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Equation 10 · Part 3 · The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol

Symbol Δ

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

What this part means

the number of tokens.

Its job in the formula

Δ is one of the signed contributions combined to compute the quantity on the left.

Where the article explains it

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

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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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