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Because the API is stateless between calls, step i in the loop must include, as ordinary input, everything that made it into the transcript through step i-1 : the tool results, the intermediate reasoning, the earlier outputs. Call the amount a single step permanently adds to that growing transcript $\sigma$ — roughly $\delta$ plus $\bar{o}$ , since both the new material a step introduces and the output it produces become part of what every later step has to resend. Without caching, step i ’s input is not just $\tau$ + $\delta$ ; it is $\tau$ + $\delta$ plus the accumulated weight of every prior step, (i-1)$\sigma$ : $c_i = p_{\text{in}}\big(\tau + \delta + (i-1)\sigma\big) + p_{\text{out}}\,\bar{o}$. Summed over an n -step trajectory, that gives

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