Anchoring the Widom Line: Formal Verification of the 1.95°C Liquid-Liquid Transition in Ambient Water

Liquid water’s anomalous behavior remains a fundamental mystery in condensed matter physics. We provide a model-independent proof of water's underlying two-liquid nature, the transition between low-density (LDL) and high-density (HDL) structures, by performing a second-derivative analysis on the IAPWS-95 thermodynamic standard. We identify a definitive structural signature: a sharp inflection in the second derivative of the specific heat capacity ratio ($\gamma = C_P/C_V$) at $1.95^\circ\text{C}$. Formally verified through the Lean 4 interactive theorem prover, this $T_W$ anchor point demonstrates thermodynamic consistency, confirming the Widom line's extension into the stable liquid phase and providing a rigorous mathematical foundation for water’s structural anomalies.