Adversarial Barrier in Uniform Class Separation
Milan Rosko
TL;DR
The paper identifies a structural obstruction to Uniform Separation within predicative arithmetic by leveraging a uniform, arithmetically representable classifier-interface extracted from Kleene realizability in HA. It shows that while instancewise solvability and provability align for $\Sigma^0_1$ predicates, demanding a single interface whose commitments are internally certified leads to a diagonal self-reference that cannot be uniformly satisfied in $\mathsf{HA}$ (and its arithmetical extensions under standard assumptions). Through a uniform refuter construction and a diagonal fixed-point argument, it derives a reflection-triggered contradiction, yielding an Adversarial Barrier that blocks uniform separation at the level of logic rather than semantic content. The results imply that the uniform version of class separation cannot be established in $\mathsf{HA}$ or in theories extending it, offering a barrier distinct from classical limits like Baker–Rudich or algebrization.
Abstract
We identify a strong structural obstruction to Uniform Separation in constructive arithmetic. The mechanism is independent of semantic content; it emerges whenever two distinct evaluator predicates are sustained in parallel and inference remains uniformly representable in an extension of HA. Under these conditions, any putative Uniform Class Separation principle becomes a distinguished instance of a fixed point construction. The resulting limitation is stricter in scope than classical separation barriers (Baker; Rudich; Aaronson et al.) insofar as it constrains the logical form of uniform separation within HA, rather than limiting particular relativizing, naturalizing, or algebrizing techniques.