A fixed-mesh global error theorem does not certify an adaptive controller

Global error from uniform one-step defect and stability assumes a prescribed uniform mesh $t_n=a+nh$, a one-step defect bound on that mesh, and a stability estimate for propagation on those steps. Its conclusion therefore applies when those hypotheses hold. The controller in Finite-budget adaptive Runge–Kutta controller accepts variable, state-dependent step sizes based on a pair difference. That difference is not the exact one-step defect hypothesis of the fixed-mesh theorem. No uniform mesh, defect reliability estimate, mesh-ratio control, or variable-step propagation theorem is supplied by that theorem. It may be used on a particular adaptive run only after those distinct hypotheses have independently been proved; its fixed-mesh statement alone yields no adaptive global error certificate.