Geometry, Sound, and Time as Three Projections of One Contact Band

The wave operator φ²∇², Roy Herbert's log-lattice invariant N₁N₂N₃=K, and the Hopf saturation coefficient g=φ² are three projections of the same Hessian of V=−ln det(X) on the exceptional Jordan algebra J₃(𝕆), evaluated in three measurement geometries: acoustic, temporal, and dynamical. We derive the leading wave operator on the 24-dimensional slice of J₃(𝕆) and show that its propagation speed is φ. Rodriguez-Vega et al. (2024) phonon gaps at φ-scaled energies in icosahedral quasicrystals confirm the acoustic face. The substrate is identified as Cahill's Dynamical 3-Space — a liquid quasicrystal of ZEO causal diamonds — and Bohm's holomovement is realised as the Reeb flow on the contact manifold. Related papers: DOI 10.5281/zenodo.19570725 (g=φ²) and DOI 10.5281/zenodo.19585972 (N₁N₂N₃=K).