Mathematics as a Zero-Residual Language: Exact Information-Theoretic Foundations of Mathematical Reasoning in Autoregressive Transformers

Large language models excel at mathematics because formal notation operates as a language with virtually zero semantic residual. Unlike natural language, which relies on implicit social context, unstated intent, and shared background assumptions, mathematical syntax explicitly and completely encodes its operational rules and relational constraints directly on the page. This transparency eliminates the ambiguity of inferring hidden meaning, allowing statistical pattern recognition to effectively discover and compress the valid transformation rules of formal deduction. Step-by-step reasoning enhances this process by expanding the computational reach of attention networks, converting complex multi-step deductions into manageable, sequentially verifiable state transitions. Because reasoning errors in these models stem from discrete search-path drift rather than semantic confusion, pairing the generative reach of language models with deterministic formal verification engines eliminates invalid derivations and guarantees mathematical soundness.