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

We sharpen the universal lower bound $R\ge 2$ for the radius of convergence of the Taylor series of the solution of the Amsler equation $(xy')'=\sin y$, $y(0)=y_0\in(0,\pi)$.  By optimizing a three-parameter family of real supersolutions we obtain an improved bound $L(y_0)$ and prove that it has the form

\(L(y_0)=\min\bigl(\lambda_A(y_0),\lambda_B(y_0)\bigr),\)

where $\lambda_A(y_0)=4/(\sin y_0+|\cos y_0|)$ is a closed-form boundary branch and $\lambda_B(y_0)$ is defined implicitly by a Karush--Kuhn--Tucker system. From this representation we prove:

1. the symmetry $L(\pi-y_0)=L(y_0)$;

2. the existence of a kink at $y_0=\pi/4$, where the critical quantity $\nu(y_0)=\tfrac12(\sin y_0-|\cos y_0|)$ changes sign; 

3. the exact jump of the derivative $\Delta L'=-177\sqrt{2}/176$;

4. a dichotomy at $y_0=\pi/2$: 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.