Equation 20 · OpenAI Codex as a Software-Engineering System: Models, Harnesses, and Evidence

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h_{k+1}=f(h_k,o_k,a_k)

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.

Inputs and operationsf(h_k,o_k,a_k)

Result or conditionh_k+1

How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value 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.

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h_{k+1}

Symbol h_k+1

$h_k$ +1 is part of the quantity the equation computes from the expression on the right.

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f

Symbol f

f is an input to the expression that computes the quantity on the left.

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h_k

Symbol h_k

$h_k$ is an input to the expression that computes the quantity on the left.

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o_k

Symbol o_k

$o_k$ is an input to the expression that computes the quantity on the left.

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a_k

Symbol a_k

$a_k$ is an input to the expression that computes the quantity on the left.

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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addition

Add the term after the plus sign to the term or group before it.

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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.

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How to interpret it

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What the article says around this equation

Long sessions pose a lossy-state problem. When a transcript exceeds context capacity, a harness may compact earlier interaction into a summary. If $h_{k+1}$ =f( $h_k$ , $o_k$ , $a_k$ ) compresses state, reliability depends on whether f preserves commitments, failed hypotheses, file identities, and unresolved constraints. More context tokens do not remove this requirement; they postpone it.

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Sources cited in the article section

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