Symbol i
i 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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Why this formula appears here
Model it simply. Suppose a session runs for k turns, and each turn adds on average 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 i tokens of accumulated result history, so the total billed across the whole session is
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Equation 9 · AI Infrastructure
The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.
Model it simply. Suppose a session runs for k turns, and each turn adds on average 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 i tokens of accumulated result history, so the total billed across the whole session is