Discretization-Induced Maximal-Canard Threshold Shifts for a Symmetric Two-Stage Runge–Kutta Family

This research compendium contains the complete extended preprint and its compilable LaTeX source, the pinned Python environment and programs for two exact symbolic checks and the local fixed-section van der Pol computation, the machine-readable joint-(r,h) data including 50/80-digit and boundary/mesh-phase checks, the generated vector figure, manifests, and SHA-256 checksums. The analytic proof is contained in the preprint. The numerical computation illustrates the local section-defined threshold law; it is not the selected-manifold proof and does not compute a global periodic-orbit canard explosion. The symbolic programs verify finite identities displayed in the proof and do not replace the analytic remainder estimates. For the declared symmetric two-stage Runge–Kutta family, the preprint proves an actual map–flow maximal-canard threshold displacement and identifies the parameter value that cancels its leading bias across the normalized analytic fold class.