Equation 12 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

What does this equation mean?

R_{\mathrm{no\ recovery}}=\prod_{i=1}^{n}p_i.

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operationsprod_i=1^np_i

Result or conditionR_no recovery

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.

Read it piece by piece

R_{\mathrm{no\ recovery}}

Symbol R_no recovery

$R_n$ o recovery is part of the quantity the equation computes from the expression on the right.

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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

the probability stage i succeeds conditional on all prior required state being correct.

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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

Suppose a task requires n materially dependent stages. If $p_i$ is the probability stage i succeeds conditional on all prior required state being correct, then $R_{\mathrm{no\ recovery}}=\prod_{i=1}^{n}p_i$ . The equal- p simplification illustrates scale: 0.99^{100} $\approx$ 0.366 . This is not a model of real coding trajectories—the stages are correlated, checkpoints exist, and some errors are reversible—but it defeats one intuition. “Usually correct” local behavior does not imply dependable long execution.

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