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# mathematics → training data quality → llm reasoning → steven byrnes → verifiability

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Pretraining Data Not Verifiability Is Why Llms Are

Steven Byrnes argues that LLMs perform well at math not because math is easy to verify, but because mathematical literature is almost entirely correct—giving the model clean training data to learn from. The "verifiability" explanation is circular since LLMs themselves do the verifying.
  • The common claim that "math is easy to verify" doesn't explain LLM math ability, because verification still relies on the LLM's judgment, which is what we're trying to explain in the first place.
  • Mathematical literature is >99% accurate, so imitative learning on this data naturally produces correct mathematical reasoning without needing heavy reinforcement learning or formal verification systems.
  • The theory predicts which domains LLMs should excel at: those with high-quality, accurate training data (like math and coding) versus those with noisier training signals (like subjective judgments in fiction or social domains).
Last saved Sep 21, 2026 · 2 min read
steven byrnes training data quality llm reasoning verifiability mathematics