Three Anomalous Properties of Water Derived from a Single Functional Equation with Zero Adjustable Parameters

Water's anomalous properties - its density maximum at 3.98 °C, its structured hydrogen-bond dynamics spanning five decades in frequency, and its apparently unique suitability as biology's solvent -have resisted zero-parameter first-principles explanation. We show that all three follow from a single algebraic identity, the Recognition Composition Law (RCL), with no adjustable parameters. The RCL uniquely forces the cost function J(x) = 1/2 (x + x⁻¹) − 1, from which a proved chain yields the golden ratio φ, spatial dimension D = 3, the 8-tick recognition period 2ᴰ = 8, and the coherence quantum Ecoh = φ⁻⁵ ≈ 0.090 eV. Three independent results follow. (i) The tetrahedral H-bond spectrum: Bethe-lattice coordination shells on the φ-ladder predict seven mode-frequency ranges; five of seven observed rungs match exactly, and three "dark modes" at rungs 7, 9, 11 constitute falsifiable predictions. (ii) The density anomaly: the cooperative H-bond transition temperature Tc = φ⁻⁵/(8kB ln φ) ∈ (267, 275) K is the unique φ-ladder scale in the liquid range, within 4% of the observed 277.15 K. (iii) Substrate necessity: water is the only common solvent satisfying the viable energy band [Ecoh/φ, Ecoh · φ], tetrahedral coordination, and central atomic number Z = 8 simultaneously; NH3, HF, CH3OH, and H2S are each excluded by at least one proved constraint. All results are machine-verified in Lean 4 with zero sorry obligations and zero new axioms. Seven specific acoustic dissociation frequencies from 20 GHz to 100 THz, testable with existing THz spectroscopy, are predicted.