Equation 9 · Part 1 · Claude Code, From First Principles: The Agentic Loop, Permissions, and What Actually Executes

Symbol c_t+1

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

c_{t+1}

What this part means

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

Its job in the formula

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

Full expressionSymbol c_t+1Article meaning

The passage around this formula

Claude Code executes $a_t$ — subject to a permission check described in the next section — against the real filesystem, shell, or network, producing an observation $o_t$ : a diff applied, a command’s stdout and exit code, a search result. That observation is appended to the context for the next step: $c_{t+1} = c_t \oplus (a_t,\ o_t)$ . The loop repeats until $\pi$ 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: $a_t$ is not a hypothetical action but a call to a real interpreter with real side effects on a real machine, $o_t$ is not simulated feedback but…

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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.

Open the illustrated subscripts: which member of a family? guide →

See this notation across published equations →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.