A Limit of AI, II: A Continuous Rate-Distortion Proof of the Certainty-Scope Conjecture

Source abstract

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 admissibilityconstrained quantity, the main theorem establishes that every decoder must incur at least the capacityimposed 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.