Equation 12 · Part 8 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

Starting index or lower bound: i=1

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

i=1

What this part means

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.

Its job in the formula

i=1 appears in the bound of this product. The bound states where the repeated operation starts, ends, or which values it includes.

Full expressionStarting index or lower bound: i=1Article 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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