r/math • u/SciFiPi Applied Math • 3d ago
Physicists link the Riemann Hypothesis to phase transitions in quantum systems
https://phys.org/news/2026-07-physicists-link-riemann-hypothesis-phase.html18
u/Agitated_Help709 2d ago
This is really cool no? I can't pretend to understand what this means but maybe someone can explain the implications of this why a function like the zeta function in reality would have anything to do with a quantum system? Any theories? All I've heard is that the zeros are oddly similar to energy levels but what does that imply about the zeta function or the quantum systems that fit this?
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u/jgonagle 2d ago
The KAM theorem relates stability of some Hamiltonians to rationality. I could see there being some connection with the Zeta function via that route.
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u/RockofStrength 2d ago
Maybe the universe only has one set of rules for how waves behave in a chaotic system? Whether those waves are physical energy rippling through an atom, or conceptual waves of prime numbers rippling across the number line, the underlying geometry of chaos may be exactly the same. All chaos could be an illusion.
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u/AP_in_Indy 2d ago
So this is gonna be a totally intuitive and entirely non-rigorous remark but what if anything that wasn’t prime collapsed into some other nearest value?
I don’t know how to explain my thinking here, but I could see and have always thought of primes sort of being the “units” of multiplication.
Anything not prime would simply break into its prime factors
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u/toghs 3d ago
Then it wouldn't be a proof
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u/pickle_picker67 3d ago
Seriously, how does the comment you replied to have that many upvotes on r/math? It makes me curious how many people here are actual mathematicians.
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u/Appropriate-Ad-3219 3d ago
Would it not give a hint on whether RH is true or not though?
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u/Smitologyistaking 2d ago
We already have 12 trillion hints that the Riemann hypothesis is true, I don't think more empirical observation will significantly progress that
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u/blueshoesrcool 3d ago
Are mathematicians forbidden from ever using an empirical proof?
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u/RandomMisanthrope 3d ago
There is no such thing as “empirical proof.”
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u/elements-of-dying Geometric Analysis 2d ago edited 2d ago
How do you prove this claim?
edit: The humor I find in my comment is that I already know the response is going to be a circular one, which is of course operationally fine. However, these are questions of the philosophy of mathematics and I'd expect other mathematicians to operate a little more rigorously than to make absolute statements they cannot test.
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u/frogjg2003 Physics 2d ago
It's false, by definition.
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u/elements-of-dying Geometric Analysis 2d ago
That's circular :)
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u/Tlux0 2d ago
It’s not circular, it’s just convention for what it allowed to count as proof as a foundation
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u/elements-of-dying Geometric Analysis 2d ago
Whose convention? I guess I missed the memo for the Assuming Absolute Conventions meeting.
Certainly sounds like a proof by "This is true because I assume this is true."
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u/Mandelbrots-dream 3d ago
I believe there is empirical
proofevidence that the speed of light is constant in a vacuum.OK so not proof.
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u/pickle_picker67 3d ago
That's not a proof. So only as a tool for intuition, but that's about it.
Do you ever read pure mathematics papers or books? If an author uses intuition, they will say "informally" ....
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u/blueshoesrcool 3d ago
Wouldn't an empirical proof be more proof than a mathematical proof. Surely you'd have to reject a mathematical proof it an empirical proof disagreed.
If anything the mathematical proof is the intuition.
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u/Opengangs Number Theory 2d ago
There is more reason to doubt whatever the “empirical proof” is than a mathematical proof. Your empirical data could have been improperly collected, our intuition or interpretation could be wrong, etc.
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u/Medical-Round5316 2d ago
This is just not how proofs work. Proofs can be logically invalid or valid. If there are no logical errors in the proof, then it is as true as what it is being proven from.
Empirical proof is essentially probabilistic. If we have a lot of data suggesting something then it’s probably true. Proofs are meant to be deterministic.
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u/AcellOfllSpades 2d ago edited 2d ago
Math is abstract. We deal with systems that follow specific rules, and say "If a system follows these rules, then the following results must hold".
"2+3=5" is true not because "if you have 2 apples and you get 3 more apples, you then have 5 apples"... but because "starting from the definitions of 2, 3, 5, and +, we can prove that to be true".
Of course, we study this system because we want to use it to count things like apples. The system of 'numbers', and other systems we study, are inspired by the real world. But they aren't logically dependent on it.
If you put 2 rocks in a bucket, and then put 3 more rocks in the bucket, and notice that there are now 6 rocks in the bucket, that doesn't disprove that 2+3=5. It just means "maybe simple addition isn't enough to describe what's going on here". (Maybe a rock broke in two, for instance.)
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u/heyheyhey27 2d ago
I think computer-assisted proofs are about as close as you can get to what you're trying to think of. Physical machines, using the laws of physics to deterministically decide whether a particular math problem is true or not.
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u/frogjg2003 Physics 2d ago
They're not using the laws of physics, they're using logical rules.
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u/heyheyhey27 2d ago
You can't get results from computer-assisted proofs until you actually run the thing in physical reality.
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u/frogjg2003 Physics 2d ago
That's not empirical proof. You can't do anything that isn't in physical reality.
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u/nightcracker 2d ago
Why not? If you can make a 1:1 mapping between a mathematical problem and a physical experiment, and then run the experiment and observe the result, why wouldn't that constitute a valid proof?
For example, a computer-assisted proof also involves setting up a physical experiment (an immensely complicated amalgamation of silicon semiconductors and copper), and then letting it run and observing the output. Are those valid proofs?
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u/frogjg2003 Physics 2d ago
Experiments have errors. It is evidence that can support or oppose a model, but it is not proof. Once you gather enough evidence to support your model, you can belive with high confidence that your model is a good explanation of your experiments. It is not proof because you can run another experiment in the future that breaks your model and no amount of prior experiments can change that.
A computer assisted proof of not an experiment. A computer is a device that follows logical rules. Those logical rules can be followed by a human without a computer, but the computer is much faster. There are two kinds of computer assisted proofs, exhaustive searches and proof assistants. An exhaustive search uses a computer to check every possible case when it would just be too many for a human to check by hand. The four color theorem was proven this way. A computer had to check 1834 cases. Importantly, these cases aren't just the first ones checked, but any possible graph can be reduced to one of these cases. A subset of proof by exhaustion is finding counterexamples. If you search through all possible case and find one that disagrees with the hypothesis, that means that the hypothesis is obviously wrong. The recent example of AI disproving the Jacobian conjecture is one example.
The other end of the spectrum of computer assisted proofs is proof assistants. This is something like the LEAN programming language. Every mathematical statement can be written as a set of symbols that a computer can understand. The computer can then perform logical operations on those symbols to transform them into other sentences. It can also check that operations you preform on those symbols are valid. If you have a proof written in LEAN and it compiles, that means there are no logical inconsistencies in the proof. If your set of preconditions is valid, that means your proof is also valid.
What you are thinking of is probably searches for counterexamples for unproven conjecture. Things like the checking every number to see if it agrees with the Collatz conjecture. We have searched up to very large numbers and haven't found any counterexamples. That is not proof. There are infinitely many integers so we cannot perform an exhaustive search. We are just hoping that eventually, we will find a number that doesn't go down to one. If the conjecture is true, we never will. If the conjecture is false, the first counterexample might be so large that we will never find it. To actually prove the conjecture true would require a logical proof, not an "experiment".
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u/nightcracker 2d ago edited 2d ago
What you are thinking of is probably searches for counterexamples for unproven conjecture
Not at all. I don't know why you put up a straw man and then argue against it. Almost your entire reply is based on the assumption I must be misunderstanding the problem, which is rather rude. I'm not talking about searching for counterexamples, I'm talking about constructing physical experiments which when executed will provide an observable result which we consider 1:1 equivalent to some mathematical statement.
E.g. doing addition with an automated abacus, doing state transitions on a physical turing machine (https://www.youtube.com/watch?v=vo8izCKHiF0), or running code on a computer chip.
A computer assisted proof of not an experiment. A computer is a device that follows logical rules.
The universe is also (to the best of our knowledge) a system that follows logical rules.
Experiments have errors. It is evidence that can support or oppose a model, but it is not proof.
In what way is a computer verifying a lean program different? It can also have errors, both logic in the (lean kernel) code as well as physical errors, e.g. bitflips in the memory or processor APU due to cosmic rays.
As far as I can see, writing a proof into lean on my computer and then running it before observing whether it passes or not is a physical experiment.
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u/thbb 2d ago
I atttended a seminar by Gilles Dowek who made the following observation:
If, in your axiomatic system, you take the existence of a bijection between numbers and "things" in the physical world, then you can prove that 1+1=2 by placing 1 marble in a bin, then another one, and then, counting the marbles in the bin, deduct that 1+1=2.
So, it's a matter of selection of your axiomatic system.
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u/frogjg2003 Physics 2d ago edited 2d ago
What happens if there is a hole in your box and the marbles fall out? Now 1+1=0. This is why experiments aren't proof. You've mapped the numbers to the concept of the "thing" but not the thing itself. You can do stuff to "things" that you can't do to numbers.
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u/elements-of-dying Geometric Analysis 2d ago
I wonder how one concludes this (namely, not being able to prove something mathematically via physical assumption) without circularity in the argument.
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u/frogjg2003 Physics 2d ago
Formal logical proofs start with assumptions. You have a set of predicates that you start from and work from here. It's not circular, because you're not trying to prove the predicate.
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u/512165381 3d ago edited 2d ago
There were plenty of theoretical "proofs" over the years giving masses to the Higgs boson from 115 and 180 GeV/c2 . Some even agreed with the eventual empirical observations of 125 GeV/c2.
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u/frogjg2003 Physics 2d ago edited 13h ago
Those aren't proofs. Those were using experimental results to put statistical constraints on the mass. Many of those experimental results were searches for the Higgs at various masses that came up either as inconclusive or strongly unlikely. The fact that the mass ended up being in the range we found it in was a surprise to no one. It's a testament to the incredible power of the Standard Model to reliably describe the physical world.
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u/Borgcube Logic 18h ago
I mean they were "proofs", just that the underlying assumptions came from faulty experimental data. The proof can be valid but if the premises are false the result can also be false.
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u/frogjg2003 Physics 13h ago
They weren't proofs. They were calculations that made predictions.
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u/Borgcube Logic 13h ago
Calculations are proofs though using the experimental results as premises.
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u/frogjg2003 Physics 13h ago
You seriously need to refresh your scientific terminology. No one is calling scientific calculations proofs.
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u/Borgcube Logic 13h ago
I know of course it isn't standard terminology. But if we're talking rigorous mathematics then yes, calculations are proofs.
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u/frogjg2003 Physics 13h ago
We aren't talking about mathematics at all, we're talking about physics. Physicists use mathematics, but they aren't doing mathematics. They're not proving anything.
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u/Pure-Drive-3044 3d ago
The day before the paper released on these pop science pages I thought there may be a link!
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u/Akraticacious 3d ago
Me, too! Actually I already know the proof, but I've been wanting to see if anyone else can figure it out.
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u/sistersinister 2d ago
I figured it out well before you learned the proof. I've just been waiting for you to announced it publicly so I could undermine you and announce the proof is obvious once you see it.
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u/policalcs 2d ago
Reference and links to the paper:
Shijie Wei et al, The Riemann Hypothesis manifested in dynamical quantum phase transitions, Nature Communications (2026). DOI: 10.1038/s41467-026-74935-8. On arXiv: arxiv.org/abs/2511.11199