Deng Yu Wins the Fields Medal: The Problem He Solved Has Stumped AI

From Deng Yu, Hilbert's Sixth Problem, to the Step AI Has Yet to Truly Take

Author | Xing Meng, Partner at 5Y Capital

The Fields Medal is everywhere today — you've probably seen the names Hong Wang and Deng Yu. What's interesting is that videos of Hong Wang have been circulating like wildfire these past couple days, while Deng Yu has stayed remarkably quiet, almost invisible. Yet it's Deng Yu I've been thinking about, over and over. Not because this is a story about a Chinese mathematician — but because the 125-year-old problem he cracked with Zaher Hani and Xiao Ma happens to strike at a growing blind spot in AI development today, one that nearly everyone is sidestepping. It's worth discussing even more than "another Fields Medal."

The problem is Hilbert's Sixth Problem.

I first heard of this concept from Ziming Liu, a newly appointed assistant professor at Tsinghua's Institute for AI. He said his vision was to solve Hilbert's Sixth Problem. At the time I didn't understand what made this problem so important, or why the first five were being skipped.

After learning more, I found that textbooks summarize it as "establishing physics in an axiomatic way." That's not wrong, but it's so grand it becomes meaningless — you can't feel any connection to AI entrepreneurship or AI investment today. More specifically, what it aims to do is this: starting from the fundamental laws describing individual microscopic particles, rigorously and without skipping a single step, derive the macroscopic equations that describe gases and fluids.

I prefer to translate it into plain language:

When we've already figured out every basic unit of the world, what makes us think that stacking enough of these units together will automatically yield the laws of the next level up?

Knowing how one molecule moves doesn't mean knowing how a gas flows. Knowing how one protein folds doesn't mean knowing how a cell responds to a drug. Knowing how one consumer fills out a survey doesn't mean knowing where a market ultimately goes. Even knowing how parameters update in a small model doesn't mean we truly understand generalization, grokking, and emergent abilities in large models.

These questions seem to belong to physics, biology, social science, and machine learning — completely unrelated. But they share the same underlying structure:

How does correctness at one scale become correctness at another?

This is Hilbert's Sixth Problem's true significance for the AI era. It's also likely the problem that many of today's so-called "world models" have yet to solve.

How a Bunch of Little Balls Become a Flow of Water

First, what exactly did Deng Yu's team do?

Imagine a box containing countless tiny hard spheres. Each sphere obeys nothing but the simplest Newtonian mechanics: uniform linear motion when not colliding, changing direction according to conservation of momentum and energy when they hit. That's it.

If there are only ten balls in the box, fine — number each one, record its position and velocity, calculate step by step.

But in a real gas, a puff of air contains roughly 10²³ molecules. Even with infinite computing power, precisely tracking every molecule's position and velocity would barely help us "understand a gust of wind." Because when we look at wind, we don't care whether the 784-quadrillionth molecule is moving left or right at this instant. What we really want to know is: how dense is the gas here? Which direction is it flowing overall? What's the temperature? The pressure?

So facing the same physical world, humanity developed three completely different descriptive languages:

Newtonian hard-sphere dynamics (microscopic) → Boltzmann equation (mesoscopic) → Euler equations / Navier–Stokes–Fourier equations (macroscopic)

At the bottom is the microscopic: Newtonian hard-sphere dynamics, tracking each particle's position and velocity.

In the middle is the mesoscopic: the Boltzmann equation. It no longer numbers particles, but instead statistics "how many particles near a certain position are moving at a certain velocity."

At the top is the macroscopic: the Euler and Navier–Stokes–Fourier equations. At this level, even the complete velocity distribution is discarded, leaving only a few macroscopic quantities: density, average velocity, temperature.

From an information perspective, this is an extremely aggressive compression: from the massive state of 10²³ particles, compressed down to a few continuously varying fields in space.

But the hardest part of this isn't "taking averages." Averaging is easy. What's truly difficult is proving this:

After you've thrown away the vast majority of microscopic details, what guarantees that the discarded information won't suddenly come back at some future moment and mess up the macroscopic result?

In other words — if I only keep density, velocity, and temperature, can next second's density, velocity, and temperature really be calculated from just these current macroscopic quantities alone? Or after one step, do I have to go back to the bottom and look up the complete history of every molecule all over again?

In physics and mathematics, this problem has a specific name: closure. A macroscopic theory only truly stands on its own when it achieves closure; otherwise it's not a theory, just a pretty interface wrapped around microscopic simulation.

Remember this word. It will come up again and again.

The Hard Part Isn't Collisions, It's the "Memory" They Leave Behind

The biggest obstacle from Newtonian particles to the Boltzmann equation is correlations between particles.

If all particles remained independent from start to finish, everything would be easy. The problem is they collide.

A and B collide, and their states are no longer independent. Then B hits C, C hits D. After a while, A might encounter D again through some other path. At this point A and D have no direct initial relationship, yet they've been quietly connected by a chain of collisions:

A → B → C → D → ... (correlations propagate, branch, and reconverge along collision chains)

Like throwing a stone into water, ripples spreading outward in circles. What starts as a small relationship between a pair of particles can, given enough time, weave into an extremely complex web of relationships: collision chains branch, branches reconverge, particles that have collided may collide again.

The Boltzmann equation achieves dramatic simplification through a key approximation — often called "molecular chaos":

Two particles about to collide can be approximated as independent of each other before the collision.

This isn't saying they truly have no history. Of course they've had past collisions, may have indirectly influenced each other through several detours. What Boltzmann really needs is: as particles become more numerous, smaller in size, and the gas sufficiently dilute, these tangled historical correlations' influence on the current collision ultimately becomes negligible.

In 1975, Oscar Lanford first proved this rigorously.

Here "proof" doesn't mean running a big simulation and seeing if a bunch of little balls moves like a gas. He proved a genuine limit theorem: as the number of particles goes to infinity and the particles themselves become infinitesimally small, the statistical distribution of the Newtonian hard-sphere system converges to the solution of the Boltzmann equation.

But Lanford's result had a fatal limitation: it only held for an extremely short time, roughly a small fraction of one mean collision time.

The problem isn't that real gases stop obeying the Boltzmann equation after this time, or that particles can only collide a finite number of times. The problem is — his proof method itself broke down.

To deduce a particle's present, you have to trace back who it collided with; to understand that collision partner, you have to keep tracing who it collided with before. Each additional step forward in time, the number of possible collision histories explodes. For a short time, "having this many complex collisions" is itself a low-probability event, and this low probability can still suppress the combinatorial explosion of collision histories; but as time stretches out, the combinatorial number of histories grows exponentially, and traditional proof methods can no longer sum these terms.

Anyone who's done long-horizon rollouts will find this dilemma familiar: a model that predicts accurately for one step, ten steps, doesn't mean that when you roll it out ten thousand steps, the errors, bifurcations, and correlations can still be controlled. Lanford proved the direction was right, but he only took a very short step past the bridgehead.

Deng Yu's team's 2024 work broke through precisely this time barrier that had been stuck for nearly half a century.

They proved: as long as the target Boltzmann equation itself has sufficiently regular solutions during this period, then the rigorous derivation from Newtonian hard-sphere systems to the Boltzmann equation can be extended to any given finite time, no longer trapped in Lanford's short stretch.

The "arbitrarily long" here needs careful understanding. It doesn't mean they unconditionally proved it to the end of the universe. The precise meaning is: you specify how long a finite period, and as long as the Boltzmann equation doesn't develop singularities, blow up, or lose regularity during this time, the microscopic particle system will follow it convergently throughout this entire period.

The 2025 follow-up paper took one more step forward.

From the Boltzmann equation to fluid equations like Euler and Navier–Stokes–Fourier, the mathematics community had already accumulated substantial work. The real bottleneck was: the first bridge from Newtonian particles to Boltzmann could previously only be crossed for a very short distance, so the front and back theoretical segments could never connect. The 2025 paper extended the 2024 results to two and three-dimensional periodic spaces, then spliced together with existing fluid limit theory, connecting the complete chain for the first time:

Newtonian hard-sphere system → Boltzmann equation → compressible Euler equations / incompressible Navier–Stokes–Fourier equations

So the simplest understanding is: In 2024, they lengthened the "microscopic → mesoscopic" bridge; in 2025, they connected the entire "microscopic → mesoscopic → macroscopic" bridge.

Of course, boundaries need to be clear: what they solved is the classical version of Hilbert's Sixth Problem — "from Newtonian mechanics, through Boltzmann theory, derive fluid equations" — not the axiomatization of all physics. It has a clear scope of application: dilute gases, hard-sphere models, specific scaling, regular solutions exist. It's still far from complex liquids, long-range interactions, chemical reactions, and real boundary conditions.

And I precisely think "the boundaries are stated crystal clear" is itself extremely important. This is precisely the difference between mathematics and much AI narrative: a truly credible model must not only tell you under what conditions it holds, but honestly tell you under what conditions it doesn't. We'll come back to this at the end.

The Essence of World Models Isn't Remembering More, But Knowing What to Forget

Deng Yu's team's work struck me as deeply relevant to AI because it puts a fact we often underestimate right on the table:

The world at different scales often requires completely different representations.

From particles to fluids, it's not simply magnifying the particle model by a trillion times. Entering the macroscopic scale, even the variables used to describe the world change: position and velocity at the microscopic level, probability distribution at the mesoscopic level, density, flow velocity, and temperature at the macroscopic level.

Crossing to the next scale isn't a quantitative increase, it's a change of language.

Today when people talk about world models, they often unconsciously understand it as an increasingly large end-to-end simulator: more data, bigger parameters, longer videos, farther rollouts, more agents — as if sufficiently fine-grained bottom-layer simulation will automatically make higher-level patterns emerge, and we'll automatically know how to use them.

Hilbert's Sixth Problem tells us: this step has never been free.

A truly useful world model, beyond prediction, must find the "sufficient state" of the world at some scale: this state must be simple enough that it doesn't need to record all details; yet complete enough that it can independently determine what happens next. Density, velocity, and temperature are one such sufficient state for the world of fluids — they discard almost all molecular-level detail, yet can still describe a gust of wind, a flow of water, even a shock wave in air.

So the truly sophisticated aspect of world models may not be remembering more, but rather — quite the opposite:

Knowing what can be safely forgotten, and being able to prove that forgetting it won't lead to errors.

"Safely" is the key word. The model doesn't just compress information; it must also know whether the compressed variables might, through feedback, correlations, or long-term evolution, come back and alter results. This, ultimately, brings us back to closure.

A few sharper questions to summarize the difficulty at this level:

Which underlying differences no longer matter? Which relationships must be preserved? Which microscopic perturbations will be averaged out? Which microscopic correlations will instead be amplified into macroscopic structures? And the most critical one — do the high-level variables we retain suffice to independently determine the future?

If the answer is "yes," we have a true macroscopic world model. If every step requires going back to check every molecule's complete state, then what we have isn't a theory, just a microscopic simulator wrapped in a thin skin.

The following sections are what I want to emphasize: this "cross-scale" problem is appearing identically in several of today's hottest AI directions.

AI for Drug Discovery: From Binding a Target to Curing a Person, How Many Layers in Between?

Drug development is the most typical example.

Today's AI can already perform quite well on many single-point problems: predicting protein structure, predicting molecule-target binding, generating candidate compounds, estimating how a genetic perturbation affects cellular expression, predicting how certain cells respond to drug action.

These are all important. But a drug's ultimate value isn't determined at the molecular level — it's determined in the patient. Between them lies an extremely long scale chain:

Molecule → binding to target → affecting pathway → changing cell → reshaping tissue → acting on organ → patient efficacy and side effects

A molecule that binds tightly to some protein doesn't mean it will change cell fate as expected. Changing cell fate doesn't mean it can repair tissue. Even if effective in the target organ, it doesn't mean it won't be toxic in another organ.

Every step up, new variables and new feedback emerge in the system. Knock out one gene, the system may compensate through another pathway; inhibit one protein, the cell may simply switch metabolic modes; kill some tumor cells, and you may create selective pressure for the remaining ones. Biological systems are rarely linear.

So the hardest part of drug development is often not predicting any single layer more accurately, but: how to reliably transmit an intervention at one scale to higher scales.

I first heard this concept from Xi-Qiao Feng, the first academician of cellular dynamics in China, and this is precisely where the true weight of today's virtual cell direction lies. The goal of AI Virtual Cell isn't simply reconstructing a cellular atlas, but learning cells' dynamic responses under different conditions and perturbations — researchers have explicitly stated the need to establish unified representations across measurement modalities and scales, and to validate models through perturbation prediction.

But note: even if we truly had a very good virtual cell, virtual patient remains distant. Because from a tissue model's perspective, a cell becomes a "particle" at the bottom layer again. Cells communicate, compete, migrate, and change each other's states. No matter how good a cell model is, stacking a million of them together won't automatically yield a credible tissue.

A level's macroscopic is often just the next level's microscopic.

Life isn't a one-time micro-to-macro, but an entire ladder of scales ascending level by level. Every crossing is a brand new Hilbert's Sixth Problem.

Agent Society: Simulating One Person and Simulating a Society Are Two Different Capabilities

Another highly typical direction is Agent Society.

A batch of companies including Aaru and Simile, plus a string of academic works, are attempting to use large numbers of generative agents to simulate consumers, voters, enterprises, policies, and social behavior. Aaru defines itself as a multi-agent population simulation system; Simile proposes expanding from individuals, long-term journeys, and interactions, step by step to entire markets and even society. Academia can already build an agent for each of over a thousand real participants based on in-depth interviews, and test their ability to reproduce those individuals' attitudes and behaviors.

The first step in such work is making individual agents sufficiently like the people they represent — can they reproduce a person's preferences, experiences, speech patterns, and decision habits. This step alone, done well, is already valuable.

But the leap from "one person" to "a group of people" hides an enormous logical jump:

"If we can accurately simulate every individual, then putting enough individuals together will naturally accurately simulate society."

This sounds self-evident, but it's almost exactly the same error as "knowing every particle's Newtonian motion naturally means knowing fluid motion."

Suppose a model can very accurately answer "will you buy this product?" Add up a hundred thousand such answers, and at best you get faster, cheaper market research. It's still not a market model.

Because a market is never the sum of a hundred thousand independent answers. It also depends on: whether consumers change their minds seeing others buy; whether competitors cut prices; which way platform algorithms distribute traffic; how KOL and friend influences propagate; how scarcity changes attractiveness; and — after companies take action based on the prediction itself, whether the original prediction still holds.

A influences B, B influences C, C in turn influences A. These feedbacks and correlations are the most lethal parts of a social system. They're full of the same "collisions" and "re-collisions" as in gases.

So the truly hard problem for Agent Society was never "how human-like is each individual agent," but:

When countless agents begin influencing each other, does the whole system still resemble a real society?

For evaluating such companies, I think we need to break it down into at least four layers, and these four layers are progressive:

Layer one, individual validity: does the agent resemble the person it represents?

Layer two, interaction validity: does the interaction between two agents resemble two real people in a real situation?

Layer three, population validity: after large numbers of interactions, do the emergent opinions, prices, norms, and behavioral distributions approach real markets and society?

Layer four, intervention validity: when we change prices, policies, products, or communication structures, can the model predict how the real world will change?

The layer with truly enormous commercial value is four. But most benchmarks are still stuck at layer one, at most layer two.

This precisely exposes an extremely common misalignment in AI investment:

The model's benchmark happens at a lower layer, commercial value happens at a higher layer, and the greatest risk hides between these two layers.

Grokking and Emergence: AI Itself Has a Hilbert's Sixth Problem

I once discussed grokking with Ziming Liu. He described his research problem as a kind of "AI version of Hilbert's Sixth Problem," and I found the analogy precise.

Grokking is an extremely counterintuitive phenomenon: a model may memorize the training set perfectly very early on, yet still perform poorly on the test set; after training for a very, very long time, it suddenly shifts from "memorization" to "understanding", and generalization ability jumps up all at once.

Ziming Liu and his team's research approach is to first see the internal mechanisms clearly in very small, very simple models — the dynamics of representation learning, phase diagrams — then study how this mechanism extends upward to larger models as data scale, parameter scale, regularization, and training conditions change.

This isn't the same as particles to fluids, of course, but structurally highly similar. Because neural networks also have different scales:

Microscopic level is how each parameter changes, how each gradient updates, how each neuron and feature forms.

Higher level is whether the model has formed structured representations, whether it has shifted from memorization to generalization, whether grokking suddenly occurs, whether emergent abilities appear after scale crosses some threshold.

Today we're accustomed to describing macroscopic results with scaling laws: more parameters, more data, more compute, loss drops according to some pattern. But scaling laws mainly tell us "how results change," not "what's actually happening inside."

It's a bit like: we've observed how temperature and pressure change, but haven't yet truly derived them from molecular collisions.

So AI's own Hilbert's Sixth Problem might be stated thus:

Can we, starting from parameters, gradients, and training dynamics, explain why macroscopic abilities like representation, generalization, reasoning, and emergence arise? And can we judge — when do mechanisms observed in small models extrapolate to large models, and when are they merely illusions at limited scale?

This matters enormously, because AI research has an extremely common methodology: first discover a phenomenon in a toy model, give it a beautiful explanation, then assume it still holds in ten-billion, hundred-billion parameter models. But small model to large model is itself a scale transition — it needs to be proven, not just analogized.

The reverse is equally true: observing a beautiful empirical curve in a large model doesn't mean we understand the underlying mechanism. Like seeing fluid flow doesn't mean you've derived Navier–Stokes from Newtonian mechanics.

Weather, Robotics, and Corporate Organizations — All Hitting the Same Wall

The same problem appears in a pile of other directions too.

Weather and turbulence. We can't resolve every tiny vortex, so processes smaller than the grid scale must be compressed into a closure model. Machine learning is being used to learn how these subgrid processes feed back into large-scale flow fields — but a closure that fits beautifully on training data, when put back into long-term physical simulation, may rapidly become unstable, even violating energy conservation, symmetry, and generalization under extreme conditions.

Autonomous driving and robotics. Predicting the next video frame doesn't equal understanding a long-term interactive physical world. From pixels to objects, from objects to scenes, from scenes to other participants' intentions, from single-vehicle behavior to urban traffic — it's likewise a continuous ladder of scale transitions. A single-vehicle model that's accurate at every step, when placed in a system of numerous human drivers, pedestrians, and other autonomous vehicles, may produce completely different macroscopic results.

AI in enterprises. We often hear a kind of ROI derivation: every programmer becomes 30% more efficient, so company R&D efficiency increases 30%. But organizational output is never the simple sum of individual outputs. More code written may increase review burden; faster demand production may intensify priority confusion; easier communication for everyone may bring more meetings; local efficiency gains may be completely eaten by organizational coordination costs. This is the same error as "smarter agents will necessarily form a smarter agent society." From individual capability to system output, organizational structure, workflows, incentives, information flow, and accountability mechanisms still lie in between.

Reading this far, that core judgment is worth hammering in again:

Scaling a simulator is not the same as simulating a scale. Making the simulator bigger doesn't mean you've entered a new scale. A million particles is not a fluid model. A million cell models is not a human body model. A million consumer agents is not a society model.

Scale expansion may bring new laws, but it won't automatically tell you what those laws are, much less automatically prove they're reliable.

Every World Model Has Its Own Hilbert's Sixth Problem

Hilbert's Sixth Problem matters not just because it existed for 125 years, nor just because someone finally took a major step forward. But because it proposes an extremely profound worldview:

The world is not a flat plane with a single resolution, but is composed of layers upon layers of different scales.

At different scales, the important objects differ, the descriptive variables differ, the effective laws differ, even the effective boundaries differ. Microscopic laws don't automatically become macroscopic laws just because you enlarge the system — a genuine theoretical transition is needed in between, requiring new representations, new assumptions, and knowing which details can be discarded and which correlations must be kept.

Today, we're using AI to simulate more and more worlds: molecules, cells, materials, weather, robots, human behavior, markets and society, even AI models themselves. And every direction, when pushed to the end, will probably hit the same problem:

Can our success at one scale be brought to the next scale?

If we can't prove this, what we have may just be a very strong local predictor, not a true world model.

We may predict a protein with extreme accuracy, yet still not know why a patient recovers; simulate a consumer with extreme accuracy, yet still not know why trends form; understand clearly how a small model groks, yet still not know why intelligence emerges at larger scales.

A bigger model doesn't equal a bigger world.

A true world model isn't cramming all details into an infinitely large neural network until it replicates the entire universe. It needs to know at every scale: what must be remembered, what can be forgotten, what will disappear in averaging, what will be amplified through interaction, what can emerge upward — and why.

Being able to complete these translations between scales — this is where the truly difficult, and truly valuable, part of world models begins.