On the Limits of Prompt-Conditioned Language Models as General-Purpose Learners

Large Language Models (LLMs) are frequently portrayed as general-purpose solvers capable of solving arbitrary tasks. We argue that this view overlooks a fundamental constraint: language is a compressed and capacity-limited interface for conveying task information. Modelling User--System interaction as a bilevel \emph{cheap-talk} game, we analyse how latent tasks are encoded into prompts and reinterpreted under alignment and safety constraints. We introduce a conceptual decomposition separating task inference from execution and derive PAC-Bayes bounds that distinguish finite-sample estimation error from irreducible structural limitations. Our first main result establishes an \emph{expressivity floor}: language acts as a capacity-limited communication channel, and whenever the informational complexity of a task family exceeds the capacity of that channel, distinct tasks become unavoidably indistinguishable to the Solver, inducing a strictly positive error floor that cannot be eliminated by additional data, optimisation, or model scaling alone. We then establish an \emph{objective-misalignment floor}: when alignment constraints restrict the admissible output set, the User-ideal distribution may lie outside the feasible class, inducing an irreducible distortion. Together, these results yield a formal negative conclusion: prompt-conditioned LLMs are not universal problem solvers through prompting alone, as there exist task families for which correct behaviour is provably unattaina