$$n_{\max} \approx \frac{\ln \tau}{\ln q}$$

Equation 7 · Foundation Models

What We Still Cannot Do: Open Problems in Frontier Model Systems

This equation gives an approximation: it relates the quantities while allowing an approximation.

so the horizon that can be sustained at a target success rate $\tau$ is $n_{\max} \approx \frac{\ln \tau}{\ln q}$. The derivative of $n_{\max}$ with respect to q is steep near q = 1 , which is the good news — small per-step gains buy disproportionate horizon. The bad news is the same expression read the other way: sustaining an order of magnitude more steps requires driving per-step error down by an order of magnitude, and every one of the six problems above is a term in that per-step error which scale is not reducing. Verification, calibration, and memory are precisely the mechanisms that break the exponential by resetting accumulated uncertainty. Without them, capability gains are spent buying a slowly lengthening…

Meanings in this article

• $n_{\max}$: the derivative of.
• $\tau$: the target success rate.
• $q$: the respect to.

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