Equation 29 · Part 1 · OpenAI Model Systems from First Principles: Weights, Post-Training, and Inference Compute
Symbol p_k
What this part means
only achievable if something can identify the successful attempt.
Its job in the formula
is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
Full expression→Symbol p_k→Article meaning
Where the article explains 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.
The passage around this formula
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 ] .
Learn the underlying idea
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Sources cited in the surrounding passage
These citations provide research context; check each source for the exact claim it supports.