<p>Floridi’s Certainty–Scope Conjecture holds that no artificial intelligence system can be simultaneously reliable on every input and broad enough to cover the full richness of unstructured data: reliability and breadth are in structural tension, and the tension is imposed by the finite information-processing capacity of any mechanism implementing intelligence. The Conjecture has until now been a philosophical thesis without a proof. This article supplies one, in the continuous regime in which the semantic targets that matter philosophically actually live. The article is the continuous half of a coordinated diptych whose discrete companion proves a matching Fano-type converse in the finite classification setting. The technical machinery is classical: taking certainty to be expected distortion under a grounded semantic target, scope to be rate–distortion complexity, and mechanism capacity to be an admissibility-constrained quantity, the main theorem establishes that every decoder must incur at least the capacity-imposed distortion floor. The structural content lies in the operational consequences. A matched-Gaussian reverse water-filling rule gives the optimal allocation of a fixed budget across heterogeneous experts; a multi-task distortion floor formalises the intuition that no system is excellent at everything at once; a side-information theorem bounds the lift obtainable from retrieval or tool use. Four critical responses to the original formulation—from Lissack, from Immediato (two articles), and from Watson and Sterkenburg—are addressed, with a worked statistical-learning instantiation in the Bayesian regime that aligns the framework structurally with the Russo–Zou/Xu–Raginsky mutual-information generalisation-bound literature. The rate–distortion formulation is shown to be invulnerable to the objections raised against earlier Kolmogorov-complexity-based formulations.</p>