What early exponential growth can't tell us about runaway processes

This is an essay about a simple but (I think) underappreciated point:

If a process follows a logistic curve, then early behavior is almost indistinguishable from exponential growth, which means you generally can’t infer its eventual carrying capacity from early data.

In other words, very different long-term outcomes can all be consistent with the same initial “runaway” phase, even with very good data.

I use Project Hail Mary as a framing device (basically mathematical fanfiction), but the real issues are inference and model choice.

Curious whether people think this is a fair critique, or whether I’m overstating how ill-conditioned such problems are in practice.

https://mathenchant.wordpress.com/2026/04/18/sorry-mary