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

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.