Numerator: k(k+1)

The passage around this formula

Model it simply. Suppose a session runs for k turns, and each turn adds on average $\Delta$ 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 i$\Delta$ tokens of accumulated result history, so the total billed across the whole session is $B(k) = \Delta \sum_{i=1}^{k} i = \Delta\,\frac{k(k+1)}{2}$. 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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