What does this equation mean?

Prompt caching is the documented fix for exactly this, and it is worth being precise about what it fixes and what it does not. Anthropic’s automatic caching, recommended for multi-turn use, places a single cache marker at the end of a request; the system caches everything up to that marker, and the next request’s identical prefix is read from cache rather than reprocessed [ 2 ] . Anthropic’s documentation walks through exactly this pattern for a growing conversation: at each new turn, the system prompt and all prior turns are read from cache, and only the newest exchange is freshly written to cache for next time [ 2 ] . A cache read costs a tenth of the base input price; a five-minute cache write costs 1.25 times it [ 1 , 2 ] . Substituting those multipliers into the model — reading the accumulated history at the discounted rate $\gamma$ $\approx$ 0.1 instead of the full rate, and writing only the newest increment at the write rate w $\approx$ 1.25 — gives $C_n^{\text{cached}} \approx n\,p_{\text{out}}\,\bar{o} + n\,p_{\text{in}}\,w\,\sigma + p_{\text{in}}\,\gamma\,\sigma\,\frac{n(n-1)}{2}$. The quadratic term survives. Its coefficient shrinks by the cache-read discount, roughly tenfold, but the shape — cost rising with the square of step count — does not go away, because caching discounts the price of resending the growing history; it does not stop the history from having to be resent. OpenAI’s own cached-input pricing lands at the same roughly tenfold discount on every current GPT-5.6 tier [ 7 ] , so the same qualified conclusion holds on that stack too: this is a property of a stateless, replay-everything request format, not a pricing quirk of one vendor.