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Equation 44 · Part 1 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

Symbol d_i

diLi>Ki+Fi.d_i L_i > K_i+F_i.
did_i

What this part means

the detection probability.

Its job in the formula

did_i is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Where the article explains it

Let an unchecked latent error at stage i create expected downstream loss LiL_i , a check cost KiK_i , detection probability did_i , and false-alarm cost FiF_i .

The passage around this formula

Checkpoint placement is a value-of-information problem. Let an unchecked latent error at stage i create expected downstream loss LiL_i , a check cost KiK_i , detection probability did_i , and false-alarm cost FiF_i . A check is favorable when, approximately, diLi>Ki+Fid_i L_i > K_i+F_i. High-value checkpoints occur before expensive or irreversible transitions: after specification, before schema migration, after dependency changes, before broad refactors, before network or credential escalation, and before merge or deployment. Cheap deterministic checks should run earlier and more often than expensive human review.

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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The article lists its research sources here.