← Mathematical compendium

Published equation contexts

∣ct∣>τ⋅Cmax  ⟹  ct→c^t,∣c^t∣≪∣ct∣|c_t| > \tau \cdot C_{\mathrm{max}} \implies c_t \to \hat{c}_t, \quad |\hat{c}_t| \ll |c_t|

Why this formula appears here

The growth is monotonic and the response is a threshold rule, which is worth writing down because it is the one piece of real mechanism in this section rather than product description. Treat |ctc_t| as the token size of the context at step t and CmaxC_{\mathrm{max}} as the model’s context limit. Compaction is triggered once accumulated context crosses a configurable fraction τ\tau of that limit: ∣ct∣>τ⋅Cmax  ⟹  ct→c^t,∣c^t∣≪∣ct∣|c_t| > \tau \cdot C_{\mathrm{max}} \implies c_t \to \hat{c}_t, \quad |\hat{c}_t| \ll |c_t|. Concretely, Claude Code “clears older tool outputs first, then summarizes the conversation if needed,” preserving “requests and key code snippets” while allowing “detailed instructions from early in the conversation” to be lost [ 2 ] . This is a lossy operation by design, and what…

Read the full article-specific guide →

Read the representative guide

ctc_t

Symbol c_t

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

Read this term in its guide →
τ\tau

Symbol τ

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

Read this term in its guide →
CmaxC_{\mathrm{max}}

Symbol C_max

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

Read this term in its guide →
c^t\hat{c}_t

Symbol hatc_t

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

Read this term in its guide →

How to interpret it

Read this expression with the definitions, units, and assumptions supplied by the article.

Research cited beside this formula

Learn the mathematics behind it

Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

∣ct∣>τ⋅Cmax  ⟹  ct→c^t,∣c^t∣≪∣ct∣.|c_t| > \tau \cdot C_{\mathrm{max}} \implies c_t \to \hat{c}_t, \quad |\hat{c}_t| \ll |c_t|.

Equation 17 · AI Agents & Systems

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

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

The growth is monotonic and the response is a threshold rule, which is worth writing down because it is the one piece of real mechanism in this section rather than product description. Treat |ctc_t| as the token size of the context at step t and CmaxC_{\mathrm{max}} as the model’s context limit. Compaction is triggered once accumulated context crosses a configurable fraction τ\tau of that limit: ∣ct∣>τ⋅Cmax  ⟹  ct→c^t,∣c^t∣≪∣ct∣|c_t| > \tau \cdot C_{\mathrm{max}} \implies c_t \to \hat{c}_t, \quad |\hat{c}_t| \ll |c_t|. Concretely, Claude Code “clears older tool outputs first, then summarizes the conversation if needed,” preserving “requests and key code snippets” while allowing “detailed instructions from early in the conversation” to be lost [ 2 ] . This is a lossy operation by design, and what…

Equation guide → · Article →