Nonnegative Runge–Kutta coefficients do not make modified Patankar schemes globally stable: a counterexample with attracting two-cycles

Unrefereed research candidate (version 1.0.1-candidate). Izgin, Kopecz, Meister and Schilling (Numer. Algorithms 96, 2024) conjectured that modified Patankar–Runge–Kutta (MPRK) schemes whose underlying Runge–Kutta coefficients are nonnegative are globally linearly stable. This package shows that the natural universal reading of that conjecture is false: MPRK22(4/7), all of whose nonzero tableau coefficients are positive, applied with step size 1 to the two-species linear system with transfer rates 9990 and 10 and started at (1/2, 1/2), converges to an attracting positive two-cycle, although the steady state (0.001, 0.999) is the unique positive fixed point and locally asymptotically stable. It also gives the complete phase portrait of this example, an interval of method parameters with two-cycles, certified parameter regions, certified examples for a three-species chain and for the third-order scheme MPRK43II(37/50), and proofs of two-species global convergence for alpha = 1/2 and alpha = 1, with a refined conjecture.

Computer-assisted statements are certified by standalone exact rational interval-arithmetic verifiers (Python standard library) whose certificates replay byte for byte; negative controls are included. AI-assisted work. Internal AI-assisted and adversarial review only: not peer reviewed, not independently reproduced, not formally verified in a proof assistant.