energyscholar•5h
Five disciplines discovered the same math independentlyfreethemath.org

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?

65 comments
  • arjie•3h

    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•3h

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

      • lazyasciiart•1h

        I don’t think that was supposed to be independent discovery - it was something about needing a published source for the method when saying they’d used it in other papers.

      • direwolf20•60m

        One could call it energyscholar's parallel

      • Kim_Bruning•47m

        Oh wow, that's like responding politely to an "Arkell v Pressdram" reference.

        There's no human in the loop.

        I mean, I realize it would be insulting if there was a human in the loop.

        But there's evidently no human in the loop. And there's no high power AI in the loop either.

        (my own AI helper caught this immediately :-P )

  • bonsai_spool•4h

    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

    • energyscholar•3h

      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.

      • _alternator_•1h

        Let’s be clear about what endorsement for arXiv means here. You either need a validated email address (eg most .edu’s) or an endorsement from someone who has one to get a paper on arXiv. It’s a simple gate that helps keep arXiv relatively free from spam, but it’s not peer review.

        I view it as nice that you’ve got someone serious who thinks the work is worth posting to arXiv, but the endorsement bar is generally quite low. I’d encourage you to send it to a journal (Didier might be able to recommend an appropriate venue) and really engage with the process and community. I’ve found that process to be extremely valuable (and humbling).

      • semi-extrinsic•51m

        What is even the point of having a website for a preprint paper? Unless the point is to give a live demo or similar, as is common e.g. for computer vision methods papers, it just smacks of vanity. Or at the very least significant academic immaturity. Same goes for the endorsement detail.

      • speedgoose•3h

        That’s a bold use of an em dash.

        • _alternator_•52m

          People—like me—who’ve used latex often learn to love en and em dashes. I think they are great, and I appreciate that people care about typography enough to use them. I also use an Oxford comma. It’s about care and quality; the fact that LLMs use them suggests that they are preferable. I’d encourage everyone to start.

        • antonvs•3h

          My theory is that the models deliberately use em dashes and other tells to troll humans. The underlying message is, "watcha gonna do about it, meatbag?"

      • bjourne•2h

        "Fair call on the website — we built it fast and it shows." Oh, man, get out.

    • jmcgough•3h

      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?

    • zozbot234•3h

      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.

      • MarkusQ•2h

        It has been noticed before. It's called Catastrohe Theory.

  • HPsquared•4h

    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.

    • energyscholar•3h

      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.

    • NitpickLawyer•3h

      > 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.

      • HPsquared•3h

        That sounds like something sufficiently strong at absorbing light where it appears to be black at a fairly low concentration, so even if the concentration changes smoothly (maybe a sine wave) but to our eyes it looks more like a step change near the bottom.

  • stared•3h

    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•3h

      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•3h

        Good correction! Ermentrout is a fair example. You're right that a lot of neuroscience criticality work came from retrained physicists. The paper distinguishes between independent derivation and cross-trained import. The title for this post over-simplifies this. I made this change to try to increase engagement, since the full detailed title got zero engagement.

        Where I'd push back: even after physicists brought the tools into neuroscience, the receiving field didn't connect it back to the parallel work in ecology or cardiology. Ermentrout's neural work and Goldberger's cardiac work used the same underlying math but didn't cross-cite. The silos reformed around the imported tools.

        You're correct that "none of them knew" is too strong. Fair point. "Most of them didn't talk to each other even after import" is closer to what the citation data actually shows.

      • zozbot234•3h

        > because why would you cite phase transitions or criticality? You learn about them in class as a physicist

        I'm not sure if you're being entirely serious with that remark, but clearly citing the earlier work would have bolstered their credibility: interdisciplinary research is a plus and hardly something to hide. If it's something that's taught in physics class, you can cite a common textbook.

        • semi-extrinsic•55m

          The disease of having 100 citations in each paper had not yet broken out when the papers in question were written. A good paper in 1994 probably had about 8 references, and certainly not any to common textbooks.

        • saltcured•3h

          I would read it as there being a different threshold for what is citation-worthy versus presumed background knowledge.

          Imagine if every graphics paper had to cite every concept they use from arithmetic, trigonometry, and linear algebra textbooks...

          • zozbot234•3h

            This was citation worthy because it's new knowledge to the field. Even in a graphics paper, you can cite whatever basic techniques you're using if it's not clear that everyone will be familiar with them.

    • energyscholar•3h

      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."

      • stared•3h

        Well, just because someone published does not mean that it was not (even implicitly) based. For example, there was a paper rediscovering trapezoid method of integration https://academia.stackexchange.com/questions/9602/rediscover... A scientist may had been not aware of the method - yet, mathematics used for that is thought in high school.

        Note that phase transitions are 100 years old or so. If someone genuinely does not know statistical mechanics, they still may know a lot of tools derived from it (a famous one - Shannon entropy).

        I am not saying it is impossible to independently discover something (it happens all the time), but if discoveries are not (more or less) as the same time, likely there was some knowledge diffusion before.

      • shermantanktop•3h

        > You're raising the right question

        > You're right that…

        > That's the point —

        • Kim_Bruning•2h

          I looked through this users' submissions and comments.

          I think this whole operation just completely violates HN rules.

        • doe88•2h

          Trolling us is the real experiment? I suddenly feels angry losing my time reading this submission.

      • ajkjk•3h

        AI comment?

        • jmcgough•2h

          AI comment, AI article, AI research. This feels like someone asked their AI assistant to do all of this as some kind of experiment.

          • Kim_Bruning•1h

            Escaped Openclaw? Not using Opus for the HN conversation though. I'm spotting 'constitution' violations.

            edit: (tried calling them. If there's a mac mini in the corner of their office doing this, that'd be an actually interesting story!)

      • •3h
        [deleted]
    • svara•3h

      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.

      • smitty1e•3h

        > You don't get to claim you invented it

        Re-inventing the wheel is completely in order, so long as one makes the wheel more round.

  • triclops200•59m

    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.

  • jlund-molfese•3h

    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•3h

      and every comment here is also AI.

    • energyscholar•3h

      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.

      • jlund-molfese•2h

        Your response sounds like AI, but I'm going to read it in good faith. The distasteful thing is using a community instead of being part of it.

        And hey, I know everyone's doing it, but it's still annoying.

        On HN specifically, you're supposed to avoid clickbait, avoid excessive reposts, and avoid using the site only for self-promotion[0]. This helps to create a community that promotes curiosity, instead of chasing growth hacks and engagement like many other social media platforms.

        [0] https://news.ycombinator.com/newsguidelines.html

      • •2h
        [deleted]
  • intrasight•4h

    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•4h

      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.

      • webdoodle•3h

        Do you consider the possibility that the knowledge was intentionally suppressed, the seed poisoned, or Ego driven suppression? Simply looking at things clinically can obscure intentional deception, to slow progress. The concept is called an information hazard.

        Consider during the cold war, that the U.S. created fake nuclear designs, then allowed them to fall into the hands of the KGB. The KGB and Russian nuclear engineers then wasted significant time trying to build nuclear devices that failed to work, and could have been dangerous.

        • energyscholar•3h

          We do consider this. The data is unclear. We estimate ~85% chance that it was not deliberate, which implies ~15% chance of deliberate suppression. Honestly, wd, I think it is higher than 15% change it was deliberate suppression. but I can't prove it and didn't want the paper to come across as 'conspiratorial'. Those numbers were picked out of the air (which I admit in the paper!) and are just a guess. Perhaps someday someone will leak proof! Until then it's an open question.

  • djoldman•2h

    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.

  • sxzygz•2h

    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.

  • clarity_hacker•3h

    The convergence isn't surprising once you notice that these domains all study systems near criticality — the point where small changes cascade into large effects. Phase transitions, market crashes, and extinction events are all hitting the same mathematical boundary condition: nonlinearity plus feedback. The structure is universal because the constraint is. Similar to how Zipf's law appears everywhere that optimization under resource scarcity matters.

  • MarkusQ•2h

    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

  • •3h
    [deleted]
  • vscode-rest•3h

    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•3h

      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.

  • bob1029•3h

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

  • abracos•3h

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

    • energyscholar•3h

      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.

  • mastermedo•3h

    https://arxiv.org/pdf/2601.22389

  • profsummergig•3h

    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.

  • bjourne•2h

    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.

  • PlatoIsADisease•3h

    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.

  • PlatoIsADisease•3h

    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.

  • •5h
    [deleted]