-component signals: stateless order-independent projection, component-wise reliability-weighted selective alignment, bounded layer-consistent representation, and empirical robustness to cycle slips. Validated analytically, on synthetic multi-harmonic signals, and on 64 real LibriSpeech utterances, the projection achieves stateless independence to machine precision and reduces low-reliability component displacement up to 4.7-fold on real speech. The layered representation of a six-octave signal remains 19 to 41 times narrower than unlayered logarithmic mapping. In comparison against a classical phase-locked loop (PLL), GTFS exhibits zero cycle-slip events across all test conditions where the PLL fails. GTFS provides a bounded, memoryless, and cycle-slip-free alternative to conventional phase-tracking, with results validated on real acoustic data and properties expected to extend to vibration-based and communication signal-processing domains.