Five disciplines discovered the same math independently

[energyscholar]

Author here. We found the same mathematical structure appearing independently in physics (phase transitions), finance (market crashes), ecology (extinction cascades), neuroscience(neural criticality), and network science (cascade failures).

Each field derived it from first principles. Each named it differently. Minimal cross-citation. The affiliated scientific paper traces this convergent discovery and asks: if the same structure keeps emerging, what does that tell us about how we organize knowledge?

[arjie]

It reminds me of “Tai’s method” of integration - an approximation discovered in 1994.

https://academia.stackexchange.com/questions/9602/rediscover...

I think I found it in that other world that is the past on Slashdot - which was a Hacker News from another era https://m.slashdot.org/story/144664

[energyscholar]

I agree that's a good parallel. I had not seen it before. Thanks for the link.

[bonsai_spool]

I wish authors would use their own voice instead of an LLM, especially in a rhetorical piece. I like the history of science, and might have otherwise read the authors' paper, but the use of LLM-isms throughout this page makes me worry that the arxiv submission will show the same lack of care/effort.

Here's the manuscript at any rate, somewhat hard to find on the webpage:

Convergent Discovery of Critical Phenomena Mathematics Across Disciplines: A Cross-Domain Analysis https://arxiv.org/abs/2601.22389

[jmcgough]

I felt the same way reading the linked webpage. Reads like minimally edited LLM output, which makes me question how much effort was put into the research itself. Was the research all LLM too? How much of the paper was LLM?

[energyscholar]

Fair call on the website — we built it fast and it shows. The paper itself is a traditional literature review and citation analysis. I am one of two human authors. We use standard methodology. Didier Sornette endorsed it for arXiv.

Thanks for pulling out the direct link. I'll change the site to make it more prominent. This is my first serious attempt at social media engagement. Thanks for pointing out flaws and where there's room for improvment.

[zozbot234]

OP's comments in this thread are also pure clanker speak, which is disappointing and shows a lack of awareness of what HN is for.[0] It would be nice if an established scholar in this area of mathematics (complex systems) could comment re: this proposed correspondence and whether it has been noticed before. To be sure, similarly duplicative developments, gratuitous differences in terminology, etc. are discovered all the time, this isn't huge news. Statistics and ML is a well-known example.

[0] I haven't actually tried this, but I'm pretty sure that even just telling the robot "please write tersely, follow the typical style for HN comments" would make the output less annoying.

[HPsquared]

Phase transitions are a really nice way to explain to someone how a complex system can appear to flip from one state to another. Especially the importance of looking at the right variable. If you look at water at 99°C or 101°C (at standard pressure) it appears like a sudden change. But if you consider energy balance, it's not like it just flips: it takes substantial energy input to boil water. If you measure energy input, you see a gradual change of phase (mass fraction slowly turning from liquid to vapour) as more energy is supplied. But then you can also have superheated water in the microwave and it's just waiting to (partially) boil... So many analogies.

[NitpickLawyer]

> it's not like it just flips.

Does this apply to that cool chem trick where a solution goes from black to transparent and back again a few times? I don't know enough to know if that's relevant or not, but I remember seeing that and be puzzled about how "sudden" the reaction appears.

[energyscholar]

Exactly right. The phase transition analogy is powerful precisely because it's not just analogy — the same mathematical operators that describe water at criticality also describe markets approaching crashes, ecosystems approaching collapse, and cardiac rhythms approaching fibrillation.

What surprised us was how many fields derived this independently. The superheated water intuition you describe maps directly to what ecologists call "critical slowing down" and what financial engineers call "increased autocorrelation near instability." Same math, three different names, minimal cross-citation.

[kjshsh123]

3Blue1Brown has a good video on phase transitions.

https://youtu.be/itRV2jEtV8Q?si=qm51bvuo-ZIT_Pjk

Veritasium has a good video on how criticality applies to other things too,

https://youtu.be/HBluLfX2F_k?si=nK51yQVlNXz5bNSA

[stared]

I have serious doubts that these discoveries were truly independent.

Phase transitions and statistical mechanics have a long history in physics. Over time, physicists and applied mathematicians began applying these techniques to other domains under the banner of "complex systems" (see, for example, https://complexsystemstheory.net/murray-gell-mann/).

Rather than independent reinvention, it seems much more likely that these fields adopted existing physics machinery. It wouldn't be the first time authors claimed novelty for applied concepts; if they tried this within physics, they’d be eaten alive. However, in other fields, reviewers might accept these techniques as novel simply because they lack the background in statistical mechanics.

[asgraham]

I know for a fact [1] that the neuroscientific discoveries were not independent of physics: the people doing the developing were largely former physicists. They likely didn't cite anything because why would you cite phase transitions or criticality? You learn about them in class as a physicist. I strongly suspect the ecology results weren't independent either, but all the theoretical ecologists I know are relatively young (if mostly former physicists) so no first person accounts.

The part of this that could totally be true is that a clinical application somewhere along the way "independently" "reinvented" it. There's a hilarious collection of peer-reviewed journal articles out there inventing a "new" method of calculating the sizes of shapes and areas under the curve. The method involves adding up really small rectangles. (I think a top comment already mentioned the Tai article [2])

[1] source: my doctoral advisor was a really really old theoretical neuroscientist who trained as an electrical engineer and mathematician. If you want a more concrete example, the work of Bard Ermentrout on neural criticality starting in the 70's or 80's. He read a lot of physics textbooks.

[2] https://science.slashdot.org/story/10/12/06/0416250/medical-...

[energyscholar]

You're raising the right question, and the paper addresses it directly. The transfer wasn't as clean as "physicists applied their tools to other fields."

Some specific cases: Wissel (1984) derived critical slowing down for ecology independently and was ignored for 20 years. The actual import to ecology came via economist Buz Brock, not a physicist. Nolasco & Dahlen (1968) derived period-doubling for cardiac tissue before Feigenbaum's universality result. Jaeger (2001) derived the edge-of-chaos condition for recurrent neural networks without citing Bak, Kauffman, or Langton.

The complex systems movement you reference existed. The paper documents that it didn't actually solve the transfer problem. The cross-citation analysis shows the gaps persisted through the 2000s and 2010s.

You're right that some domains imported rather than reinvented. The paper maps where each transfer was independent, where it was imported, and where it was partial. That's the point — the pattern is messier and more interesting than either "all independent" or "all imported."

[svara]

I mean, introducing a technique from one field in another is innovative.

You don't get to claim you invented it, but a lot of progress happens by finding connections between things that are individually well known.

[intrasight]

It tells me that knowledge takes time to propagate.

Good math is universal, which means it's probably been discovered millions of times across the universe.

[energyscholar]

The propagation time is the interesting part. Critical slowing down was in physics textbooks by the 1970s. Ecology didn't import it until 2003 — via a chance conversation at a conference bar. Cardiology took until the 1990s. The FDA approved the resulting cardiac test in 2001.

That's not normal diffusion. Those are 30-year gaps for math with direct life-safety applications. The paper asks why, and finds structural explanations in how we organize knowledge.

[jlund-molfese]

It’s kind of lame to post the same clickbait three times in under 24 hours. I guess it’s nothing new, but feels inorganic.

[ajkjk]

and every comment here is also AI.

[energyscholar]

Dude, I'm sorry to offend you. And sharp of you to notice. I failed to get substantive engagement the first two times (1 point and 4 points) so I tried again. This time I got some engagement.

Re. the title, I started with a boring conservative title and got precisely zero engagement, so I changed the title to be a bit more clickbaitish. Just like most of the other titles in New. Did I do wrong?

As I said, this is my first serious attempt at social media engagement and I'm just learning how it works.

[sxzygz]

You and your coauthor need to write up a detailed account of your “Metatron model”. This paper, if it were to count as research, should be how other phenomena can be simulated by choices of parameters for your model.

Otherwise, you’ve just described yet another synthetic model that exhibits criticality (without proof no less). Which is not particularly interesting, unless your model subsumes other phenomena.

[triclops200]

This phenomena was also described/characterized in prior Hegelian literature as part of the law of quantitative into qualitative change, though, not formulated mathematically at the time. Interestingly enough, and In the context of how cross discipline this discovery has historically been, iirc, Lenin played around with mathematically characterizing the phenomenon, though, I am not aware of the extent to which he did. Very universal phenomenon for sure.

[djoldman]

Contrarian view with a dusting of generative AI spiciness:

Generative AI may be just the type of thing to connect these types of previously solved problems across disciplines.

[PlatoIsADisease]

I'm usually pretty pro-blog. I like when people have an interest in things. No ads, just someone wanting to prove their intelligence and popularity. But... OP... You didn't even explain the math.

Anyway, none of this is that surprising since deduction takes higher level ideas and tests them on lower level to prove the hypothesis.

If anyone wants to read Karl Popper, this will seem significantly less noteworthy.

[vscode-rest]

I’m no mathematician (studied up to diff eq, linear algebra, discrete), but from glancing through the paper I do not really have an ability to apply this concept to a problem of my own, though it does seem useful.

Do you think this is something that should be taught generally? In which class would it fit? It feels generally diffeq-ish.

[energyscholar]

Good question. It's closest to dynamical systems, which usually lives in applied math or physics departments. But that's kind of the problem — it gets taught as theory in one department and never reaches the engineers and clinicians who'd actually use it.

If you've done diffeq and linear algebra you have the prerequisites. Appendix B (page 17 of the paper) is our attempt at making it practical — worked examples rather than proofs. Would be curious if it lands for someone with your background.

We plan to do a follow-up paper that provides a standard format for this math that could be taught across domains. That doesn't belong in this first paper. First priority was to show the pattern and get people thinking about it.

[bjourne]

Can you in plain English explain exactly what unifies these discoveries? I have a hard time seeing what unifies traffic congestion with eigenvalue analysis of ESNs. While many systems contain thresholds, a traffic jam is not chaotic in the same way that an epileptic seizure is.

[MarkusQ]

And they don't even seem to have noticed Catastrophe Theory, which was based on the study of exactly this.

https://en.wikipedia.org/wiki/Catastrophe_theory

[abracos]

Is the main goal to see if LLM can do this sort of research and cross-pollination?

[energyscholar]

No, the goal is documenting the convergence pattern itself. We did use LLMs as research tools — acknowledged in the paper — but the cross-domain analysis and citation mapping are human work.

I'll explain how we got to this point. I had previously mentored my friend, Robin Macomber, in math & physics for several years. Robin Macomber independently discovered a variation of criticality math and asked me to evaluate. After due consideration I recognized a pattern: his work echoed that of Kenneth Wilson's renormalization group theory, which I'd previously studied. I then conducted a detailed survey of all academic fields that touched on criticality (using an LLM!) and found, to my great surprise, that this same math had been independently discovered many times in many domains. So I wrote a paper about it.

[bob1029]

https://en.wikipedia.org/wiki/Giant_component#Giant_componen...

[mastermedo]

https://arxiv.org/pdf/2601.22389

[profsummergig]

There's a Taleb vs. Sornette debate (argument) on YouTube.

I thought Taleb won (complex system outcomes, in the sociopolitical realm, cannot be predicted). But then I'm a Taleb fanboy.

Sornette (my first and last exposure to him) came across as a relic from a different age. Pitifully out of touch.

[PlatoIsADisease]

Everything you mentioned is a simplified system that applies in specific defined cases.

Its almost like the math came first, then the problem later.

You might want to read about induction vs deduction, this is deduction. I don't totally agree with Karl Popper, but at least he can explain why we see this math in multiple places.