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

What does this equation mean?

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

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Inputs and operationsE_Z[prod_i=1^np_i(Z)]

Result or conditionR

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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R

Symbol R

R is computed from the expected values combined on the right.

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Z

Symbol Z

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

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i

Symbol i

i appears in the bound of this product. The bound states where the repeated operation starts, ends, or which values it includes.

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n

Symbol n

n appears in the bound of this product. The bound states where the repeated operation starts, ends, or which values it includes.

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p_i

Symbol p_i

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

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=

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

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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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superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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i=1

Starting index or lower bound: i=1

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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n

Ending index or upper bound: n

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

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How to interpret it

Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

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: $R=\mathbb{E}_Z\left[\prod_{i=1}^{n}p_i(Z)\right]$ . 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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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: $R=\mathbb{E}_Z\left[\prod_{i=1}^{n}p_i(Z)\right]$ . 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 test.

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