Chapter 6 - A New Structure
Chapter 6 - A New Structure
The next morning.
Siwoo woke up a little later than usual. When he went downstairs and sat at the dining table, Hyejin examined his face first.
While she was checking if the area under his eyes was dark, if his fingertips were trembling, or if there were any traces of a nosebleed, Taesik saw that gaze and chuckled.
"You didn't do any secret research yesterday, did you?"
"...I didn't."
"Then what's with that look?"
"It's a look of injustice."
"You really didn't?"
"Really."
"Suspicious..."
"...Appa."
"Why?"
"You have no trust in your son."
"It's because of your track record."
Siwoo couldn't refute that.
In reality, until just a few days ago, he had wanted to exceed the two-hour time limit. He felt like he could see the answer if he just looked a little longer, if he just dug a little deeper.
But after collapsing once, he realized that seeing and enduring are different things.
No matter how close the answer looked, it was over if his body collapsed.
Hyejin served rice and said firmly.
"Two hours again today."
"Yes."
"And if you go over?"
"I won't."
"Good."
Taesik muttered from the side.
"Your nagging has increased."
"I can hear everything."
"I said it so you'd hear it on purpose."
After the meal, Siwoo went up to his room, and on the desk were the notes he had written the day before.
[Not a wall, but a baseline]
[Not crossing, but aligning]
[Not a language problem]
[Not a coordinate problem conversion, but an alignment]
They were short sentences, but for some reason, they kept catching his eye. Until yesterday, he thought it was translation.
Geometry explains the same object in the language of geometry, Algebra in the language of algebra, and Topology in the language of topology, so he thought they just needed to be translated into each other.
But looking at it again, translation alone wasn't enough.
Just like changing Korean to English, it was possible to move formulas into another language, but it wasn't perfect, so the meaning changed little by little.
Just like how maps of the same city don't overlap perfectly if the scales are different.
Siwoo took out a sheet of paper, drew two circles, put dots inside, and connected them with lines.
Overlapping parts, misaligned parts, disappearing parts, and remaining parts. Siwoo looked at the drawing and said softly.
"It's not translation."
Siwoo pressed the stopwatch, and with a tik sound, the two hours began.
This time, instead of writing formulas from the start, he drew a picture first.
Three circles: Geometry, Algebra, Topology. Next to them, a small square, and arrows connecting them. It was sloppy at first.
But as he drew a few more lines, a strange shape began to appear.
The three disciplines looked like they were standing in different places, but in reality, they were measuring the same object with different standards.
The problem wasn't the language; it was the standard.
Because the standards were different, the results were slightly off even when looking at the same thing, and people treated that misalignment like an error or an exception.
But if that misalignment itself was the core, the story changes.
The arrows on the paper increased.
Structure A, Structure B, Structure C. It wasn't that they were moving separately, but that they were moving together within a single framework.
[Common Standard]
He wrote that and immediately crossed it out. "Common Standard" was too ordinary.
This wasn't a problem of making the standards one. It wasn't about forcibly making different standards into one, either.
It had to be a way to keep each standard while placing them on the same space simultaneously.
Only then would the meaning of Geometry not disappear, the structure of Algebra not collapse, and the sense of Topology be maintained.
Siwoo thought for a moment and wrote a new sentence.
[Common Coordinate System]
In an instant, something began to click in his head.
Siwoo's eyes changed, and he took out a new sheet of paper.
Circles, lines, arrows, and simple formulas were connected all at once.
There weren't many formulas; in fact, they were decreasing. As complex lemmas and long-winded assumptions disappeared, what remained was a surprisingly simple structure.
If this was really correct, it meant people had been going too far around for decades.
After an hour passed, Siwoo completed the first form.
It was a way to put the three disciplines on one common coordinate system; it wasn't a proof yet.
But the parts that kept misaligning in existing papers found their place naturally within this framework.
The parts he thought were blanks didn't disappear; instead, they became connection points.
He didn't know exactly what he had drawn yet.
But it was definitely a shape that didn't exist in any existing paper.
He checked again.
First condition maintained, second condition maintained, third condition modifiable, fourth condition—this was the problem.
The fourth condition was still precarious.
If he touched it wrong, the whole structure would collapse. Siwoo glared at the paper and muttered.
"This is the threshold."
Siwoo erased the word "wall."
It wasn't a wall. Because it breaks if you try to cross it, you have to align it precisely.
Like a threshold, if the position where you step is slightly off, the balance collapses.
Siwoo wrote again under the fourth condition.
[Not crossing, but aligning]
The remaining time was 31 minutes.
It was too close to stop here, but dangerous to push to the end.
Not going to the end, but seeing only what was necessary—that was the method he needed right now.
For the last 20 minutes, Siwoo organized the structure.
Common Coordinate System, baseline, threshold, blank connection points, and alignment. He tied these words into one flow.
There was no complex proof, but instead, one short sentence remained that anyone could understand.
[The blank is not an error, but a connection point]
The moment he wrote that sentence, Siwoo knew.
He had found something today. It wasn't a completed answer, but there was a path where he thought there wasn't one.
There was a door that wasn't marked on the map.
A new structure that didn't exist in any existing paper was lying on the paper, and Siwoo attached a title to the bottom of the last page.
[Common Coordinate System Draft]
At that moment, the stopwatch rang, and he put down his pencil.
He wanted to write more, but he took his hand off and organized the papers.
Siwoo selected only a few important pages, scanned them, and sent an email to Jonathan.
[Progress Report]
After pressing the send button, Siwoo got up from his seat immediately.
Jonathan Reed was reading the printouts Andrew sent without even skipping lunch.
At first, he thought it would be another short memo, but his hand stopped at the first page.
On the second page, he pulled his chair in, and on the third page, he took off his glasses.
When he reached the last page, he couldn't say anything for a while.
"Common Coordinate System Draft..."
Jonathan read the sentence in a low voice.
This wasn't a proof or a paper, but it wasn't a scribble either.
It was an attempt to put the approaches that had been scattered until now into one framework, and it was completely different from existing methods.
He called Andrew immediately, and as soon as Andrew arrived at the lab, he received the paper and his face stiffened without even reading a few pages.
"Professor..."
"Did you see it?"
"This is..."
"Yes."
"It might not just be a problem of the Hodge Conjecture."
Jonathan didn't answer, but he was thinking the same thing.
If this structure was correct, it wasn't just that the approach to one difficult problem would change.
The very way of connecting different fields could change.
It was still rough, dangerous, and had many gaps, but the direction was too clear.
Andrew read the last sentence again.
[The blank is not an error, but a connection point]
"Professor, about this sentence."
"I stopped there too."
"It might be better if it were wrong."
Jonathan chuckled.
"That's not something a mathematician should say."
"If it's right, things get too big."
"They've already gotten big."
That afternoon, at MIT.
Professors gathered in a small mathematics conference room.
Jonathan distributed Siwoo's materials, and the professors read quietly at first.
But a few minutes later, their expressions began to change.
Someone frowned, someone made marks on the paper, and someone read it again from the beginning.
The first to speak was the Algebraic Geometry professor.
"It's not a proof."
Jonathan nodded.
"There are many holes."
"That's true."
"Then why is it so bothersome?"
Andrew said.
"We had the same reaction."
Another professor held up the paper.
"This child sees the parts we discarded as errors as connection points."
"What if that's wrong?"
"It's an ordinary delusion."
"What if it's right?"
The conference room became quiet, and no one could easily answer that question.
If it's right, the very way of looking at the problem for decades is shaken.
If it's right, this child isn't solving the problem, but changing the shape of the problem.
Jonathan walked to the blackboard and copied the structure Siwoo had drawn.
Common Coordinate System, alignment of the three structures, baseline, and connection points. As the blackboard filled up, the atmosphere in the conference room became heavier.
At one point, a professor said in a low voice.
"We need to hear this directly."
"I agree."
"Should we call the child?"
Jonathan thought for a moment.
"Not yet."
"Why?"
"That child is currently adjusting his speed. If we pull him, he might break."
"Our job is not to push him."
"Then?"
"We should open the path, but let the child decide the speed."
That evening, when night fell, a call came from Andrew, and Siwoo closed his magazine and picked up the receiver.
"Hello."
"Siwoo, it's Andrew."
"Hello."
"I saw the material today."
"Yes."
"About the term 'Common Coordinate System'..."
"Is it strange?"
"It's strange."
"Is it wrong?"
"It's too accurate to say it's wrong, and too dangerous to say it's right."
Siwoo thought for a moment.
"Then we don't know yet."
Andrew laughed.
"Yeah, we don't know yet."
"Then it's fine."
"Why?"
"Because if we don't know, we can just look."
Silence flowed over the phone for a moment, and Andrew laughed strangely after hearing those short words.
If you don't know, you can just look. It was a phrase mathematicians feared even after risking their whole lives.
There are people who stop because they don't know, and there are people who dig in because they don't know.
Yoon Siwoo was the latter.
"Siwoo, don't do any more today."
"I'm not."
"Really?"
"I was looking at a magazine."
"What magazine?"
"The one with dinosaurs and space."
Andrew laughed again.
"That's good. Keep watching that."
"Yes."
After hanging up, Andrew sent a short message to Jonathan.
-He is resting today.
It seems right not to touch him forcibly.
Jonathan read the message and nodded.
He was standing in front of the blackboard, and Siwoo's Common Coordinate System structure remained on the blackboard as it was.