ZLT-P50M-O: The Observability Precondition — Observer Term, O-H Non-Separability, and the Euler-Lagrange Equation for Hysteresis

ZLT-P50M assumed that the observer verifying policy coherence Π > 0 has zero structural cost. This paper proves that assumption is false in any regime where H > 0 and the observer is not orthogonal to the system ([Ô,Ĝ] ≠ 0). The cost is real, Lagrangian-formable, and structurally consequential. Three theorems are derived from L_ZLT^obs = L_ZLT + (κ/2T_eff)·Ḣ² + ν(t)·δΠ(t)² + c_O·χ_obs(H): T1 (Observer Term Divergence): A_ZLT^obs → ∞ under self-observation at saturation, reproducing P0F as a variational consequence. T2 (O-H Non-Separability): O_eff(H) = O₀·(1−H/H_max)₊ — observability and hysteresis are not separable. T3 (H Euler-Lagrange): second-order equation of motion for H; in the overdamped limit reduces to P50K's first-order decay law. Two corollaries derive P0A (C1) and OD-1 (C2) as theorems rather than postulates. Seven cross-domain DRCs span quantum measurement, clinical medicine, economic information asymmetry (Nobel 2001), labor markets (P82 Grade A), neural networks (P50M-E Grade A, H_NN=0.59), behavioral public goods games (Herrmann et al. 2008, pre-registration doi:10.5281/zenodo.19577461), and LLM inference (P50L-B, McNemar N=40, p