# Hyperbolic trend with upcoming singularity fits METR capabilities estimates. 

*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*](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.](https://www.lesswrong.com/posts/GAJbegsvnd85hX3eS/thoughts-on-extrapolating-time-horizons) 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](https://www.lesswrong.com/posts/iPPe7qbZNcu5MsfPo/can-we-model-technological-singularity-as-the-phase) 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](https://en.wikipedia.org/wiki/Critical_exponent)).  For hyperbolic growth, such as [human population till 1960](https://www.lesswrong.com/posts/bYrF8rXFYwPqnfxTp/1960-the-year-the-singularity-was-cancelled), we have $q = 1$. [Here](https://www.lesswrong.com/posts/nfeRiz99pJGFjBATM/singularity-and-phase-transition-2-a-priori-probability-and) 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.](https://colab.research.google.com/drive/1AZM75HphIMrQfgK5uKsGPZEdZIuJWi6t?usp=sharing) 

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/3d2a0a948cbea21f01bb1e771225c75b8591eb945620f108.png)

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).

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/bd6f18731f9099fc3c71338d66da7421c518c660461e0580.png)

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

If we increase $q$, the mean square error decreases until $q = 40$. Here, for example, the plot for $q = 2$. 

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/f9625d8f0c1bade956a95fe9a35028cffe50ffa5b98052b5.png)

And for $q = 3$

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/e9ac1eedd074e5fa943f69891fcd134d33f5bf8c552d558f.png)

We see that unlike $q = 1$ it captures very well both initial and final time points. Going for higher $q$ improves the fit even more.  However, we should not go for too high $q$, since usually in physical systems the critical exponents ($q$ here) are of order 1. We can limit it in the other way: Nikola Jurkovic [suggests](https://www.lesswrong.com/posts/GAJbegsvnd85hX3eS/thoughts-on-extrapolating-time-horizons) that as soon as model can do a month-long task with 80% accuracy, it should be able to do any cognitive human task. In our terms, it means that singularity (when the frontier model can do tasks of any length) should be very close to the 1 month threshold. Let's set a condition that the time between 1-month threshold and singularity should be also not more than a few months. This leave us with only $q$ not bigger than 3, and 3 is already quite a stretch. 

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/1657100accd1eacfe0dd2d7454453ebb0d3e4b9c59e30574.png)

(The vertical dashed line is a singularity, the horizontal is a 1-month threshold. It is slightly more than 3 months from this threshold to singularity, so already a stretch for $q = 3$). This distance to singularity is growing with $q$

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/321dcea5fa45ca5c0c84b5fa1a40a602d58d82359199da32.png) 

q                                   t_c                      Δt (years)  

1                           2025-11-06                   0.0035  

2                           2026-05-06                   0.0872  

3                           2026-11-15                   0.3034  

4                           2027-05-29                   0.6159  

5                           2027-12-11                   0.9905

Now, the meatiest question: when is the singularity, if progress is still hyperbolic? It appears to be very soon for small $q$, from October of 2025 for $q =1$ to November of 2026 for $q = 3$ (funny, the later date almost precisely coincides with [human population singularity](https://www.science.org/doi/10.1126/science.132.3436.1291) Nov, 13, 2026).  

Does it sound realistic? Let's point at a few facts to make the statement less dramatic. Singularity in time horizons is not yet a technological singularity. It just means that AI can do any cognitive task a professional software engineer can do. It is even not necessarily AGI in a strong sense - I highly doubt that METR evaluation can go beyond standard tasks to the tasks of the level "once in a few years research breakthrough". AI may be able to saturate the benchmark "task duration measured by a professional software engineer" without reaching ability to create breakthrough research. So even if we observe AI doing all software engineering jobs, it has a dramatic and transformative effect on society, but it is not necessarily an AGI capable of self-improving yet. I think, with this disclaimer, automated average software engineers in a very short term is not something totally implausible.

What do I make of it? First, I am updating towards shorter timelines. Second, I think it would be a good practice to look at the trends in AI not only from an exponential point of view,  but also from divergent power-law point of view, as it may be a better fit, and, maybe, a better predictor.