Runge–Kutta Bias in Local Maximal-Canard Thresholds: A Chain-Tree Condition

This research compendium contains the complete 106-page extended preprint and its compilable LaTeX source, together with exact symbolic checks, numerical programs, machine-readable data, vector figures, environment specification, manifests, and SHA-256 checksums. The preprint proves an actual local map–flow maximal-canard threshold law on compact domains of symmetric two- and three-stage Runge–Kutta families. The threshold functional detects only the chain-tree coordinate of the order-three B-series defect, so class-wide leading-bias cancellation is equivalent to b^T A c = 1/6 while the independent bushy-tree condition may fail. Version 1.3.3 strengthens the dynamics-first presentation of the result. It explains earlier why the fixed-section threshold is a natural local fold observable, presents the defect–transport–splitting–root mechanism before the technical architecture, and states explicitly that a global canard-explosion law requires separate control of the model-dependent return map. The theorem, method domains, asymptotic coefficients, joint wedge, remainders, root tube, applications, and numerical data are unchanged. The finite-boundary van der Pol and rational Rosenzweig–MacArthur computations illustrate local fixed-section threshold laws; they do not construct selected invariant manifolds numerically or compute a global periodic-orbit canard explosion. The symbolic programs verify finite identities and do not replace the analytic remainder estimates.