The Oxygen Octave – Foundations: Quantitative Correlations Across Vibrational Systems This edition expands the Structural Oxygen Model with new empirical correlations across vibrational, thermodynamic, and geometric domains. Building on prior sections linking φ : √2 harmonic invariants to oxygen’s molecular geometry, the updated framework introduces two experimentally testable modules: (1) Vibrational Coherence Transition (Vapor → Liquid Water) — Raman and IR data demonstrate that O–H stretching modes preserve proportional coherence during phase transitions, suggesting that molecular energy reorganizes rather than dissipates. (2) Pressure-Dependent Coherence in H₂O — The redshift of O–H vibrational frequency under compression follows discrete harmonic domains (φ, √2, 3/2), confirming that geometric invariants persist even under perturbation. Together, these results reinforce the hypothesis that oxygen acts as a universal coherence mediator, capable of maintaining structural proportion across molecular, biological, and potentially cosmological scales. A new visual synthesis (Figure 6.5A – Cross-Domain Continuity Bridge) links O₂, O₃, and H₂O to mitochondrial Complex I–IV via shared harmonic ratios, providing a unified template for coherence transfer. All sections have been revised for precision, with extended quantum-geometric analysis and updated figure discussions. Data Availability: All datasets referenced in this work are publicly accessible via NIST, HITRAN, JPL, and IUPAC. AI-based ratio extrapolations (Grok 2025) are available upon request. Researchers, collaborators, or institutions interested in testing or expanding this hypothesis are invited to connect and explore possible lines of validation together Jaime OjedaJaime.ojse@gmail.com