What does this equation mean?

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)$\sigma$ 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; only the newest increment written each turn — a roughly constant w$\sigma$ tokens per step, not a growing one — consumes ceiling headroom. Caching is therefore doing two separate jobs that are easy to conflate: it discounts the price of the growing history, and independently, it exempts that same growing history from the throughput governor. A workload that caches for the price discount and does not realize it is also buying rate-limit headroom is undercounting what caching is worth.