addition

The passage around this formula

Two structural consequences follow directly from that documented pricing, and they are analysis, not disclosure, since Anthropic states the multipliers but does not publish this derivation. Model a session as a sequence of turns. At turn i, the agent resends the accumulated transcript, c sub i-minus-one, as a cache read; adds a batch of new tokens, delta sub i, as a cache write, from a tool result or a newly opened file; and produces o sub i output tokens. Writing p-in and p-out for the base input and output prices, the billed cost of that turn is $C_i = 0.1\,p_{\mathrm{in}}\,c_{i-1} \;+\; 1.25\,p_{\mathrm{in}}\,\Delta_i \;+\; p_{\mathrm{out}}\,o_i, \qquad c_i = c_{i-1} + \Delta_i$. Summed across an N-turn session, the accumulated-context term is a triangular sum that grows in proportion to the square of…

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