Autumn 2036, the last class of the afternoon. Ms. Zhou projected the numbers of seven problems onto the white screen. The old projector skewed the colors; the text came out bluish.

“This semester’s project,” she said. “Pick one of the seven Millennium Prize Problems and build a complete project around it. Research the background, solve it with the educational model, and give a demonstration. Five bonus points on the final grade.”

Someone in the classroom let out a quiet cheer.

“Don’t celebrate too early. You also have to submit a process log, and the failed attempts can’t be deleted.”

Lin Xiao lowered her head and copied down the problem numbers. She had only ever heard of the Riemann Hypothesis; she couldn’t even pronounce the names of the others.

On the way home, Xu Yihang walked beside her. “I’m picking the Riemann Hypothesis — it’s the most famous. What about you?”

“Haven’t decided yet.”

At home, Lin Xiao opened her tablet. The educational model greeted her: “Hello, Lin Xiao. What can I help you with?”

She typed: “Of the seven Millennium Prize Problems, which one is most connected to everyday life?”

A few seconds later, the answers appeared. When she saw the Navier–Stokes equations, she stopped: they had to do with weather forecasting, aircraft wing design, blood-flow simulation. The model explained one of its core questions like this: under specified conditions, will fluid motion that starts out smooth develop places where it can no longer stay smooth within a finite time?

“What does ‘no longer smooth’ mean?”

The model played an animation. Water flowed around an obstacle, curling into tiny vortices. The image kept zooming in; somewhere, the color grew brighter and brighter.

“For example, the velocity at some point might become unbounded. But the animation only aids understanding — it can’t replace a proof.”

Lin Xiao remembered a video of an airplane stalling she’d seen in science class last year.

“That’s the one.”

In the research that followed, she learned the problem was divided into four propositions: A, B, C, and D. In 2026, OpenAI had organized more than ten thousand agents, run them for 88 hours, and announced the completion of proofs for C and D. The verification records released afterward filled several screens of material.

Lin Xiao scrolled down for a long time without reaching the bottom.

What ten years ago took more than ten thousand agents three and a half days, her tablet could now handle as an after-school project.

Still, it wasn’t as simple as asking once. The educational model had stripped out scientific advances after 2020 and locked away part of its autonomous planning ability. Ms. Zhou said it was like a driving school’s car: some things were deliberately left for you to do. The fact that you’d never use a manual gearshift in real life doesn’t stop you from stalling here.

On Saturday afternoon, Lin Xiao cleared half her desk and officially started running the model.

The first twenty minutes went smoothly. The model moved forward like an old horse that knew the road, trotting along. She followed the diagrams, occasionally asking it to explain a term.

Then it slowed down.

Path 1 failed. Path 2 failed. Path 3 failed.

At the third failure, Lin Xiao’s finger moved to the pause button.

On the screen, lines of formulae kept flowing. The model was modifying a parameter she didn’t recognize.

She pulled her hand back.

Maybe this was normal. It was a hard math problem — who succeeds in three tries?

Path 4 failed. Path 5 failed.

She went to the kitchen to cut an apple. By the time she came back, the counter had jumped to 8.

The ninth attempt failed too.

“Stop for a moment.”

The model stopped: “The current construction does not yet satisfy all conditions. Attempting parameter adjustments.”

“Hold off on the adjustments,” Lin Xiao said. “Draw out what you’re doing right now. I want to see it.”

A diagram appeared on the screen: a rotating vortex structure, stretching under a force field. Arrows and a few curves beside it.

She pulled up the diagrams from the earlier failed paths and laid them side by side.

The nine images were almost identical — only the size and angle differed slightly.

“You didn’t try nine times,” she frowned. “You tried once, and then repeated it eight times.”

As soon as she said it, she second-guessed herself. The model had changed so many formulae — maybe the diagrams just couldn’t show it.

“Wait.” She added, “Is it that you draw them too alike, or is there really no difference?”

The model marked the identical parts on all nine images, then listed below the parameters that had changed.

“They use the same construction. The parameter adjustments are not entirely repetitive, but all nine attempts so far are stuck on the same estimate.”

“Which one?”

A passage of formulae lit up.

Lin Xiao couldn’t understand it.

“What is this estimate for?”

“It must simultaneously guarantee that the growth at the center is fast enough while the other quantities still satisfy the problem’s constraints. Under the current structure, these two requirements conflict.”

“Is there hope if you keep adjusting parameters?”

“Cannot be ruled out. But no improving trend is visible so far.”

Lin Xiao glanced at the clock. More than forty minutes had passed.

“Save this path first,” she said. “Don’t delete it. Is there an approach with a different structure?”

The model listed three alternative directions.

She had it draw each one. In the second diagram, the external force no longer stretched the same vortex continuously, but acted in stages — like a hand pushing a swing every little while.

“Can you try this one?”

“Yes. The external force must first be checked against the problem’s conditions.”

“Then try it.”

The model started the derivation again.

Seven minutes later, the verifier turned red.

Lin Xiao stared at the screen, her heart sinking.

Maybe she shouldn’t have stopped it just now. The original path had been running so long — maybe one more adjustment would have done it.

She opened the old log, then closed it.

“Was this failure the same kind as before?”

“No. This time the external-force construction didn’t satisfy the specified conditions.”

“Then check that first. Don’t compute all the way to the end just to find out.”

The model listed the conditions separately and redesigned the driving scheme. Lin Xiao wrote a line in her log:

Switching directions can fail too. First ask whether it’s the same kind of failure.

For the next half hour or so, she didn’t stare at the formulae. Each time the model finished a section, it handed her a diagram, along with a note on what that diagram omitted.

Finally, the verifier turned green.

Lin Xiao sat up straight.

“Did the simulation pass, or the proof?”

“The formal proof check passed. The simulation is for demonstration only and plays no role in the proof.”

“Then put that sentence into the presentation too.”

Only then did she go get the apple. The cut half had turned brown.

On presentation day, she first projected the screen recording of the failures onto the white screen. The counter ticked in the corner.

“At the third failure, I wanted to stop it. But I didn’t know whether that was a normal part of the process, so I didn’t. By the ninth, I asked it to draw out all the earlier attempts.”

The nine diagrams appeared side by side.

Someone said from below: “They all look the same.”

“That’s what I thought too,” Lin Xiao said. “But it could also be that the drawings were too simple. So I asked it to mark where things changed.”

The parameters and markers lit up one by one.

“These places did change. But it kept getting stuck on the same requirement, and it wasn’t getting better.”

She opened the new diagram, explained the difference between the two constructions, and finally demonstrated the fluid simulation. The bright spot at the center grew brighter and brighter, while the total-energy curve beside it barely moved.

“This visualization helps us understand local change versus global constraints — it’s not the computer actually drawing an infinity. The proof check is on another page.”

She switched over. Under the green marker stretched a verification log too long to see the end of.

“I can’t explain every line of this,” she admitted. “What I can explain clearly is how the two constructions differ, and why I asked it to stop and check.”

When she finished, Ms. Zhou asked: “What do you think was the most valuable thing you did in this project?”

“Putting the nine diagrams side by side.”

“And what if the next version of the model does that by itself?”

Lin Xiao had been ready to keep talking, but at these words she stopped.

The classroom was very quiet.

“Then we wouldn’t have to wait for me to count to nine,” she said.

A few classmates laughed.

“Then what would you do?”

Lin Xiao looked at the diagram on the white screen.

“I don’t know yet.”

Ms. Zhou didn’t answer right away.

“But before this project,” Lin Xiao went on, “I always believed that as long as it was still computing, it meant the answer was close.”

Ms. Zhou nodded and made a note on her tablet.

At ten that night, her mother came out of the study holding a glass of water and found Lin Xiao sitting on the sofa.

“How did the presentation go?”

Lin Xiao opened the simulation and handed it over: “Mom, try it.”

Her mother dragged a few sliders. The bright spot at the center shrank, then brightened.

“What’s this problem called?”

“The Navier–Stokes equations.”

“Oh, that one.” Her mother paused. “The one from ten years ago.”

“You remember?”

“I remember. I was at work that day; it kept getting forwarded in the group chats. Someone said weather forecasts would never be wrong again, and a while later someone else came out to explain that wasn’t what it meant.”

“It wasn’t.”

“You understand it better than I do now.” Her mother smiled. “Back then I even forwarded the news to your dad. He asked me what had been proved, and I couldn’t answer.”

Lin Xiao took the tablet back and exited the simulation. The nine diagrams appeared on the screen again.

“Actually, I wanted to stop it at the third attempt,” she said.

“Why didn’t you?”

“I figured it understood better than I did.”

Her mother looked at the screen for a while. “And then?”

“Then I figured it was okay to just ask.”

Her mother set the glass on the coffee table and scooted closer to her.

“Walk me through it from here. That little swirl just now made sense to me, but these I don’t get.”

Lin Xiao opened the first diagram again. She started with the arrows, got stuck on the curves beside them, and went back to check her own log.

The two of them huddled over the tablet. The steam from the water glass slowly faded away.