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i−1i-1

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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 σ\sigma — roughly δ\delta plus oˉ\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 τ\tau + δ\delta ; it is τ\tau + δ\delta plus the accumulated weight of every prior step, (i-1)σ\sigma :

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i−1i-1

Equation 9 · AI Agents & Systems

The Economics and Physical Limits of Running AI Agents at Scale

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

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 σ\sigma — roughly δ\delta plus oˉ\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 τ\tau + δ\delta ; it is τ\tau + δ\delta plus the accumulated weight of every prior step, (i-1)σ\sigma :

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