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Published equation contexts

Ci=0.1 pin ci−1  +  1.25 pin Δi  +  pout oi,ci=ci−1+ΔiC_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

Why this formula appears here

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 Ci=0.1 pin ci−1  +  1.25 pin Δi  +  pout oi,ci=ci−1+ΔiC_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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pinp_{\mathrm{in}}

Symbol p_in

pip_in is one of the signed contributions combined to compute the quantity on the left.

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ci−1c_{i-1}

Symbol c_i-1

cic_i-1 is one of the signed contributions combined to compute the quantity on the left.

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Δi\Delta_i

Symbol Delta_i

Deltaia_i is one of the signed contributions combined to compute the quantity on the left.

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poutp_{\mathrm{out}}

Symbol p_out

pop_out is one of the signed contributions combined to compute the quantity on the left.

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Published contexts (1)

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Ci=0.1 pin ci−1  +  1.25 pin Δi  +  pout oi,ci=ci−1+Δi.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.

Equation 1 · AI Agents & Systems

The Economics, Energy, and Physical Limits of Claude Code and Agentic Development Tools

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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 Ci=0.1 pin ci−1  +  1.25 pin Δi  +  pout oi,ci=ci−1+ΔiC_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…

Meanings in this article

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