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

What the math gives to ML

The paper provides a nonstandard, observable-specific order condition for Runge-Kutta discretizations of fast-slow systems: although two independent third-order defects appear locally for any second-order method, fold passage annihilates the bushy-tree component and retains only the chain-tree component. Consequently, the maximal-canard threshold bias scales as K_theta(J) h^2 epsilon^2, and its leading coefficient vanishes when b^T A c = 1/6, even without classical third order. This mechanism can transfer to neural ODEs and continuous-depth networks whose useful output is an event, separatrix, attractor, or bifurcation threshold rather than pointwise trajectory accuracy.