Equation 20 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

What does this equation mean?

p_i^*=p_i+(1-p_i)d_i c_i r_i.

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Inputs and operationsp_i+(1-p_i)d_i c_i r_i

Result or conditionp_i^*

How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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p_i^*

Symbol p_i^*

the simplified effective stage reliability.

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p_i

Symbol p_i

the direct success probability.

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d_i

Symbol d_i

$d_i$ is one of the signed contributions combined to compute the quantity on the left.

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c_i

Symbol c_i

$c_i$ is one of the signed contributions combined to compute the quantity on the left.

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r_i

Symbol r_i

the probability repair succeeds without creating another material defect.

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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addition

Add the term after the plus sign to the term or group before it.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

An agent can recover only if the system generates a discriminating observation. Let stage i have direct success probability $p_i$ . If failure occurs, let $d_i$ be the probability it is detected, $c_i$ the probability the cause is diagnosed correctly, and $r_i$ the probability repair succeeds without creating another material defect. A simplified effective stage reliability is $p_i^*=p_i+(1-p_i)d_i c_i r_i$ . The formula is intentionally conservative and incomplete. Multiple retries, repair costs, correlated defects, and false alarms require richer models. Its value is conceptual: “the agent can iterate” improves reliability only through detection, diagnosis, and repair.

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Sources cited in the article section

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