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Published equation contexts

ct+1=ct⊕(at, ot)c_{t+1} = c_t \oplus (a_t,\ o_t)

Why this formula appears here

Claude Code executes ata_t — subject to a permission check described in the next section — against the real filesystem, shell, or network, producing an observation oto_t : a diff applied, a command’s stdout and exit code, a search result. That observation is appended to the context for the next step: ct+1=ct⊕(at, ot)c_{t+1} = c_t \oplus (a_t,\ o_t). The loop repeats until π\pi emits a stop rather than a tool call. Nothing in this is exotic; it is the standard shape of a tool-using agent. What matters for this article is what each symbol is standing in for in a real, shipped product: ata_t is not a hypothetical action but a call to a real interpreter with real side effects on a real machine, oto_t is not simulated feedback but…

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ct+1c_{t+1}

Symbol c_t+1

ctc_t+1 is the next indexed value computed from the current indexed quantity and the update terms shown on the right.

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Published contexts (1)

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ct+1=ct⊕(at, ot).c_{t+1} = c_t \oplus (a_t,\ o_t).

Equation 9 · AI Agents & Systems

Claude Code, From First Principles: The Agentic Loop, Permissions, and What Actually Executes

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Claude Code executes ata_t — subject to a permission check described in the next section — against the real filesystem, shell, or network, producing an observation oto_t : a diff applied, a command’s stdout and exit code, a search result. That observation is appended to the context for the next step: ct+1=ct⊕(at, ot)c_{t+1} = c_t \oplus (a_t,\ o_t). The loop repeats until π\pi emits a stop rather than a tool call. Nothing in this is exotic; it is the standard shape of a tool-using agent. What matters for this article is what each symbol is standing in for in a real, shipped product: ata_t is not a hypothetical action but a call to a real interpreter with real side effects on a real machine, oto_t is not simulated feedback but…

Meanings in this article

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