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Equation 10 · Part 2 · The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol

Symbol k

B(k)=Δ∑i=1ki=Δ k(k+1)2.B(k) = \Delta \sum_{i=1}^{k} i = \Delta\,\frac{k(k+1)}{2}.
kk

What this part means

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

Its job in the formula

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

The passage around this formula

Model it simply. Suppose a session runs for k turns, and each turn adds on average Δ\Delta tokens of new tool result to a transcript that is resent in full on every request, because nothing is evicting or compressing it. The tokens billed on turn i include roughly iΔ\Delta tokens of accumulated result history, so the total billed across the…

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Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the surrounding passage

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