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Published equation contexts

Cncached≈n pout oˉ+n pin w σ+pin γ σ n(n−1)2C_n^{\text{cached}} \approx n\,p_{\text{out}}\,\bar{o} + n\,p_{\text{in}}\,w\,\sigma + p_{\text{in}}\,\gamma\,\sigma\,\frac{n(n-1)}{2}

Why this formula appears here

Prompt caching is the documented fix for exactly this, and it is worth being precise about what it fixes and what it does not. Anthropic’s automatic caching, recommended for multi-turn use, places a single cache marker at the end of a request; the system caches everything up to that marker, and the next request’s identical prefix is read from cache rather than reprocessed [ 2 ] . Anthropic’s documentation walks through exactly this pattern for a growing conversation: at each new turn, the system prompt and all prior turns are read from cache, and only the newest exchange is freshly written to cache for next time [ 2 ] . A cache read costs a tenth of the base input price; a five-minute cache…

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CncachedC_n^{\text{cached}}

Symbol C_n^cached

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

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poutp_{\text{out}}

Symbol p_out

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

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oˉ\bar{o}

Symbol baro

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

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pinp_{\text{in}}

Symbol p_in

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

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ww

Symbol w

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

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σ\sigma

Symbol σ

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

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γ\gamma

Symbol gamma

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

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Its accuracy depends on the assumptions and range of use described in the article.

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Published contexts (1)

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

Cncached≈n pout oˉ+n pin w σ+pin γ σ n(n−1)2C_n^{\text{cached}} \approx n\,p_{\text{out}}\,\bar{o} + n\,p_{\text{in}}\,w\,\sigma + p_{\text{in}}\,\gamma\,\sigma\,\frac{n(n-1)}{2}

Equation 25 · AI Agents & Systems

The Economics and Physical Limits of Running AI Agents at Scale

This equation gives an approximation: it relates the quantities while allowing an approximation.

Prompt caching is the documented fix for exactly this, and it is worth being precise about what it fixes and what it does not. Anthropic’s automatic caching, recommended for multi-turn use, places a single cache marker at the end of a request; the system caches everything up to that marker, and the next request’s identical prefix is read from cache rather than reprocessed [ 2 ] . Anthropic’s documentation walks through exactly this pattern for a growing conversation: at each new turn, the system prompt and all prior turns are read from cache, and only the newest exchange is freshly written to cache for next time [ 2 ] . A cache read costs a tenth of the base input price; a five-minute cache…

Equation guide → · Article →