A Limit of AI, I: A Discrete Information-Theoretic 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, 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 nontrivial 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.