Parameter-free Universality: Standard QM vs RealQM

New article about a basic aspect of mathematical modeling of physics:

• Universality in Quantum Mechanics: A Claim That Cannot Be Tested, and One That Can.

Recall Einstein's ideal model of physical reality as a parameter-free mathematical model which can say something about reality, like a thought experiment without experimental input, like computing the circumference $2\pi$ of a circle of radius 1 from the equation $x^2+y^2=1$, but of greater scope such as computing the drag of a body through a slightly viscous fluid from the shape of the body alone, or the spectrum of a Hydrogen atom. 

Pure Mathematics: The Next Chess?

Will AI turn theorem proving into an intellectual game? 

For centuries, chess was regarded as one of the highest expressions of human intelligence. Great chess players were admired for their extraordinary powers of logical reasoning, imagination and strategic thinking. Then came computers. In 1997, IBM's Deep Blue defeated world chess champion Garry Kasparov. Today, ordinary chess engines play far beyond the level of any human grandmaster. Chess survived, but something fundamental changed. Being the world's strongest chess player no longer meant being the world's strongest chess-playing intelligence.

Could pure mathematics now face a similar transformation? It has traditionally been regarded as an even higher form of intellectual achievement than chess. Unlike chess, mathematics has played a fundamental role in the development of science and civilization. But here we must distinguish applied from pure mathematics. Applied mathematics develops methods for understanding, predicting and controlling the physical world. Newton's calculus, Fourier's analysis and modern numerical computation exemplify this tradition. Their ultimate value lies in what they make possible. Pure mathematics, by contrast, investigates abstract structures and logical relationships without requiring applications. Its achievements are judged primarily within mathematics itself. 

During the twentieth century, pure mathematics increasingly developed a culture centered on solving difficult problems. Fermat's Last Theorem, the Poincaré conjecture and the Clay Millennium Problems became celebrated intellectual challenges. The resemblance to chess is striking. Both activities operate within precisely defined logical frameworks. Both reward extraordinary ingenuity. Both have hierarchies of increasingly difficult problems. Both celebrate champions who succeed where others have failed. 

But is solving a difficult mathematical problem necessarily more intellectually significant than winning a difficult chess game? The answer depends on whether the mathematical achievement reveals something fundamentally new or merely overcomes a formidable technical obstacle. A profound mathematical concept may transform our understanding of an entire field. A difficult proof may simply establish the truth of a previously formulated conjecture. The distinction is between creating new understanding and demonstrating exceptional problem-solving ability. 

Now AI enters the mathematical arena. Just as computers surpassed human chess champions, AI may eventually surpass human mathematicians in proving difficult theorems. If that happens, the production of proofs may become increasingly routine. A mathematical conjecture could become something like a chess position: a challenge presented to a machine capable of finding the solution. The prestige attached to solving difficult problems would then face the same challenge that confronted competitive chess. Yet mathematics has something chess does not necessarily possess: the possibility of discovering entirely new structures and connections, both within mathematics and with the physical world. 

Perhaps this is where its future lies. The central question will no longer be who can prove the most difficult theorem, but who can formulate the most illuminating mathematical ideas. Chess did not disappear when computers became superior players. It became a game in which humans increasingly learn from machines. Pure mathematics may follow a similar path. 

There is, however, a fundamental distinction. Chess has objective rules for winning, but no claim that winning contributes to understanding reality. Pure mathematics has objective rules for proving, but the significance of what is proved requires a separate judgment. A chess victory is meaningful within the game. A mathematical proof establishes a truth within a mathematical framework, but its importance cannot be measured by proof difficulty alone. 

If pure mathematics increasingly rewards the conquest of difficult problems without asking what new understanding they bring, it risks becoming an intellectual sport governed by its own rules of prestige. AI may now expose this weakness by making increasingly sophisticated theorem proving a machine activity. 

Perhaps the greatest contribution of AI to pure mathematics will not be the theorems it proves, but forcing mathematicians to confront a question long overshadowed by the celebration of difficult proofs:

• What makes a mathematical theorem worth proving?

AI Collapse of the Academic System in Mathematics?

Anybody can now pick up an open math problem (conjectured theorem without proof) using AI and let AI solve the problem, by proving the theorem to be correct or wrong, and send it to a math journal for publication with the persons name. Thus anybody can today ask for a position as a professor in mathematics. A new situation. 

The math community meets the new situation with two ideas seeking to maintain the old system by introducing a requirement typically fulfilled in the old system with human mathematicians composing proofs of theorems as raison d'etre for having a chair as professor with salary, namely that the person putting name on a math article/proof  (produced by AI) should be able to, at least partly, understand the proof. 

There would thus be two categories of mathematicians both with impressive publications lists, one who understand, at least partly, the proofs in the list, and another who do not. And those who claim to understand, at least partly, would get paid. 

Maybe a perfect system, but who would be able to decide if somebody understands, at least partly, or does not really understand so much? That would require some form of committee at the math department with members who really understand the proof at hand. The alternative would be honest self-evaluation: (i) understand everything, (ii) a bit and (iii) not much.  

So the classical academic system over centuries built on named published work, is a facing a serious challenge. Has the individual scientist lost her/his name and then along with that existence? 

In experimental science there would still be a niche for individual experimental work like an archaelogist  digging in the ground for some old bones, but even that could better be done by a machine. 

Role of Analytical Mathematicians to Understand AI Proof?

The Advisory Group on Mathematics and Artificial Intelligence at Institute for Advanced Study has issued a statement on Oct 6 about the Oct 6: On OpenAI’s Release of Mathematical Results. Let me cite from this document that gives lots of input to questions about the role of analytical/pure mathematicians in an AI future:

• As announced a few weeks ago, OpenAI has released a large collection of mathematical results generated by an internal model, reporting solutions to hundreds of open questions. 
• This is an important event for mathematics, with consequences both for mathematics and for the mathematical community that extend far beyond the individual results.
• The future of mathematical research cannot consist only of understanding results produced by AI labs. 
• Mathematicians must be able to formulate their own questions, develop their own approaches, and explore directions that have not been selected as examples of an AI system’s capabilities. 
• We reaffirm our published recommendations on responsible release. 

AI vs Numerical PDE - Turbulent Euler/Navier-Stokes - Data Bank

Today the mathematics community has been struck by an OpenAI repository with mathematical proofs of mathematical theorems:

• Several hundred claimed resolutions of long-standing conjectures across number theory, algebraic geometry, representation theory and mathematical physics...
• Milne's rationality conjecture, Goldfeld's conjecture, Nagata's conjecture, Bloch's conjecture for complex surfaces, Fujita's freeness conjecture, Hilbert's tenth problem over ℚ, the irrationality of Catalan's constant. Those are not routine results; several have been open for decades.
• But nothing whatever in numerical analysis, finite elements, error control or computational PDE. That is not a subject being served unevenly. It is a subject that does not appear.

Here is a reflection on this new revolutionary state of affairs: 

• AI as Computational Mathematician Rather Than Theorem Prover

It seems AI as LLM can do analytical mathematics since it is a symbolic language, and then better than human mathematicians, like computer chess vs Karpov. But computational mathematics is a collection of well defined algorithms not asking for AI as LLM, and so can serve mainly for coding as a language, data collection and output evaluation. 

This means that computational mathematics appears as the winner under the AI advance, while the effect  on analytical mathematics may be profound. 

I will now test if AI for Euler/Navier-Stokes can deliver a data bank of representative turbulent flows with dual-based output error control, by running the same code with data collected by AI, and report the result.    

Nobel Prize in Physics 2026

The Nobel Prize in Physics 2026 is another Prize assigned to the neutrino as a "ghost particle" invented by Wolfgang Pauli to describe an apparent loss of energy in beta-decay which nobody could explain. Pauli was not happy with his baby because it had no features at all, no mass, no charge, nothing. This is what the Nobel Prize home page has to say

• The neutrino is the shyest particle in the universe, has no electric charge and almost no mass. In general, it passes unnoticed through matter – seldom does a neutrino make its presence felt by colliding with an atomic nucleus. 
• Every second, without you noticing, 65 billion neutrinos from the Sun flow through your little finger nail.

The 2026 Prize went to detection of a couple of high energy neutrinos from outer space in a one kilometer ice-cube in the South Pole ice mass, not directly because the neutrino is too shy to show but indirectly as a little flash of light supposedly the trace of neutrino flying by. 

Another Parturient montes, nascetur ridiculus mus as theoretical physics, the King of Science, with mathematics the Queen.

This is what RealQM says about neutrinos and beta-decay.

New Explanation of Periodic Table based on Coulomb Alone

RealQM as a new model for atoms based on non-overlapping electron unit charge densities interacting by Coulomb potentials has delivered a new explanation of the Periodic Table based on the numbers 2 and 8 carrying the physics of both period length 2, 8, 8, 18, 18, 32, 32, 50, 50, and period doubling as (8, 8), (18, 18), (32, 32) and (50, 50) in the following form:

• 2,  8, 18=2+8+8, 32 =8+8+8+8, 50 = 2+8+8+8+8+8.

Note that this is a fundamentally different explanation the the textbook one since 100 years based on the s, p, d, e, f orbitals of the eigenfunctions of the Hydrogen atom with numbers 8=2+6, 18=2+6+10, 32=2+6+10+14 without physics. Also recall that the magic/golden  numbers of the nucleus are 2 (alpha-particle) and 8 (O-16) as the most stable nuclei. RealNucleus explains why. 

Read the explanation in this article under revision for IJQC.

Euler's Equations: Symbolic vs Numerical Mathematics

Here is another summary of the situation actualized by the OpenAI 57 page symbolic/analytical proof of blow-up of solutions to the unforced Euler equations:

• Euler Equations: Symbolic vs Numerical Mathematics

The mathematics community is in shock after the OpenAI 167 page proof of blow-up for forced Navier-Stokes equations presenting a solution to the Clay Navier-Stokes Millennium Problem far beyond the horizon of human mathematicians. Same shock as delivered by computer chess 20 years ago.  

What will the impact be on symbolic mathematics as the Queen of Science? What will be left to human mathematicians to do? Something like speed chess? Best proof of given proposition in 5 minutes? And for mathematics education? 

The article pleads for a synthesis of symbolic and numerical mathematics with a role for human mathematicians. What do you think? Game over? Synthesis possible? Math education tomorrow?

Per Enflo 1944 - 2026 Mathematician and Pianist

Per Enflo on September 28 in the middle of the next step on the daily walk with his wife Lena in Östervåla Uppland, under inspection of plants on the ground in the spirit of Linné and with inspiration from the sky finally closing his constructive proof of the Invariant Subspace Problem for Hilbert Spaces, took a last breath sending a shock wave to family, friends and mathematical community. Per Enflo is certainly the most famous Swedish mathematician all times by having solved named major problems, also concert pianist expressing the true meaning of the music of Mozart, Beethoven, Schubert and Chopin.

I met first Per during my post doc years 1974-76 at the math department of the University of Chicago, when Per visited as the new shining star of Functional Analysis, with offers from all the big universities, after having solved one of the key open problems in that area formulated by its founder Stefan Banach in the 1930s, as a 9 page Counterexample to the Approximation Problem in Banach Spaces published in Acta Math in 1973, which took the math community with storm. Watch the documentary movie about Banach with Per in the main role (and me with little side role) and Per's home page.

Per then followed up in 1981 with an 101 page counterexample to the Invariant Subspace Problem in Banach Spaces, published in Acta Math after 6 years of refereeing, to return 40 years later to the case of Hilbert spaces with an explicit construction of an invariant subspace. 

It took 43 years before we met again, in Stockholm in 2008 when Per had rejoined with his love from youth Lena and returned to Sweden after 25 years in the US, and Per welcomed me and my wife Ingrid to his piano trio concert at the Mazer Musical Society. We found each other on the spot into a 20 year long friendship along with our wifes, with music, math and love. It is very sad that Per with his very kind person and amazing talent is no longer here. In the Swedish math community we shared experiences of exclusivity as a special bond.   

Per was a master of constructive mathematics, constructing an invariant subspace for any continuous linear operator T in Hilbert space H, by constructing a sequence of vectors converging to a vector for which repeated application of T does not span H and thus forms an invariant subspace. Per was also a master of constructive music as combination of body/hand and soul/mind with ability to on the spot transpose any given sheet music to any key. We could meet in hands-on constructive math/piano but also lofty speculation, never any argument.

On Sept 8 and 26 another shock wave into the math community had been sent by OpenAI as constructive proofs of blow-up of solutions to the equations of Navier-Stokes (167 pages) and Euler (57 pages) performed by AI. 

Per expressed that he had been lucky to not meet this seemingly formidable competition before, with AI able to construct proofs of any number of pages, beyond understanding by human mathematicians. Per could thus pass on to a heaven of math without shattered beliefs with his usual happy "that is also ok" and with his Musical Legacy completed in 15 CD on Spotify.

Pictures from Aug 20 2026:

 

OpenAI vs Clay Navier-Stokes vs Numerical Analysis

Here are two new articles connecting to the recent hype of AI doing mathematics instead of mathematicians:

• Constructive Solution of the Clay Navier-Stokes Problem
• AI vs Numerical Analysis

Questions to ponder

• Does AI pull the carpet under mathematics education?
• What is the meaning of AI proofs of mathematical theorems?
• Is the main role of a mathematician to prove theorems?
• What is the role of numerical analysis?