No Universal Hyperbola: A Formal Disproof of the Epistemic Trade-Off Between Certainty and Scope in Symbolic and Generative AI

We formally disprove a recently conjectured artificial intelligence trade-off between epistemic certainty and scope in the universal hyperbolic product form in which it was published. Certainty is defined as the worst-case correctness probability over the input space, and scope as the sum of the Kolmogorov complexities of the input and output sets. Using standard facts from coding theory and algorithmic information theory, we show, first, that when the conjecture is instantiated with prefix (self-delimiting, prefix-free) Kolmogorov complexity, it leads to an internal inconsistency, and second, that when it is instantiated with plain Kolmogorov complexity, it is refuted by a constructive counterexample. These results establish a general theorem: contrary to the conjecture's claim, no universal "certainty-scope" hyperbola holds as a general bound under the published definitions.