Equation 1 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

What does this equation mean?

\tau=(s_0,o_0,a_0,s_1,\ldots,a_{T-1},s_T),

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Inputs and operations(s_0,o_0,a_0,s_1,ldots,a_T-1,s_T)

Result or conditionτ

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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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\tau

Symbol τ

τ is part of the quantity the equation computes from the expression on the right.

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s_0

Symbol s_0

$s_0$ is one of the signed contributions combined to compute the quantity on the left.

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o_0

Symbol o_0

$o_0$ is one of the signed contributions combined to compute the quantity on the left.

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a_0

Symbol a_0

$a_0$ is one of the signed contributions combined to compute the quantity on the left.

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s_1

Symbol s_1

$s_1$ is one of the signed contributions combined to compute the quantity on the left.

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a_{T-1}

Symbol a_T-1

$a_T$ -1 is one of the signed contributions combined to compute the quantity on the left.

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s_T

Symbol s_T

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

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=

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

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subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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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

Let a task generate a trajectory $\tau=(s_0,o_0,a_0,s_1,\ldots,a_{T-1},s_T)$ . 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

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Sources cited in the article section

[2] Introducing GPT-5.3-Codex ↗

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