Symbol w
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That exemption changes the shape of the rate-limit exposure the same way it changed the shape of the dollar cost. In the uncached model above, step i sends (i-1) tokens of retransmitted history as ordinary, fully metered input — every one of which counts against the per-minute ceiling, every single step, growing linearly with i within a single trajectory. A long, rapid, uncached loop can approach its own organization’s input-tokens-per-minute ceiling from its own growth alone, with no other caller involved: the trajectory throttles itself. With per-turn caching applied, the accumulated history is read from cache and, per the documentation above, does not count toward the limit at all;…
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Equation 37 · 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.
That exemption changes the shape of the rate-limit exposure the same way it changed the shape of the dollar cost. In the uncached model above, step i sends (i-1) tokens of retransmitted history as ordinary, fully metered input — every one of which counts against the per-minute ceiling, every single step, growing linearly with i within a single trajectory. A long, rapid, uncached loop can approach its own organization’s input-tokens-per-minute ceiling from its own growth alone, with no other caller involved: the trajectory throttles itself. With per-turn caching applied, the accumulated history is read from cache and, per the documentation above, does not count toward the limit at all;…