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Equation 23 · Part 5 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

Symbol p_i

R=EZ[∏i=1npi(Z)].R=\mathbb{E}_Z\left[\prod_{i=1}^{n}p_i(Z)\right].
pip_i

What this part means

pip_i appears inside an expected value, so its contribution is averaged under the distribution or condition shown by that operator.

Its job in the formula

pip_i appears inside an expected value, so its contribution is averaged under the distribution or condition shown by that operator.

The passage around this formula

The product formula is most misleading when failures share a cause. Let Z represent a latent condition such as an ambiguous requirement, poisoned dependency, missing platform constraint, or flawed test oracle. Conditional on Z , stages or parallel agents may appear independent; marginally, their errors are correlated: R=EZ[∏i=1npi(Z)]R=\mathbb{E}_Z\left[\prod_{i=1}^{n}p_i(Z)\right]. Running five agents against the same incomplete specification does not create five independent opinions. They can converge on the same wrong interpretation because they share context, model family, tools, and evaluator. Even model diversity may leave the common-mode cause untouched if every run receives the same poisoned observation or passes the same weak…

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A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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