Why does a long conversation get expensive faster than it gets long?

An answer that works

Because every turn resends everything before it. The conversation grows linearly and the tokens billed grow with the square of the turn count: forty turns of a conversation that ends at 25,800 tokens bills about 564,000, roughly 22 times its own final size. Prefill latency follows the same curve, so turn 40 is also about ten times slower to start than turn 1. Prompt caching and compaction fix both, and caching only works if history is appended rather than rewritten.