A Limit of AI, I: A Discrete Information-Theoretic Proof of the Certainty–Scope Conjecture

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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, 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 finite-classification regime—the simplest non-trivial setting in which the Conjecture admits an exact statement, a sharp converse, and a matching achievability result—and is the discrete half of a coordinated diptych whose continuous companion proves a matching rate–distortion converse for non-finite semantic spaces. The technical machinery is classical: taking certainty to be Bayes-optimal accuracy and scope to be the class count n , a Fano-type argument yields a converse binding certainty and scope against mechanism capacity, with the n -ary symmetric channel attaining the exact certainty form with equality; the simplified explicit upper bound has asymptotic slack $$\ln 2/\ln n$$ on the certainty scale. Six structural extensions carry the philosophical weight. Multi-task composition compounds scope multiplicatively, so joint certainty decays as O (1/ T ); fixed-bandwidth side information yields vanishing gains; static post-processing of the mechanism’s output adds zero augmented capacity, placing chain-of-thought (so idealised) on the wrong side of the bound; sequential tool use requires logarithmically growing query counts; routing does not lift the local constraint; and context-wise feasibility need not compose into global feasibility. A synthetic bottleneck experiment illustrates the bound’s bite, with an analytical account of harder regimes via the Bhattacharyya bound. The diptych’s discrete-side engagement with Lissack, Immediato, and Watson and Sterkenburg is recorded.