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Equation 10 · Part 9 · The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol

subscript

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

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

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}. Doubling the length of a session does not double the bill; it roughly quadruples it, because the earlier results are being paid for again on every one of the later turns. This is not a hypothetical — it is the direct arithmetic consequence of the same design choice OpenAI documents for its own tool schemas: whatever is…

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.