A Rigorous Classification of ODE Reductions of Nonlinear Wave Equations via the Ansatz \(\omega = f(\phi)\)

I classify all smooth non‑constant solutions of the second‑order wave equation \(\omega_{uv}=H(\omega)\) of the form \(\omega(u,v)=f(\phi(u,v))\) that constitute a genuine ODE reduction. The definition is formulated through the existence of an auxiliary variable \(\xi(u,v)\) such that the coefficients \(\xi_u\xi_v\) and \(\xi_{uv}\) depend only on \(\xi\), with \(\xi_u\xi_v\not\equiv0\), and \(\phi\) is a strictly monotone function of \(\xi\). From this definition we prove rigorously that the geometric coefficients \(P=\phi_u\phi_v\) and \(Q=\phi_{uv}\) must themselves be functions of \(\phi\) alone --- the constrained condition. Under that condition a complete integration shows that \(\phi\) is necessarily a strictly monotone function of either a linear expression \(\xi=\alpha u+\beta v+\delta\) (with \(\alpha\beta\neq0\)) or a factorizable bilinear expression \(\xi=(\alpha u+\beta)(\gamma v+\delta)\) (with \(\alpha\gamma\neq0\)). The reduced ODE for \(g(\xi)=f(\phi(\xi))\) is the generalized oscillator equation \(g'' = \pm H(g)\) in the linear case and the generalized Amsler equation \((\xi g')' = c\,H(g)\) in the bilinear case. The so‑called liberated form \(\phi_{uv}+k(\phi)\phi_u\phi_v = L(\phi)\) is shown to be a universal reparametrisation that alone does not guarantee a reduction; an explicit counterexample is given. The classification is complete for smooth data; the extension to complex holomorphic functions is discussed as a remark. The proof is self‑contained and uses only smoothness and elementary calculus, providing a direct alternative to the classical Lie symmetry approach. We additionally prove that the only solution that can be simultaneously represented as a function of a linear phase and of a bilinear phase on overlapping open sets is the constant vacuum \(\omega\equiv\omega_0\) with \(H(\omega_0)=0\). This result clarifies the role of the constant solution as the unique possible bridge between the two reduction families.