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Equation 22 · Part 3 · The Token Tax of Giving a Model More Tools

Symbol m

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

What this part means

the session that only ever actually needs.

Its job in the formula

m occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Where the article explains it

Applied to the same 58-tool, roughly 55,000-token example described above, that reframes the earlier formula: instead of the full T(n) ≈\approx τ0\tau_0 + nsˉ\bar{s} , a session that only ever actually needs m of the n available tools pays approximately τ0\tau_0 + σ\sigma + msˉ\bar{s} , where σ\sigma is the search tool’s own fixed overhead.

The passage around this formula

…tokens)” [ 5 ] . Applied to the same 58-tool, roughly 55,000-token example described above, that reframes the earlier formula: instead of the full T(n) ≈\approx τ0\tau_0 + nsˉ\bar{s} , a session that only ever actually needs m of the n available tools pays approximately τ0\tau_0 + σ\sigma + msˉ\bar{s} , where σ\sigma is the search tool’s own fixed overhead. The fraction of the original tax avoided is ρ(n,m)  =  1−σ+msˉτ0+nsˉ\rho(n, m) \;=\; 1 - \frac{\sigma + m\bar{s}}{\tau_0 + n\bar{s}}. which grows toward its ceiling as m shrinks relative to n . Anthropic reports a measured outcome…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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