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Published equation contexts

ρ(n,m)  =  1−σ+msˉτ0+nsˉ\rho(n, m) \;=\; 1 - \frac{\sigma + m\bar{s}}{\tau_0 + n\bar{s}}

Why this formula appears here

Anthropic’s Tool Search Tool is a direct, documented implementation of that idea. The mechanism is described plainly: “You provide all your tool definitions to the API, but mark tools with deferlr_loading: true to make them discoverable on-demand,” with a lightweight search tool — built-in options include BM25-based and regex-based search — querying an index of names and descriptions, and the full definition of a matching tool injected into context only once the model actually requests it [ 5 ] . The search tool itself is not free, but its overhead is small and fixed rather than proportional to catalogue size: “Only the Tool Search Tool loaded upfront (~500 tokens)” [ 5 ] . Applied to the same…

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nn

Symbol n

n occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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τ0\tau_0

Symbol tau_0

tau0u_0 occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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τ0+nsˉ\tau_0 + n\bar{s}

Denominator: tau_0 + nbars

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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ρ(n,m)  =  1−σ+msˉτ0+nsˉ\rho(n, m) \;=\; 1 - \frac{\sigma + m\bar{s}}{\tau_0 + n\bar{s}}

Equation 22 · AI Infrastructure

The Token Tax of Giving a Model More Tools

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

Anthropic’s Tool Search Tool is a direct, documented implementation of that idea. The mechanism is described plainly: “You provide all your tool definitions to the API, but mark tools with deferlr_loading: true to make them discoverable on-demand,” with a lightweight search tool — built-in options include BM25-based and regex-based search — querying an index of names and descriptions, and the full definition of a matching tool injected into context only once the model actually requests it [ 5 ] . The search tool itself is not free, but its overhead is small and fixed rather than proportional to catalogue size: “Only the Tool Search Tool loaded upfront (~500 tokens)” [ 5 ] . Applied to the same…

Meanings in this article

  • mm: the session that only ever actually needs.
  • σ\sigma: the search tool’s own fixed overhead.
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