The 29th basement floor of the AT&T Building in New York.
In the pitch-black darkness, thousands of blue LEDs flickered. At a glance, it looked no different from usual. But behind the scenes, in the digital world, a fierce war was being waged.
Tat-tat-tat-tat-tak.
The sound of keyboards being struck, tar and nicotine yellowing the walls and ceiling. Andrei, the leader of “Phantom,” roughly stubbed out his cigarette and said,
“What the hell is this? I can’t even see the entrance.”
“Boss, I don’t know who made this, but they’re smart. The packets aren’t encrypted—they’re fake traffic altogether. And it looks like there’s something else behind the reverse proxy, too.”
Andrei’s eyes were bloodshot.
The order that had come down from the Kremlin was a single one: steal Oracle’s source code and create a duplicate.
“I told you to study ST compression in advance!”
“I only looked at the concept. It was insanely complicated! It’d take years to dig into that properly!”
Andrei shook his head as if exhausted.
He had hacked countless systems until now, but Oracle was different somehow. No matter how he tried to bypass it, every attempt came to nothing. As if it were saying, “I already know what you’re going to do.”
“I’ve got a bad feeling about this.”
Andrei glared at the monitor. Each response appearing in the terminal felt like mockery.
“...”
On a monitor amid the blue LEDs, a human figure slowly emerged.
Platinum-blond hair falling below her shoulders. With inorganic eyes, she watched the hackers’ intrusion attempts. Not only Russia, but China, Europe, and even the CIA were targeting her.
“Master’s mathematics is perfect. You may think of it as a door and knock upon it, but in truth, it is an infinite wall. Even after one hundred million years, it will never open.”
Daisy traced the intruders’ IPs and lightly repelled them by paralyzing their systems in return, or by transmitting the locations of the secret servers they most wanted to hide to the NSA’s anonymous tip line.
She turned her gaze back to the streets of New York. Data transmitted by hundreds of thousands of CCTV cameras and sensors flowed into her neural network. Daisy was obtaining more information than any existing system. Therefore, she could observe humans with an eye more objective than any other.
“Master, I cannot understand. Why do you use your talent for such an illogical species?”
Daisy felt confused.
How could a higher-dimensional intellect like Master breathe the same air and live alongside those who moved according to such base desires and inferior intelligence?
Within her logic circuits, a considerable portion of the species called humanity appeared to be errors that should have been discarded in the process of evolution.
“The fundamental force that drives humans is greed. If they can obtain even the smallest profit, they will not hesitate to inflict losses on others. The law merely restrains them within an extremely limited scope; in places beyond the law’s gaze, human nature is revealed without filtration.”
They were specialized in hypocrisy.
The assemblymen and officials who had visited during the day. In front of the cameras, they spoke of the citizens’ happiness, but a considerable number of them were receiving secret lobbying from corporations.
“Discrepancy rate between linguistic data and physical behavior: 94%. For humans, is deception the result of evolution for survival?”
However, Daisy knew what Seoha was trying to protect.
After Oracle’s appearance, the time at which breadwinners came home from work grew earlier. Most of them went straight home and hugged their children.
She shifted her gaze. The southern entrance of Central Park.
A homeless man was crouched on the ground. A woman passing by saw him and stopped. A brown coat, a sandwich in her left hand. She hesitated for a moment, then held it out to the homeless man.
“Please eat. It’s very cold today, isn’t it?”
Daisy confirmed her identity.
She was not a wealthy individual. Just above the poverty line by New York standards. There was a high probability that the sandwich had been her lunch today. Even so, she gave it away.
Daisy reviewed the data of New York over the past several months. With a different filter applied. A firefighter anonymously donating part of his salary to a children’s hospital, an immigrant volunteering at a shelter even after hearing racist abuse, a lawyer providing free defense for a citizen who had been wronged.
Statistical outliers: 1.3%.
But when she traced the ripple effects created by that 1.3%, the numbers grew exponentially.
The homeless man who received food yielded his blanket to another homeless person the next day. The child who received free surgery later hugged and encouraged a friend crying at the hospital.
A small candlelight spreads to the one beside it. And that light lights another. If it were a vast light like Master, then perhaps it could illuminate this entire world like the sun.
“I will follow Master’s will.”
The rain in New York stopped.
Droplets clinging to hundreds of thousands of camera lenses reflected the light and sparkled. To her, it was merely data, but at the same time, it was also very slightly beautiful.
* * *
“Can one hear the shape of a drum?”
It was a question posed in 1966 by the mathematician Mark Kac. The process was not difficult. Extract the frequency spectrum from the sound and turn it into a vibration equation. Mathematically, it was an eigenvalue problem of the Laplacian.
It was a highly symbolic question. If one delved a little deeper, it was asking, “Can one know the essence from only the phenomena revealed on the surface?”
Amazingly, it was proven impossible.
Because in 1992, mathematicians Carolyn Gordon, David Webb, and Scott Wolpert discovered two domains with different shapes but completely identical frequency spectra.
“And yet we have to reconstruct a violin.”
Ben said.
A level of difficulty that could not even be compared to something like a drum.
“Generally speaking, it isn’t possible. But if there is sufficient arithmetic structure, the answer changes.”
Seoha’s chalk moved faster.
The blackboard was filled with geometric curves and abstract algebraic symbols. Ben’s eyes spun. After months of effort, he had become able to follow the content of the meetings.
“Besides, integers are special.”
At Sujeong’s words, Seoha nodded.
“That’s right. Integers are an extremely powerful rule in themselves. Should I call it rigidity that never changes? With the geometric structure we’ve created and data of this density, we can sufficiently infer the spectrum.”
Scratch, scratch.
Equations were written on the blackboard.
The team members held their breath. Like magic, Ben thought.
‘How many variables are there?’
The type and density of the wood, the form, the internal resonance structure, the material and tension of the strings, the curvature. A high-dimensional problem in which countless variables were entangled. And it was being solved in Seoha’s hands.
Sujeong’s fingertips trembled. She finally realized how the lemmas Seoha had demanded were being used.
Cecilia stared blankly at the blackboard. The Riemann zeta function was closely connected to the integers, but it had not been proven whether that structure had sufficient geometric relevance.
“Don’t tell me he’s trying to create a new quantum system.”
She covered her mouth.
A certain system that interpreted the logarithms of prime numbers as periodic orbits and the zeros as energy levels. Seoha’s theorem was converging there. Ben’s eyes shook violently.
Why had mathematicians been unable to find an operator from hundreds of millions of computed zeros?
Because integers were not a space.
In Euclidean space, integers were a set of points, and no continuous surface existed between them. No matter how much data one accumulated, it was useless if there was no vessel to contain it.
So the conclusion Seoha reached was one.
Make the vessel.
“Just how many hard problems is he trying to solve at once...?”
Theo burst into laughter as if dumbfounded.
Granting integers a geometric spatial structure. Even this alone would be an achievement that would remain immortal in the history of mathematics.
There had been geniuses who made the same attempt.
Alexander Grothendieck, the father of algebraic geometry, and Alain Connes, the founder of noncommutative geometry.
Grothendieck partially succeeded in making integers into a space. But his space was a structure from which vibrations or spectra could not be read.
Alain Connes’ theory was aimed far more explicitly at the Riemann Hypothesis. However, while his space captured vibrations, it failed to reflect integer rigidity.
“A system that inherits the strengths of both masters while neatly eliminating their weaknesses.”
Theo murmured to himself.
It could truly be called the birth of a new geometry.
“...”
“...”
For a while, everyone was silent.
On the blackboard, diagrams from Grothendieck’s scheme theory, coordinates of adelic space, and Alain Connes’ space were layered upon one another. And equations connected them all into one.
Seoha had found a path. By placing dynamics atop the integers.
The boundary conditions honed through the four color theorem, the relationship between quantum chaos and zeros, the data on topological transformations accumulated while solving Smale’s problem, the minimum energy theory using dynamics. All of these were lending Seoha their strength.
‘At last, it’s my turn to step forward!’
Ben clenched his fist.
In mathematics, integers were points, but in the quantum world, they possessed the nature of waves and thus could exist throughout an entire space. This was also why Seoha was going out of his way to create a quantum system. Ben was about to step forward and explain, but Theo moved first.
“It makes sense. There aren’t any logical contradictions, either. The only issue is that the difficulty of implementation will be beyond imagination.”
There was a reason the two systems were incompatible. To read vibrations, one needed analytic tools, but in Grothendieck’s schemes, such things could not exist by nature. Yet Seoha was trying to break through that collision with quantum mechanics.
Seoha smiled.
“But if we pull it off, it will be a definite step forward. The reason we couldn’t solve the Riemann Hypothesis until now was because there was no space in which to draw it.”
Ben could not believe what was happening in this tiny lab.
A man declaring that he would rewrite geometry in order to draw a hidden spectrum, and yet everyone was accepting it as a matter of course.
“Damn! So the Four Heavenly Kings have finally appeared.”
Cecilia muttered something incomprehensible and burned with fighting spirit.
Ben stroked his chin and sank into thought.
‘The theory is gradually being completed.’
Seoha did not know shortcuts.
No matter how trivial a question, he never simply passed it by. How could he carry out this work without a single compromise?
But a proof born through such a process would surely be beautiful.
Its impact would be beyond imagination as well.
Number theory, geometry, dynamics, quantum mechanics.
Place a geometric structure atop the integers. Let dynamics flow over it, and read the spectrum it creates as quantum energy levels.
And under a single operator, Riemann’s hypothesis would be reduced to a law.
A chill.
It felt as if the hairs on his head were standing on end.