Equation 12 · Part 4 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

Symbol p_i

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

p_i

What this part means

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

Its job in the formula

$p_i$ is an input to the expression that computes the quantity on the left.

Full expressionSymbol p_iArticle meaning

Where the article explains it

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 passage around this formula

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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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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