[Steuard]

All I have to say is that if one of my students turned in those curves as "best fits" to that data, I'd hand the paper back for a re-do. Those are garbage fits. To my eye, none of the very noisy data sets shown in the graph show clear enough trends to support one model over any other: are any of those hyperbolic curves convincingly better than even a linear fit? (No.) The "copilot code share" data can't possibly be described by a hyperbolic curve, because by definition it can't ever go over 100%. (A sigmoidal model might be plausible.) And even if you want to insist on a model that diverges at finite time, why fit 1/(t0-t) rather than 1/(t0-t)^2, or tan(t-t0), or anything else?

The author does in fact note that only the arXiv data fits this curve better than a line, and yeah: that's the one dataset that genuinely looks a little curved. But 1) it's a very noisy sort of curved, and 2) I'll bet it would fit a quadratic or an exponential or, heck, a sine function just as well. Introducing their process of doing the hyperbolic fit, they say, "The procedure is straightforward, which should concern you." And yeah, it does concern me: why does the author think that their standard-but-oversimplified attempt to fit a hand-chosen function to this mess is worth talking about? (And why put all of that analysis in the article, complete with fancy animated graph, when they knew that even their most determined attempt to find a signal failed to produce even a marginally supportive result 80% of the time?)

In short: none of the mathematical arguments used here to lead in to the article's discussion of "The Singularity" are worth listening to at all. They're pseudo-technical window dressing, meant to lend an undeserved air of rigor to whatever follows. So why should we pay attention to any of it?