Teach Only What Is Missing: Frontier-Calibrated Counterfactual Self-Distillation for Multilingual Reasoning

An instruction-tuned language model that solves a math question posed in English often fails the same question posed in a low-resource language, although its reasoning stays English-centered inside. Existing methods add a cross-lingual signal at the input, by translating the question or the training solutions, or at the answer, by preferring English-derived responses or rewarding correct ones. None of them locates where, inside the reasoning, the target-language question falls short, and none uses the model's own English-conditioned distribution as the target. We start from a simple test: a prefix of the model's own verified English reasoning, placed before the target-language question's reasoning, restores the correct answer, and the length needed differs across questions and languages. We propose Counterfactual English-Conditioned Ladder Training (CELT), a self-distillation method that trains the model toward its own English-conditioned distribution, only on the missing span. To locate the span, Frontier Measurement probes the model on the target-language question with a growing number of steps of the English trace supplied, and records the smallest sufficient count as the frontier. To close it, Counterfactual Distillation matches the next-token distributions of the adapted model under the target-language question to those of the frozen model under the English question, with full weight on the missing span, from several starting depths. To keep what the model already does and to help it read the question, we add Own-Solution Preservation and Question Reading. At inference nothing is added. Across three backbones and three benchmarks, CELT is the most accurate method in every language, with the largest margins on low-resource languages, and improves English as well, because the supervision comes from the model itself and lands where each question's reasoning is missing.