addition

The passage around this formula

Collect the pieces above into one expression and the gap between a rate card and an invoice becomes explicit. For a single request, let n be new input tokens, w cache-write tokens, r cache-read tokens, y visible output tokens, z billed-but-invisible thinking tokens, and $\tau$ the fixed tool and system-prompt tokens added regardless of content. Let $p_{\mathrm{in}}$ , $p_{\mathrm{cw}}$ , $p_{\mathrm{cr}}$ , and $p_{\mathrm{out}}$ be the published rates, and let a $\in$ (0, 1] be the fraction of attempts an application accepts without needing to reissue the request, because it was neither truncated at the maximum-tokens cap nor lost to a timeout, an overload error, or a rate limit. Cost per accepted…

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