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Equation 12 · Part 3 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

Symbol n

Rno recovery=∏i=1npi.R_{\mathrm{no\ recovery}}=\prod_{i=1}^{n}p_i.
nn

What this part means

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

Its job in the formula

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

The passage around this formula

Suppose a task requires n materially dependent stages. If pip_i is the probability stage i succeeds conditional on all prior required state being correct, then Rno recovery=∏i=1npiR_{\mathrm{no\ recovery}}=\prod_{i=1}^{n}p_i. The equal- p simplification illustrates scale: 0.99^{100}≈\approx0.366 . 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 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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