Equation 29 · OpenAI Model Systems from First Principles: Weights, Post-Training, and Inference Compute
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only achievable if something can identify the successful attempt. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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which is concave in k and saturates quickly. Two caveats destroy any naive extrapolation from it. Attempts from one model on one prompt are strongly correlated, so realised gains fall well below this bound; and is only achievable if something can identify the successful attempt. Without a verifier, extra samples buy candidates, not answers. This is precisely why the reasoning-effort control and the availability of parallel test-time compute are architectural facts about a product rather than mere quality dials — the GPT-5 system card describes a variant that makes use of parallel test-time compute, and states that all models were evaluated at high reasoning effort [ 1 ] .
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