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Equation 25 · Part 16 · The Economics and Physical Limits of Running AI Agents at Scale

Denominator: 2

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}
22

What this part means

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

Its job in the formula

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

The passage around this formula

…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…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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