Equation 8 · Part 3 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

Symbol D

R=\mathbb{E}_{g\sim D}\mathbb{E}_{\tau\sim P(\tau\mid g)} \left[V(s_T,g)\right].

D

What this part means

the system reliability under task distribution.

Its job in the formula

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

Full expressionSymbol DArticle meaning

Where the article explains it

System reliability under task distribution D is $R=\mathbb{E}_{g\sim D}\mathbb{E}_{\tau\sim P(\tau\mid g)} \left[V(s_T,g)\right]$ .

The passage around this formula

where $s_t$ is environmental state, $o_t$ the agent’s observation, and $a_t$ its action. Task success is not a property of the final message. It is an externally evaluated predicate V( $s_T$ ,g) over final state and goal g . System reliability under task distribution D is $R=\mathbb{E}_{g\sim D}\mathbb{E}_{\tau\sim P(\tau\mid g)} \left[V(s_T,g)\right]$ . Every term matters. Change the task distribution, model, harness, tools, retry budget, permissions, environment, or evaluator and the reliability estimate changes.

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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See this notation across published equations →

Sources cited in the article section

[2] Introducing GPT-5.3-Codex ↗

These citations provide research context; check each source for the exact claim it supports.