Solvability detection is one of the most challenging aspects of mathematical reasoning for Large Language Models (LLMs). While prior work has studied this capability extensively, these analyses have been limited to English. Consequently, it remains unclear whether multilingual failures arise from differences in internal Solvability Belief or from language-dependent failures to express it. To address this gap, we introduce the first multilingual benchmark of paired solvable and unsolvable mathematical problems, extending ReliableMath to French and Greek. Using this, we train multilingual probes predicting Solvability Belief and analyze the solvability detection capabilities of state-of-the-art LLMs behaviorally, representationally, and in terms of faithfulness. We find that Solvability Belief is encoded as a largely universal, language-agnostic feature, and that higher-resource languages such as English, despite achieving stronger mathematical reasoning performance, exhibit lower solvability-detection faithfulness.

In current Mixture-of-Experts architectures, routing is performed based on representations dominated by structure shared across all tokens, limiting expert specialization. We show that contrasting each token against an Exponential Moving Average of the layer’s hidden states, rather than routing on absolute magnitude, concentrates the routing signal onto a low-dimensional, highly separable subspace. Building on this, we propose the Contrastive Routing Mechanism (CoRM), which scores each expert by the gap between its affinity for the incoming token and its affinity for this shared reference state, interpreted through a distinct per-expert projection. The resulting experts have routing boundaries that align with linguistic structure significantly more than the Top-k baseline. Our experiments show that CoRM improves average zero-shot accuracy by +0.67 to +1.69 points (Top-1) and +1.38 to +1.77 points (Top-2) over standard Top-k MoE baselines on nine zero-shot reasoning benchmarks, at the minimal cost of 2.9% added parameters and 2.6% added FLOPs per token.

Despite the remarkable capabilities of large language models (LLMs) across a range of tasks, mathematical reasoning remains a challenging frontier. Motivated by the observation that humans learn more effectively when prompted not what to think but how to think, we introduce BloomWise, a cognitively-inspired prompting technique designed to enhance LLMs’ performance on mathematical problem solving while making their solutions more explainable. BloomWise encourages LLMs to generate solutions - in the form of explanations - by progressing through a sequence of cognitive operations-from basic (e.g., remembering) to more advanced reasoning skills (e.g., evaluating) - mirroring how humans build understanding. The process iterates through these levels, halting early if a convergence criterion is met: specifically, if two or more consecutive levels yield the same answer, the solution from the earliest such level is output; otherwise, the process continues until all levels are completed. Through extensive experiments across five popular math reasoning datasets, we demonstrate the effectiveness of BloomWise. We also present comprehensive ablation studies to analyze the strengths of each component within our system.