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Equation 7 · Part 8 · What We Still Cannot Do: Open Problems in Frontier Model Systems

Denominator: ln q

nmax⁡≈ln⁡τln⁡q.n_{\max} \approx \frac{\ln \tau}{\ln q}.
ln⁡q\ln q

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

ln q occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

so the horizon that can be sustained at a target success rate τ\tau is nmax⁡≈ln⁡τln⁡qn_{\max} \approx \frac{\ln \tau}{\ln q}. The derivative of nmax⁡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…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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