What does this equation mean?

That convergence on roughly one tenth of base input for a cache read — independently verified today across all three platforms [ 2 , 6 , 10 ] — supports a design rule. Let w be the write-price multiplier for the chosen retention window and r $\approx$ 0.1 the read-price multiplier. For a prefix served identically R times within that window — the first serving is the write, the remaining R-1 are reads — caching beats never caching once $R^{*} = \frac{w - r}{1 - r}$. is exceeded. At Anthropic’s five-minute rate, w = 1.25 and $R^{*}$ $\approx$ 1.28 : the prefix pays for itself starting with the first read after the write, matching the break-even Anthropic’s pricing page states for that duration [ 6 ] . At the one-hour rate, w = 2 and $R^{*}$ $\approx$ 2.11 , matching the stated two-read break-even for that window [ 6 ] . The formula’s use is computing the same threshold for a w and r that are not round numbers, and deciding, before writing code, whether a prefix’s expected reuse clears the bar for the TTL about to be selected.