Translator: Dreamscribe
"Oppa, how do I look?"
Seo-eun, wearing a cute pink dress, spun around once in front of Seo-ha.
Several days had passed since the event ended, but Seo-eun was still utterly enchanted with the dress.
"You look incredibly pretty. It's perfect for you, Seo-eun!"
"Hehehe! Really? Can I wear this outside again?"
"Of course. Let's go together next time."
"Then I'm going to practice walking prettily!"
Seo-eun lightly held the hem of her dress like a princess and mimicked a gait she'd seen somewhere. Seo-ha found it so adorable that he chuckled to himself.
"Kids! Time to eat!"
When they came down, New Year's tteokguk was laid out on the table.
"Wow, tteokguk!"
Seo-eun was the first to sit down in her chair.
The way she carefully arranged the skirt of her dress so it wouldn't wrinkle suggested she still hadn't come out of her little game.
"Let's eat."
Chul-ho hurriedly picked up his spoon so the children wouldn't have to wait long.
The tteokguk steamed gently.
Inside the milky white bone broth, thinly sliced rice cakes were arranged beautifully. The garnishes of egg jidan, green onion, and seaweed flakes added color to the dish.
"Mm. Delicious again this year."
"Thank you, Mom."
"Thanks for the food!"
Mi-young nodded repeatedly, her face glowing with pride.
For Mi-young, this ordinary routine of the family gathering together to share a meal was beyond precious. That was why, even though the publisher had commissioned her to write a follow-up essay collection, she never once slacked off on preparing meals.
"Your dad got promoted."
Chul-ho tossed the words out casually, as if it were nothing.
"Really?"
Seo-ha's eyes widened.
An office job at a small-to-mid-sized shipping company affiliated with the school; that had been the first position Chul-ho had taken in America. His main duties were inventory management and acting as the liaison with Korea, but Chul-ho, with his characteristic diligence and meticulousness, performed the role given to him at 120 percent.
"It hasn't been computerized?"
"Would anyone willing to do that kind of work be working at a place like this? The previous person probably jotted things down in a notebook somewhere."
The state of the company's management had been shocking.
'I can't stand this.'
Twenty years as a civil servant had made him a master of Excel.
In just one month, every item of inventory in the warehouse had been digitized. It was work he'd started without barcodes or any management system.
"This much was leaking out?"
The management team leader stared at the Excel sheet and muttered in disbelief.
But his work didn't end there.
'Don't underestimate a Korean civil servant.'
He wrote functions so that a reorder warning would pop up when stock ran low, and delivery delays would be marked in red. Unit prices that had been managed by hand were soon switched over to automatic calculation.
It didn't take long at all for Chul-ho to become a core member of the company.
"A round of applause for Dad!"
Clap clap clap!
At Mi-young's words, Seo-ha and Seo-eun rose from their seats and applauded enthusiastically.
Chul-ho shrugged his shoulders.
He thought it was a true relief that he could at least maintain some minimum dignity as the head of the household.
* * *
Seo-ha had read thousands of papers on the Riemann Hypothesis by now.
Among them were no small number of studies that had been hailed as truly groundbreaking breakthroughs. Analytic Number Theory, functional equations and symmetry, connections to the prime number theorem, random theory.
They were all theories that had once shown promise but were now known to everyone as failures.
Why, despite so many mathematicians' long efforts, did the Riemann Hypothesis still remain unsolved?
Had mathematical techniques been lacking?
Had the sheer volume of computation been too much to handle?
Had ideas been in short supply?
Seo-ha shook his head.
It was none of those.
Analytic Number Theory had developed brilliantly over more than a century, and the calculation of the zeros of the Zeta Function had expanded to a meaningless degree since the advent of quantum computers.
The approaches to the Riemann Hypothesis had likewise been so varied that there was practically nothing that hadn't been tried. Yet the common thread was clear. The moment anyone tried to pin down the zeros of the Riemann Zeta Function, the approach quickly revealed its limits.
'Why can the space of possible solutions not escape the region Riemann proposed?'
No one had been able to answer this question.
Seo-ha decided to throw out a slightly more fundamental question.
'Is the Riemann Hypothesis a question of true or false?'
This was not a philosophical question but a mathematical-logical one.
If it was true, why did every attempt stop at obtaining an approximation?
If it was false, why had not a single counterexample appeared to disprove the hypothesis?
Every numerical value said the Riemann Hypothesis was true, but nobody could prove it.
"In cases like this, the direction is usually wrong."
Various thoughts flashed through Seo-ha's mind.
Godel's Incompleteness Theorem.
Within a sufficiently complex axiomatic system, there exist propositions that are true but cannot be proven. Many logicians had already argued that the Riemann Hypothesis might be one such kind of problem.
Seo-ha shook his head.
'This is not the way to go.'
In the end, this kind of approach would only lend legitimacy to the claim that it could not be proven.
Seo-ha expanded his thinking into the physical world.
There is a certain relationship between the Riemann Hypothesis and the behavior of quanta. This was something he had already proven through observing quantum chaos.
'The Uncertainty Principle?'
Just as position and momentum cannot be measured precisely at the same time, could it be that the existence of the zeros and the reason for their existence cannot be captured simultaneously?
His mind tingled. His intuition was whispering that something in the Uncertainty Principle was connected to the Riemann Hypothesis.
Seo-ha set down his pencil and sat in his chair for a long time. One hour, two hours, until the sun slowly began to set.
Then suddenly, a word flashed through his mind.
'Measurement?'
Mathematicians had always treated the zeros of the Riemann Zeta Function only as results.
Why do the zeros exist?
Where are they distributed?
But no one had ever seriously asked.
'What are we measuring?'
In quantum mechanics, a wave function has no physical quantity before observation.
It exists only as a structure of possibility. And the moment observation occurs, that structure collapses and becomes fixed at a single value.
'Riemann, what on earth did you discover?'
Natural numbers have no fixed position.
But the moment they pass through the Zeta Function, they acquire period and phase.
The Zeta Function lifted the natural numbers, which are discrete (discrete * separated) objects, up onto the continuous space of the Complex Plane.
The irregularity of the primes was converted into a series, and through Analytic Continuation, the series extended into a domain where it wasn't even originally defined.
'All non-trivial zeros of the Zeta Function lie on the real part 1/2.'
There was no proof. No explanation.
Riemann had simply, as if writing a diary, recorded a single phenomenon in plain terms.
To him, it might have been nothing more than a passing observation. Who could have known? That this single paper would end up torturing countless mathematicians for such a long time.
Theo waved his hand at Seo-ha.
"There's no shortage of crazy mathematicians out there. I mean, even my Stanford cohort buddies reached out to me."
Since that day's presentation, Team Apex had been receiving looks that were half concern and half expectation.
The worry that Seo-ha might collapse just like all the mathematicians who had fallen before the Riemann Hypothesis, and the hope that, if it was him, he might actually solve this devilish problem.
"But you've got a rough idea of it, right?"
Su-jeong asked, looking at Seo-ha.
Seo-ha had always been like this. Even in choices that seemed impulsive at a glance, there was always a reason.
"I had a question."
"About the Riemann Hypothesis?"
"Yeah. Mathematical thinking usually has definitions and premises, and a flow that follows from there. But strangely? In Riemann's case, it was completely different."
"Come to think of it..."
Sri stroked his chin.
"Observation came first. One day, the phenomenon just dropped from the sky. It was so curious that mathematicians chased after it, forcing structure onto it any way they could."
"It's a pretty unusual case."
They had been asking about the cause and meaning of the phenomenon, but they had never clearly defined what it was they had observed.
Like ancient peoples who had been greatly shocked after witnessing lightning.
"So I wondered. Could there perhaps be something that forces the zeros to appear only at certain positions?"
The three of them were startled.
"Ah! Set the position aside for a moment?"
"So you're saying the zeros are the product of inevitability?"
Seo-ha nodded.
"If we assume that, it cleanly explains why no counterexample has appeared until now. Of course, proof would be impossible unless we figured out what that 'something' is."
Seo-ha walked to the blackboard and drew the complex plane.
"The Zeta Function converts natural numbers into waves. Something is definitely happening in that process."
"Like a Fourier Transform?"
"Similar, but far more powerful.
The Zeta Function seems to impose an absolute constraint on integers. Like how the string of a stringed instrument can never produce a sound above a certain frequency."
Seo-ha had also drawn inspiration from his conversation with Lawrence.
The Zeta Function hangs natural numbers on a string. Then it fixes them onto the frame that is the real part of the Complex Plane.
Theo turned the idea over in his mind. There was not a single contradiction in Seo-ha's hypothesis.
"So you're saying real part 1/2 is the maximum amplitude the Zeta Function allows... there's no way out unless the system itself breaks."
Sri shouted, as if he'd just realized something.
"Ah! Then that also explains the relationship with quantum chaos that Seo-ha proved earlier, doesn't it?"
"The spacing statistics between energy levels, Random Matrix Theory.
People had already been catching on to it for decades. I just proved that there was a rule to it."
Seo-ha drew an energy level diagram next to the plane he had drawn on the blackboard.
"This is speculation, but it's no coincidence that the patterns appearing in quantum chaos and the distribution of the Riemann zeros match."
"Huh?"
"Ah... ah!"
"I see."
The three of them reacted intensely.
"That's right. If my hypothesis is correct, it would mean both are subject to the same kind of constraint."
Could it be evidence that they were approaching a law of the universe?
Physics and mathematics were connecting in a place no one had ever expected.
"This doesn't seem like something Number Theory can handle."
Su-jeong lowered her head, her face downcast.
Number Theory dealt with static objects such as natural numbers, algebra, primes, divisors, and congruence relations. But the path Seo-ha was now trying to take was constraints and spectra.
It was fair to say there was nothing she could do in that territory.
"No. It's not something that current Number Theory can handle."
"Huh?"
"This is a problem that absolutely has to be dealt with through Number Theory."
"Why?"
"Because the key is that integers and quanta are governed by the same law. I believe that if it's you, noona, you can find a way, right?
If this is solved, Number Theory might have to be rewritten from the ground up."
Su-jeong nodded with a serious expression.
"Just leave it to me."
Creak.
Su-jeong opened the door of the research lab and stepped out into the hallway.
She had been walking perfectly normally, but the moment she turned the corner, she staggered dangerously.
She pressed a hand against the wall and barely managed to catch her balance.
"Ugh!"
Nausea surged up.
The fluorescent lights in the hallway seemed to sway dizzyingly.
Su-jeong hurriedly found the restroom and dropped to her knees in front of the toilet.
"Hurk!"
Thankfully, she hadn't eaten anything all day.
All that came up was stomach acid.
Her hands trembled and cold sweat poured down her whole body.
'I believe that if it's you, noona, you can find a way, right?'
What if I fail?
Will you just solve everything yourself like you used to?
"How long am I going to keep relying on him? I came all this way to be of help to Seo-ha!"
Shhhh...
'Yes. I'm scared.'
She admitted it.
She was afraid of disappointing Seo-ha. She was scared of being the one who dragged the team down. She was terrified of being left behind, alone, in front of the most famous unsolved problem in mathematical history.
But what scared her the most was the possibility that Seo-ha would solve this problem without her help.
Su-jeong gripped the side of the sink tightly. And then she muttered, as if repeating it to herself.
"I can do this."
Because Seo-ha said she could.
Because his eyes had never once been wrong.