The foundational impossibility results of distributed computing -- the Fischer-Lynch-Paterson theorem, the Two Generals Problem, the CAP theorem -- are widely understood as discoveries about the physical limits of coordination. This paper argues that they are nothing of the sort. They are consequences of a category mistake: treating Forward-In-Time-Only (FITO) information flow as a law of nature rather than recognizing it as a design choice inherited from Shannon's channel model and Lamport's happened-before relation. We develop this argument in six steps. First, we introduce the category mistake framework from Ryle through Spekkens' ontic/epistemic distinction in quantum foundations. Second, we identify FITO as the hidden axiom that unifies the classical impossibility results. Third, we apply Spekkens' Leibnizian principle to show that FITO-based models contain surplus ontological structure. Fourth, we develop the counterfactual: what changes when FITO is dropped. Fifth, we demonstrate that the impossibility theorems are theorems about FITO systems, not about physics. Sixth, we sketch the transactional alternative -- bilateral interactions that dissolve the apparent impossibilities by replacing unidirectional message passing with atomic bilateral transactions. The implication is that distributed computing has spent fifty years optimizing within the wrong design space.