A sharpened lower bound for the convergence radius of the Amsler equation via optimized supersolutions

We sharpen the universal lower bound R≥ 2 for the radius of convergence of the Taylor series of the solution of the Amsler equation (xy')'=sin y, y(0)=y₀∈(0,π). By optimizing a three-parameter family of real supersolutions we obtain an improved bound L(y₀) and prove that it has the form \(L(y_0)=min(λ_A(y_0),λ_B(y_0)),\) where λA(y₀)=4/(sin y₀+|cos y₀|) is a closed-form boundary branch and λB(y₀) is defined implicitly by a Karush--Kuhn--Tucker system. From this representation we prove: 1. the symmetry L(π-y₀)=L(y₀); 2. the existence of a kink at y₀=π/4, where the critical quantity ν(y₀)=12(sin y₀-|cos y₀|) changes sign; 3. the exact jump of the derivative Δ L'=-177√2/176; 4. a dichotomy at y₀=π/2: the one-sided derivatives of L either both vanish, or they are non-zero and of opposite signs (cusp). Numerical evaluation confirms the second alternative. Numerical verification on four independent grids confirms every analytic prediction to within measured numerical noise. We are explicit about what is proved, what is numerically observed, and what remains open.