What does this equation mean?

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 the turn count whenever new context arrives at a roughly steady rate, even though every token in it is billed at a ninety percent discount. Doubling the length of a session does not double its context-carrying cost; holding the per-turn addition fixed, it roughly quadruples it. This is an accounting identity that follows mechanically from the documented multipliers, not a measured benchmark, and it matches what Anthropic’s own guidance says qualitatively about why usage climbs in a long session: the full conversation is resent with every request, and each tool call adds another such request carrying that batch of results, so a one-line question in a session left open all day still draws usage for the whole conversation behind it [ 3 ] . The discount makes long sessions affordable turn to turn; it does not make them cheap in aggregate, because the thing being discounted is itself growing.