A Machine-Verified Zero-Parameter Temperature Scale Near Water's 4°C Density Anomaly: Theorem-Level Transition Scale and Hypothesis-Level Density Offset from the φ-Ladder

Water's density maximum at T_obs = 277.15 K (3.98 °C) is a central open problem in condensed matter physics. We derive, within the Recognition Science (RS) framework, a unique zero-parameter cooperative H-bond transition scale T_c = φ⁻⁵ / (8 k_B ln φ), where φ = (1 + √5)/2 is forced by cost self-similarity, E_coh = φ⁻⁵ is fixed by the Fibonacci constraint on D = 3, and the factor 8 is the forced 8-tick recognition cycle.

The theorem-level result, machine-verified in Lean 4, is the interval bound T_c ∈ (267, 275) K, together with the uniqueness statement that the adjacent φ-ladder rungs lie below freezing and above boiling. We then introduce a clearly labeled hypothesis-level finite-width correction T_max = T_c(1 + φ⁻⁸), motivated by the octave width of the crossover, which yields T_max ∈ (272.6, 281) K and contains T_obs.

Thus the present paper establishes a theorem-level zero-parameter transition scale near 4 °C and a hypothesis-level mapping from that scale to the density maximum itself. The three dedicated Lean modules introduced for this result compile with zero "sorry" and zero new axioms.