Chapter 174
Translator: Dreamscribe
"S-Seo-ha?"
Theo wore a bewildered expression.
The moment the proof hit a wall, the air around Seo-ha changed. His gentle demeanor was nowhere to be found, and something almost like madness radiated from his face.
He wasn't the only one to sense the change; a low murmur rippled through the lecture hall. But Seo-ha heard none of it.
Hummmm-
The world simply hummed like a giant beehive, nothing more.
Seo-ha took a step back and looked at the whole blackboard. Equations filled all three panels. They came into view like a vast landscape.
'I can't see it from here. Let's go deeper.'
Where did he end and where did Ducky begin? A meaningless distinction. Two consciousnesses were trying to merge into one. Seo-ha didn't resist. Before long, the boundary between them vanished.
A new field of vision opened.
There were no straight lines, no horizontals. A space of Noncommutative Geometry where everything was curved. Within the space his own equations had created, Seo-ha's body drifted.
There, he saw the glittering Zeros.
"The stars are still there."
If the Operator had diverged into imaginary numbers, they should have vanished. But they remained, simply trembling there in their places.
Suddenly Seo-ha recalled reading the waves of the Riemann Zeta Function in the music room back in his Gifted High School days. This was it. The essential nature of this space.
Real part 1/2.
A single line cut through the space. And every star sat upon that line. They were trembling, but they never strayed from the line laid down for them.
'Look.'
Ducky pointed to it.
The gaps between one Zero and the next.
Something lay within those seemingly irregular gaps. Seo-ha zoomed in his vision and looked at them.
2, 3, 4, 5, 11, 13....
They were Prime Numbers.
Each time the Zeros lined up, a Prime Number asserted its presence between them.
"Are the Zeros pulling the Primes in?"
'Or maybe the Primes are pushing the Zeros apart, creating the gaps.'
Seo-ha tried expanding the space.
The line of real part 1/2 stretched out infinitely. The Zeros glittered like stars in the Milky Way. He could feel the Primes breathing in the spaces between them.
"Beautiful."
'Because it's the rule of the world. We aren't the first to come here.'
Ducky said quietly.
"So Riemann must have seen this too."
Riemann had surely stood alone in this space. But he had no language to express it. Noncommutative Geometry had not yet been born, and Étale Cohomology did not exist. All he held in his hands was Complex Analysis and his own intuition.
And so he wrote this in his paper:
[It is very plausible to assume that all of these zeros lie on the real part 1/2.]
Darkness slowly settled over the space.
Something faint as mist gradually took on form, until it shaped itself into a human figure.
A man of lean build; he was not tall, and his shoulders were slightly stooped.
His black coat was neat but somehow worn. A pale face, deep-set eyes. He gazed at Seo-ha with an expression steeped in regret.
Seo-ha recognized him at once.
"Riemann."
He stepped closer and said something.
Seo-ha couldn't hear his words, but he could grasp their meaning.
"Were you waiting for me?"
A faint smile rose at the corner of his lips. He was relieved.
Peace, arrived at last after a long wait.
How lonely he must have been.
Seo-ha felt a pain as though his chest were being squeezed.
This man had arrived too early. When the world was not yet ready, Riemann had felt his way along the edges of the universe with intuition alone.
He held out a gaunt, withered hand.
Seo-ha thought he could make out ink stains. A hand that must have lifted a pen and written equations countless times.
Tilting his head, Seo-ha took his hand.
"Are you inviting me?"
Riemann smiled and shook his head.
And then, as if to make sure he understood, he spoke, shaping each word slowly and clearly with his lips.
Seo-ha's eyes widened.
"Carry it on...."
With Riemann's words, Seo-ha's consciousness snapped back up into the auditorium of the real world.
The cool air of the lecture hall, the sharp, heavy smell of chalk stinging his nose, and the hundreds of eyes fixed on him. Seo-ha staggered and braced himself against the blackboard.
'Carry it on?'
A theory he'd had to set his pen down before finishing. A cry to take up that noble will, the will that had longed for the order of the universe even in the face of death.
"Seo-ha!"
"Seo-ha!"
His teammates started to rush forward, but Seo-ha raised a hand and stopped them. He had to hurry and put in order what was tearing through his mind right now.
Scratch scratch.
Seo-ha's chalk began to move again.
Hundreds of structures appeared and vanished in his mind. Unsorted intuitions swirled like a storm and collided.
'Don't worry, Riemann. I can do this. There's been so much progress in mathematics since your time.'
Among all those intuitions, Seo-ha managed at last to find the right path. The space that Alain Connes and Alexander Grothendieck had dreamed of. But that alone was not enough. A new geometry was needed.
Suddenly he remembered the Zeros he'd seen in Riemann's space and the arrangement of Primes within them.
From 2 to 3, from 3 to 5.... They were not a mere sequence of numbers. They were an arithmetic flow in which the wave of each preceding Prime determined the phase of the next.
'Therefore the Operator is....'
Seo-ha picked up Ducky's thought.
"It was a metronome controlling the wave and the flow of time."
A geometry that could contain this.
Trying to fix numbers in space had been the mistake. A static space could not hold time and waves. What reflected in Seo-ha's eyes was a sea of numbers churning with vast energy.
Scratch scratch.
Realization came flooding in.
In the world of waves, order itself became truth. For through the flow of Primes ran an arithmetic time that could never be reversed.
In this strange space Seo-ha had created, even a point became part of the flow. And so the Operator, inevitably...!
"Huh?"
Andrew Wiles was the first to notice the anomaly.
A geometric space no one had seen before. A higher-dimensional concept that encompassed spacetime. Naturally, a place where an absurd amount of chaos existed.
The moment Seo-ha's Operator dropped anchor on those rough waves, the imaginary terms that had been diverging and defying all logic canceled out and vanished as if by miracle.
Whitman stroked his chin as if turning something over in his mind.
"It resonates?"
Andrew Wiles nodded.
"That space will need further verification, but if Seo-ha's logic holds, that Operator is the Hamiltonian of this geometric universe."
The professors could not collect themselves.
Seo-ha had fallen deep into thought, then begun writing furiously. From that point on, most of the professors could no longer follow the thread.
And a Hamiltonian, of all things.
The fabled key that Hilbert and Pólya had longed for: this Operator possessed the property of being Self-adjoint.
It meant that as the wave of one Prime passed into the next, energy was perfectly conserved. And if so, then naturally every Zero of the Zeta Function would be unable to stray from a single line.
A proof in the most beautiful form, one that allowed no rebuttal.
Gulp.
The air in the lecture hall shifted strangely.
Those who understood and those who did not: a few stared at the blackboard with even greater concentration, while the majority of scholars, not knowing what was happening, read the room.
"This looks like it's almost at the end now."
"To see this before I die, it truly pays to live a long life. The greatest fortune of my life."
"I should have taken my medicine before coming. I'm worried my heart might stop."
The final moment of verification that remained.
If the Zeros of the Zeta Function matched the Spectrum of this Operator, the proof would be complete.
Scratch scratch.
Seo-ha wrote a single equation on the blackboard.
The condition under which the Zeta Function equals zero. Seo-ha transcribed the Riemann Zeta Function into the language of the geometric space he had built.
The terms of the function changed one by one.
"Th-that's...!"
"Is this transformation really correct?"
"Be quiet for a moment."
Andrew Wiles stared blankly at the blackboard.
The transformed terms matched the Spectrum exactly. Seo-ha's logic was flawless. Because the structure did not permit it, nothing could escape.
Tap-tap.
The final equals sign written on the blackboard.
In that instant, Seo-ha saw it.
A carpet unfurled across a universe as vast as infinity. The line of the real part stretching out endlessly. The Zeros embedded along it glittered like stars in the Milky Way. And the Operator pulsing along the Primes within them.
It was a breathtakingly majestic and beautiful sight.
Seo-ha closed his eyes.
The faint apparition of Riemann. He turned up the collar of his worn coat and nodded as if satisfied. Then he slowly faded away into the deep darkness.
The masters who understood Seo-ha's proof all wore the same expression.
Every one of them, eyes reddened, was holding back tears. It had been a truly long span of years. To write this single line of proof, countless mathematicians had given their entire lives.
「...Therefore, all Nontrivial Zeros of the Zeta Function lie on the real part 1/2.」
Hilbert, Pólya, Hardy, Selberg, Deligne, Bombieri, Grothendieck, Alain Connes. And the countless geniuses who vanished without even leaving their names behind.
「Q.E.D.」
A solemn feeling came over Seo-ha.
If even one of them had been missing, could today's proof have existed? Ask it a hundred times, a thousand times, and the answer would still be no.
So this was a tribute to them. Seo-ha lifted his chalk and wrote one more line below.
「For Riemann and Hilbert, and for all the giants living and breathing between the lines of this proof.」
Hhk.
The moment the chalk stopped, the sound of someone weeping rose from the lecture hall.
It was Andrew Wiles.
Hair gone white, thick glasses, the stooped back peculiar to a man who had done nothing but mathematics his whole life. The old professor revered the world over was shedding tears. Whitman pulled out a handkerchief and covered his face. The other masters beside him were no different. Their lips trembled, and then their shoulders heaved as they wept.
Sob, sob.
The weeping spread.
The lecture hall, packed with hundreds of mathematicians, had become a sea of tears before anyone realized. No one wept loudly. Mathematicians do not cry that way.
Out of sorrow and gratitude for those who had passed on before they could see the proof; out of the joy that all those years of suffering had finally been repaid; and out of the respect and thanks they held for the young mathematician who had inherited every last piece of their legacy and made it the cornerstone of a new era.
Theo closed his eyes and looked upward. He could not hold back the tears that came bursting out. And in the midst of it all, a single thought rose in him.
'I'm so glad I get to share this moment.'
Clap clap clap!
Applause came from somewhere.
The old scholar rose without even wiping his wet glasses and began to clap. Then Whitman, and then, one by one, the winners of the Three Major Math Prizes seated in the front row rose from their seats.
Clap clap clap!
The applause spread backward like a wave.
When at last all several hundred people in the auditorium were on their feet, the sound grew as fierce as a storm, as if it would burst through the ceiling of the hall.
To Su-jeong's ears, it sounded like the trumpet call of a military band.
It was both a tribute to the genius named Seo-ha and something like a report of victory: that humanity had at last cracked the cipher of God.
Reporter Nam felt bewildered, not knowing how to describe the scene. He didn't understand the details, but his heart was boiling over.
Still, there were a few things even he could understand.
That something very old had come to an end; that a great many people had been waiting for this; and that the young mathematician standing before the blackboard had, in the end, answered their wait.
*****
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