Yu Deng won the Fields Medal. The profiles still did not explain what he proved

July 23, Philadelphia. The names Yu Deng and Hong Wang were announced. For the first time, mathematicians holding Chinese citizenship had won the Fields Medal.

Then everyone's social feed filled with profiles. Peking University class of 2007. Shenzhen Senior High School. Recommended admission. Transfer to MIT. Princeton doctorate. The biographies were more thoroughly rehearsed than a recruitment agency's pitch. Some articles even repeated a story about him finding the key idea in a fried-chicken shop. Deng later corrected it in an interview. It did not happen. The team worked on the problem from 2018 to 2025. Seven years.

Seven years of work compressed into a fried-chicken anecdote. That is how a news cycle digests an achievement.

This article is therefore not about his personal story. It is about the theorem, and about the trail of ideas I followed out of it. The first half belongs to their work. The second half is mine. A warning before we start: some of my later deductions were wrong at first, and the mathematics pushed back immediately. I have kept those collisions in the article because they were often the most interesting part.

First, say the problem in ordinary language

Physics has carried a crack for 154 years.

Newton's laws are time-reversible. Reverse the velocity of every particle in the universe and the whole world plays backward exactly. The equations do not need so much as a punctuation change. But in 1872 Boltzmann wrote his gas equation and proved the H-theorem: entropy only increases. That equation is irreversible.

One side says time has no direction. The other says it does. Both sides are correct physics, yet they contradict each other in plain logic. Boltzmann's colleague Josef Loschmidt threw velocity reversal straight back at him. If you derived an irreversible equation from reversible laws, something extra must have slipped into the derivation. The argument was still unresolved when Boltzmann died by suicide in 1906.

In 1900, Hilbert placed the matter sixth on his famous list of 23 problems: derive macroscopic physics rigorously from the laws governing atoms. "Rigorously" matters here. It means mathematical rigor, where even an epsilon cannot go missing.

Lanford laid the first section of foundation in 1975. He proved that the derivation works for a short time. How short? About one fifth of the mean time between collisions. In the time a particle would undergo five collisions on average, his theorem covered one fifth of the first one. Entropy had no time to grow. One corner of the foundation existed, and the building stopped for fifty years.

In August 2024, a paper by Yu Deng, Zaher Hani and Xiao Ma appeared on arXiv. It extended the result to arbitrarily long times. A second paper in March 2025 continued from the Boltzmann equation to the equations of fluid mechanics. Newton, Boltzmann and Navier-Stokes were welded into three connected levels.

That is what the Fields Medal means here. It is not simply that a Chinese mathematician won a prize. It is that the arrow of time now has a foundation.

Hundreds of pages amount to an accounting system

Most profiles stop after "solved Hilbert's sixth problem." The mechanism of the proof is where the value lies, and almost nobody explains it. So I will.

The central move is to split the distribution of particles into two parts. One part contains particles going about their business independently. The other contains particles with old accounts between them, the correlations known mathematically as cumulants. The proof needs to establish one thing: those old accounts eventually disappear.

How do you keep the books? Draw the collision history as a graph. Every collision is a node. Particle trajectories are edges. The authors call it a molecular graph. A loop appears when two particles collide, separate, travel around and meet again. They collided before and then met again. That is memory. A loop is a particle's memory of the past.

Every piece of memory must pay for one decay factor.

Much of those hundreds of pages enforces this rule. The authors devise a cutting algorithm that breaks any complicated collision-history graph into pieces, audits it loop by loop and makes sure that no memory escapes its charge. Once the books balance, old correlations vanish, the macroscopic equation closes, the H-theorem applies and entropy increases.

A 154-year philosophical fight ended through bookkeeping. The first time I understood the structure, I got goosebumps.

The key: information did not disappear. It left

Now we reach the most important sentence in this article, and the starting point for everything I tried afterward.

In a world with a fixed number of particles, not one bit of information is lost. Liouville's theorem guarantees it. So where did the information apparently destroyed by increasing entropy go?

Collisions keep pumping information out of the place you can see, the distribution of individual particles, and into the place you cannot see, the higher-order correlations among three, four or billions of particles. The information did not die. It escaped into a depth that no observer can track in practice.

You see entropy increase because you are allowed to see only the low orders.

This also explains why Loschmidt's reversed-velocity state does not count as an ordinary counterexample. It is not an arbitrary state. It carries precisely coordinated higher-order correlations among all particles. Such a state is not typical under the relevant measure, and the theorem's assumptions filter it out. Irreversibility is not hidden in the equation. It is hidden in the statistical choice of which initial states count as normal.

Up to this point, I have been describing Deng's theorem. From here on, I take that key and try it in other doors. Three opened. Half of another shattered.

First door: hash functions as Boltzmann machines of cryptography

My first thought was the hash function.

A cryptographic hash diffuses every trace of structure in the input across the bits of its output until statistical tests can find no pattern. A deterministic algorithm looks random to an observer. Read that sentence again beside propagation of molecular chaos: deterministic mechanics looks independently random to a low-order observer.

The skeleton is the same. Deng's theorem can be read as the first rigorous instance of this phenomenon in a real physical system. People have compared chaos and cryptography for decades without a substantial theorem underneath the analogy. Now there is one.

Then I got carried away and wondered whether the mechanism could produce a hash algorithm.

That was the half-door that broke. The mathematics slapped the idea down quickly. Thermodynamic "irreversibility" means that information has been hidden, but in principle a reversible simulation integrated backward can recover it. Cryptographic "irreversibility" means the information is sitting in front of you and you still cannot compute the inverse. The first is a question of measure. The second is a question of complexity. Between them stands the existence of one-way functions, an open problem even harder than P not equal to NP. A perfectly diffusive linear transformation can be statistically flawless and still fall to Gaussian elimination in seconds. Chaos is not computational hardness.

Breaking that half-door clarified something else. The statistical security of a hash and its computational security are different things. The first has a physical analogue. Physics cannot supply the second. Most popular explanations of cryptography do not make that distinction clearly.

Second door: why you remember yesterday but not tomorrow

This door opened into philosophy, and I stayed there longest.

Under determinism, both past and future are written into the equations without preference. But examine your own experience. You remember yesterday, not tomorrow. Why?

In the language of escape, the past exists as extractable low-order structure. Fossils, ledgers and the synapses in your brain are macroscopically visible traces. The future is equally determined, but it is hidden in a conspiracy of higher-order correlations and cannot be read in advance. Ontologically the two directions are symmetric. Epistemically they could hardly be more different.

Yet saying that an observer cannot see high-order correlations is not enough. Coarse-graining is itself time-symmetric. If the arrow came only from the observer's viewpoint, entropy should increase toward the past as well. It does not. The symmetry is broken by a blunt physical fact: the universe began in a special low-entropy state with very little correlation. Philosophers call this the Past Hypothesis. The empirical evidence for it is overwhelming. Every memory you possess is evidence. But no deeper principle explains the hypothesis itself. Keep asking and you hit the wall of cosmology.

The complete recipe for an arrow of time is therefore a coarse-grained observer plus a low-entropy beginning. Both are necessary. Deng's proof makes that recipe unusually clean. Across hundreds of pages, the only time-asymmetric input is the assumption of no correlation at t = 0. One place. Just one.

Third door: toward the next Millennium Prize problem

I think the last door is the most valuable because it points toward something still unsolved.

The regularity problem for the Navier-Stokes equations is one of the seven Millennium Prize problems. Can a smooth solution for a three-dimensional fluid blow up in finite time? The more I read, the more it looked like the entropy problem on another stage. Turbulence passes energy down to smaller and smaller scales. This is another escape into higher order, except that "higher" now refers to spatial scale. The question becomes whether the escape can happen so quickly that the solution dies with it.

There is a fact that many people miss. The Navier-Stokes equations are themselves a low-order truncation of the Boltzmann equation, the first term of the Chapman-Enskog expansion. They are not first principles. They are an approximation. Deng's second paper makes the interface at that truncation rigorous. The seam he welded faces the next unwelded seam directly.

I also got hit by the mathematics behind this door. For a while I thought that escape itself meant danger. Two-dimensional Navier-Stokes answered immediately. Two-dimensional fluids have cascades too, but energy escapes toward larger scales. An extra conserved quantity changes the balance, and global regularity was proved in the 1960s. The lesson is that escape alone does not decide survival. Direction, rate and the quantitative strength of the conserved quantities decide it together. Mathematics has no patience for large philosophical nouns. It wants the accounts.

Drawing the net closed

Connecting the three doors left me with one picture.

Nature is a hierarchy of descriptions. Newton is at the bottom, Boltzmann above it, fluid mechanics above that. Each level is a low-order closure obtained by coarse-graining the level below. The failure modes at each level, including failed closure, lost predictability and possible blow-up, all appear at the same place: where information crosses that level's truncation boundary and escapes into degrees of freedom the level cannot see.

Deng's work welded the two lowest seams in this tower and proved that the escape across those seams converges. Navier-Stokes regularity is the open case at the third seam.

I later discovered that physicists already have a name for this view in which the direction and rate of escape decide a system's fate. It is the renormalization group, used for half a century across phase transitions, turbulence and field theory. I did not know the sign on the door when I reached it. Finding that other people had already paved the road felt better than believing I had opened it myself. It meant the road was real.

Return to the Fields Medal. The prize was not for a biography, an anecdote or even one 126-year-old problem. It marked the first time we calculated rigorously how a universe built from laws with no direction can grow a direction.

The answer is that direction does not lie in the laws. It lies in where the information goes. At this moment, every collision among the countless molecules in your body is pushing the present into a depth you will never be able to read. You cannot return, not because the road has broken, but because the map back is written somewhere you have no permission to inspect.

The argument began in 1872. It lasted 154 years.

Part of it is now a theorem.