EDIT: Big error in this post spotted and corrected by gjm here https://www.lesswrong.com/posts/ZEuDH2W3XdRaTwpjD/hyperbolic-model-fits-metr-capabilities-estimate-worse-than . I believe the motivation part still stands, but so far no experimental confirmation.

EDIT2: I also realized that we need to take into account confidence intervals provided by METR as well. I will make a separate post on them.

Recently METR published their measurements of AI abilities to complete long tasks for GPT-5, along with the previous models. They demonstrate that there is still an exponential trend, and GPT-5 does not deviate from it. Moreover, Nikola Jurkovic from METR suggests with the same data that there is already an even faster growing exponent. This pattern, however, indicates often that we might be able to better capture the process as something faster than exponential.

A few years ago I suggested that prior to a singularity, we should expect that a particular quantity, related to technological progress, should demonstrate a power-law divergence from analogy with condensed matter physics. The ability to complete long tasks seems to be a good candidate for such a quantity. Namely, we can suggest:

$H = \frac{C}{(t_{S} - t)^{q}}$

i.e. the time horizon $H$ (the length of the task the model cam complete) diverges as the time approaches the time of singularity $t_{S}$ , and the power is $q$ (in condensed matter physics called critical exponent). For hyperbolic growth, such as human population till 1960, we have $q = 1$ . Here I speculated that if instead of looking at the number of people, we look at the amount of processed information, or research progress, or something like that, we could observe hyperbolic growth after 1960 as well (indication - the rise of computers at this time).

Up to now, all this was speculation, but I provide it as reasoning why to look for a diverging power-law rather than an exponential function. So, after all this is said, let's see what happens if we fit this function to the data. You can play with it in the Colab notebok here.

Here for example the results for $q = 1$ , standard hyperbolic growth. The mean square error is already 3 times less than for exponential fit (image below).

And it also does not capture very well the last few time points.

If we increase $q$ , the mean square error decreases...