Overparameterized models often have continuous parameter symmetries, so different parameters define the same predictor. We show that PAC--Bayesian analysis should be performed on the quotient predictor space: pushing a prior and posterior to the quotient preserves the empirical and population Gibbs risks while removing the nonnegative KL contribution caused solely by how the two distributions differ among parameterizations of the same predictor. Quotienting alone does not determine which prior to use. We construct a canonical choice of one parameterization for each predictor and account for the geometric volume of its equivalent parameterizations. This transforms a neutral reference prior into a data-independent prior that reflects the model's implicit bias. It approximates the ideal but inadmissible posterior-matched prior, which would minimize the KL term by depending on the training data. The resulting certificate is tighter exactly when this geometry-induced prior has smaller KL divergence from the learned quotient posterior than the neutral prior. We test this prediction in Fourier regression with a Hadamard parameterization and in Query-Key attention, using ordinary SGD without an explicit regularizer. The implicit-bias prior reduces the mean quotient-space KL by \(40.69\%\) and the mean PAC--Bayes certificate by \(21.40\%\) in the Fourier-Hadamard experiment. The smaller, prior-scale-dependent improvement in Query-Key attention confirms the predicted conditional nature of the effect.
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 unattainable even in the infinite-data regime. More broadly, our analysis shows the limits of prompt-based generalisation arise from information-constrained communication and alignment-constrained objectives. This suggests that interfaces beyond natural language, including multimodal observations and, external memory, may reduce the inherent LLM limitations by increasing the task-relevant information available to the System.