OpenAI Analyzes Navier-Stokes Blow-Up by OpenAI: Unphysical

I asked OpenAI to analyze the Clay Navier-Stokes Blow-Up so famously presented by OpenAI. I received a clear and simple analysis displayed below showing a flow velocity with for each given t<1,  support and magnitude shrinking to zero along with viscosity, and for each given viscosity > 0, magnitude tending to infinity as t approaches 1. 

OpenAI concludes that this tells virtually nothing about fluid motion, which is the subject of the Clay problem as announced very clearly. In particular, nothing is revealed about the mathematics of turbulent flow as the true open problem addressed by Clay. OpenAIs solution is a vanishing point singularity which has nothing to with turbulence as being distributed over large regions of the flow. 

One can argue that the OpenAI solution is produced by an intelligence that knows nothing about fluid motion, and reads the formal problem specification by Fefferman as the Devil reads the Bible. It is very strange that the SuperIntelligence of Terence Tao and Charles Fefferman cannot understand this.

Compare with my posts on the Clay problem (46 posts)

Here is the OpenAI analysis

OpenAI’s Navier–Stokes Blow-Up: What Happens When Viscosity Changes?

OpenAI’s proposed Navier–Stokes construction claims finite-time blow-up for every positive viscosity. But what does “for every viscosity” actually mean?

The manuscript constructs a flow at viscosity 1, then obtains other viscosities through the transformation

u_ν(x,t) = √ν u(x/√ν,t),
f_ν(x,t) = √ν f(x/√ν,t).

Pressure scales by ν, while time remains unchanged. Thus every member of this family blows up at t = 1. See equation (10.22) in the manuscript.

This is an exact mathematical scaling. But it changes the experiment: smaller viscosity comes with smaller velocity, smaller forcing and smaller spatial dimensions. The Reynolds number UL/ν remains unchanged at corresponding times.

Following the scaling through the argument reveals something more important. The collapsing core has radius

L_r ≍ √ν (1 − t)^1/2.

Its radial diffusion time therefore satisfies

L_r²/ν ≍ 1 − t.

Here ≍ means equality up to positive constant factors. Viscosity remains essential to the balance. It does not disappear as ν tends to zero, because the spatial scale shrinks to compensate. The same compensation preserves the competition between shear amplification and viscous damping in the oscillatory corrections.

The final blow-up estimate, along a shrinking observation path, takes the form

u_θ,ν ∼ e₀√ν (1 − t)^{−1/2 − h},
e₀ > 0,   0 < h < 1/100.

For every fixed time before 1, velocity tends to zero as viscosity vanishes. Nevertheless, for every fixed positive viscosity, it diverges as time approaches 1.

These statements are compatible because the convergence is not uniform near blow-up. A fixed large speed is reached only within a time interval proportional to

ν^{1/(1 + 2h)}

before the singularity. Large velocities retreat into increasingly small spatial and temporal regions.

Even the fixed blow-up time is a normalization choice. Another exact scaling,

ũ_ν(x,t) = ν u(x,νt),

keeps the spatial support unchanged, changes the force to ν²f(x,νt), and gives blow-up at T_ν = 1/ν.

The conclusion is precise: assuming the construction is correct, it supplies viscosity-dependent forces producing blow-up. It does not establish how changing viscosity affects a flow under fixed forcing and fixed geometry.

That distinction does not invalidate the mathematical claim. But it limits what the result tells us about physical fluid motion.

Illusion of Precision in Quantum Mechanics

Theoretical physics prides itself to be the most precise of all of science with 12 decimal places for the (anomalous) magnetic moment of the electron as an elementary particle in full agreement with extremely precise measurements.

This was Richard Feynmans joker card to meet questions about quantum mechanics as the supreme achievement of human intellect computed using Feynman diagrams. 12 decimal places! That would give at least 3 decimal places correct for the rest of physics. 

This is put into perspective in this new article.

End of Scientific Publishing? Back to Secrecy of Alchemists?

Today anybody can send in an article to mathematical journal with a proof of an old or new conjecture/theorem as a signed original piece of work (by using an AI assistant to carry out the details of the proof). There are still some journals that do not accept any AI presence, but of course it may be impossible to decide the extent of the assistance. 

This means that the name on the article looses its role as unique identifier to a publication list along with the individuality of the person asking for a university position based on the list. 

But human beings without individuality connects to all sorts of ants societies (or maybe even ants have individuality), anyway very different from human society. 

In particular, it invites to a completely new view on scientific publication. In the old days you would seek to make your discovery/proof known to the world in a scientific journal in order to add to your publication list as basis for your university position with pay. 

Today, you would rather do the opposite: If you have discovered something on your own, you would like to keep it stay with you and not give it away to everybody. Thus you would not disclose your findings but keep them secret to possibly exploit as something to sell, like software with a new capability. 

The scientist would then become more of a classical magician in possession of a collection of secret tricks.

Comment by AI:

• Modern science emerged in part by replacing the secrecy of alchemists with the openness of publicly testable knowledge.Your scenario suggests that AI, paradoxically, could drive science back toward the alchemist's model.

How to be Master not Slave of LLM: Leibniz World of Math

Large-Language Models as a Cognitive Virus by Ricard Solé, Giulio Ruffini, Francesca Castaldo, Marco Tuccio, Luis F. Seoane, Manlio de Domenico, Santiago F. Elena, David C. Krakauer, Michael Levin:

• There is a high risk of a run-away effect toward persistent dependence, with abrupt losses in cognitive competence. 
• The same framework, however, identifies conditions for cognitive immunization, based on reducing transmission and facilitating reversibility. 
• Our results highlight how LLM adoption may involve nonlinear collective transitions with important consequences for cognitive autonomy.

In other words, it remains to train cognitive capacities to not become a slave under LLM but the master of  LLM by keeping cognitive autonomy. What to do? Can school help? Mom and Dad? University? Life?

Physical training? 

Let us go back to Leibniz who formed the first vision of automated thinking and ask what advice he can give when 350 years later his vision has become true: The key to cognitive autonomy is 

• Leibniz World of Mathematics.

Do not hesitate to take the first step to autonomy and mastery. 

RealQM: Universality and Chirality

Two new articles about RealQM: 

• Tetrahedral Stereocentre from Packing: Chirality’s Geometric Origin without Hybridisation
• Universality in Quantum Mechanics: A Claim That Cannot Be Tested, and One That Can

The second connects to the 2026 Nobel Prize in Chemistry.

Will AI Kill Mathematics and so Theoretical Physics?

Field Mathematics Medalist Hugo Dominil-Copin asks if AI will kill mathematics:

The only way forward seems to be Resistance, but will it work? The mathematics Hugo is speaking about is so called pure mathematics performed in symbolic form on the black board with chalk, or pen and paper at the desk, then published in math journals as articles filled with symbols. Numerical/computational  mathematics fills formulas with concrete meaning in terms of numbers. The threat to life posed by AI is against pure/symbolic mathematics where LLM has shown to be a formidable player with symbols. 

Theoretical physics is based on symbolic mathematics in the form of equations in symbolic form, which can be given concrete meaning in terms of numbers and images depicting physics, if the equations are computable, that is solutions can be computed with available computational work. 

Fact is that the basic mathematical models of physics such as Quantum Mechanics, Quantum Field Theory, Quantum ChromoDynamics and String Theory, are based on equations which are not computable and so their study is limited to symbolic manipulation where even AI meets severe difficulties because possibilities are very limited. 

Theoretical physics thus meets another threat of being inaccessible to AI even to ASI, and so in particular to humanity. What theoretical physics needs are computable models, which is not a hopeless case since real physics must be some form of analog computational process in principle possible to mimic in digital computation.   

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.