Pythagorean Harmonic Geometry as Oxygen-Curvature Invariance

Abstract.

This research note proposes a physical basis for Pythagorean harmony: the classical musical ratios (1:1, 9:8, 4:3, 3:2) correspond approximately to stable curvature states of oxygen-based molecules (O₂, O₃, H₂O, OH⁻). Normalized bond angles (180°, ~114°, ~109°, 104.5°) form a discrete geometric set that aligns with the Pythagorean harmonic ladder within experimental tolerances. I interpret these as oxygen-curvature invariants, suggesting that ancient harmonic theory may encode empirical regularities of vibrational geometry rather than purely abstract numerology. This note also links these invariants to modern notions of curvature-driven invariance in orbital mechanics (Kepler’s equal-areas law) and registers priority for the hypothesis within the broader Oxygen Octave framework.