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Equation 12 · Part 7 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

superscript

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

What this part means

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.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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