Local maximal-canard threshold shifts under Runge–Kutta discretization: an observable-specific order condition

Abstract

This record contains the 67-page arXiv preprint and its complete compilable upload package. The source archive also contains two vector figures, a symbolic residual audit, invariant-graph threshold-validation scripts, machine-readable diagnostics, representative graph data, and reproduction notes. The paper proves an actual local map–flow maximal-canard threshold law for compact analytic classes of affinely normalizable planar folds and every fixed compact, uniformly finite-stage family of real Runge–Kutta methods of order at least two, including explicit and implicit methods. Both order-three B-series defect coordinates enter the pointwise residual, but Gaussian fold transport annihilates the bushy-tree direction and retains only the chain-tree defect. Hence b^T A c = 1/6 cancels the leading joint-fold threshold bias without elevating classical trajectory order. In the fold-matched fixed-ε formulation, the condition removes the Kθ(J)ε² term in the h² coefficient; it does not raise fixed-ε h-order. Under the stated composition hypotheses, the result supplies the local fold contribution to a global threshold displacement; an outer return can contribute at the same order. arXiv: 2608.04304Repository: https://github.com/h-lu/rk-canard-thresholdsPDF SHA-256: e4ab5cad12e56ddb160c145ac5c5a67b166c486e8a8a73bea02d150ba980b786Source SHA-256: 7801f751a0ff9c0e211d433c464fca6b660a6dd660d4742667903759ed426b46