Equation 12 · Part 9 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

Ending index or upper bound: n

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

n

What this part means

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

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.

Full expressionEnding index or upper bound: nArticle meaning

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

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

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See this notation across published equations →

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