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

A fixed-step Runge--Kutta discretization may approximate ordinary trajectories accurately while displacing the parameter at which selected attracting and repelling slow manifolds form a maximal canard. For the minimal polynomial model $x'=x^2-y+xy$, $y'=\epsilon(x-\lambda)$, we compare actual section-defined thresholds of the flow and its exact Runge--Kutta map, uniformly over all admissible choices of the attracting and repelling slow manifolds. We first expose the mechanism on a symmetric, A-stable two-stage slice and then prove it on a compact two-parameter domain of the complete self-adjoint two-stage family and on a separate self-adjoint three-stage family. The order-three B-series defect has independent bushy- and chain-tree coordinates, but the threshold functional detects only the chain-tree defect. Thus class-wide leading-bias cancellation is equivalent to $b^T A c=1/6$, while the other classical order-three condition may remain unsatisfied. On the sharp model, the leading coefficients are $(12\rho-1)/32$ and $(24\nu-1)/128$ for the declared two- and three-stage families. Uniformly as $\epsilon=r^2\to0$ in $r^2\le h\le r^{3/4}$, these coefficients multiply $h^2\epsilon^2$ in the displacement between nearby actual map and flow roots. The proof transports a finite Runge--Kutta defect to an actual section-defined root through exponential selection shielding, an exact common-target recurrence, and discrete Gaussian Melnikov quadrature. A right-fold van der Pol formula provides a classical calibration. A rational Rosenzweig--MacArthur specialization gives an actual local selected-threshold law in the original source clock, and a joint-$(r,h)$ finite-boundary proxy computation, independently checked at 192-bit precision, illustrates the predicted sign reversal and leading-bias cancellation.