Abstract

Large language models (LLMs) encode problem difficulty as an internal signal that can be linearly decoded from their residuals. Given their multilingual capabilities, we investigate whether this meta-cognitive signal is language-agnostic and how it is organized across the model’s layers by training linear probes on the AMC subset of the Easy2Hard benchmark, translated into 21 languages. We found that difficulty-related signals emerge at two distinct stages of the model internals, corresponding to shallow (early-layers) and deep (later-layers) internal representations, that exhibit functionally different behaviors. Probes trained on deep representations achieve high accuracy when evaluated on the same language but exhibit weaker cross-lingual transfer. In contrast, probes trained on shallow representations generalize better across languages, despite achieving lower within-language performance. This closely aligns with existing findings in LLM interpretability, showing that models tend to operate in an abstract conceptual space before producing language-specific outputs. Our results suggest that this two-stage organizational principle extends beyond simple semantic processing to meta-cognitive properties such as problem difficulty, highlighting an internal control signal that is not tied to surface meaning.