Equation 47 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol i
i appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.
Symbol epsilon_i
epsilo is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
Symbol epsilon
epsilon is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
Starting index or lower bound: i
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.
How to interpret it
Read this expression with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
For consequential work, “be careful” is not a stopping rule. Let the maximum tolerable probability of material failure be . A conservative decomposition assigns budgets to failure classes or transitions such that . This union-bound allocation can be loose, but it forces explicit questions. How much risk comes from specification ambiguity? How much from an untested platform? What is the permitted probability of credential exposure or irreversible data migration? Which observation would reduce the dominant term?
For background, read the article’s source list.
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