Chapter 258

Translator: Dreamscribe

「Step 1: The density of branch points in a continuous topological space cannot be measured countably.」

At first glance, the proposition seemed vague and abstract. But the meaning contained within it ran deep.

“Hmm.”

Professor Ramanathan, an authority in computer science, sat in his chair and brought a hand to his chin. He gazed at Seo-ha, sinking deep into thought.

‘Are you trying to deny computer science itself?’

Modern engineering had been built upon the proposition that ‘every computational process in nature can be simulated by a Turing Machine’. Yet Seo-ha was saying that nature was never truly discrete. That something which could neither be enumerated nor integrated existed at the very root of computation.

If that were proven, what would change?

The fundamental limits of computers and computer science.

Even if artificial intelligence advanced and quantum computers became practical, even if a machine with infinite computational resources were to exist, engineers would have to acknowledge that there were realms it could never reach.

That was the grand premise from which his 13-Step Hypothesis began.

“What do you think, Professor?”

Ramanathan asked Leonard, the dean of the physics department sitting beside him.

“Well. Let’s watch a little longer.”

His eyes were wide as he stared at the blackboard.

Ever since the 13-Step Hypothesis had been announced, Leonard had felt uneasy in the back of his mind. Seo-ha’s hypothesis conflicted with quantum mechanics, the mainstream framework of physics.

Quantum mechanics stated that energy and matter were observed in discrete levels. At least within the range humanity could measure, the universe existed in the form of particles separated into distinct units.

Physicists had long taken comfort in that fact. If something was discrete, it could be counted, and if it could be counted, then someday everything might be accounted for. That was why many scholars believed that, given sufficient computational resources in the distant future, the universe could be simulated with one hundred percent fidelity.

‘If that is true....’

Physics would lose its center of gravity and be thrown into tremendous confusion.

Cold sweat ran down his back.

Whitman, sitting in a corner at the rear of the lecture hall, wore a satisfied expression.

Even the great Newton and Einstein had attempted to explain the world only within the constraints of reality. That must have been a compromise. They had possessed no way to describe the world beyond those limits.

But Seo-ha did not accept limits. He simply accepted the raw universe he had seen exactly as it was.

Cronen gave a hollow laugh and said,

“There is something I came to realize painfully while solving the Yang-Mills Hypothesis. Our current mathematical system cannot explain the interactions of this vast universe.”

Cronen’s gray eyes followed the trail of equations written across the blackboard.

“But when I assumed Seo-ha’s hypothesis was correct, the hopelessly tangled threads of Yang-Mills unraveled with astonishing smoothness.”

It was still too early to celebrate.

Whitman nodded with a grave expression.

***

Seo-ha adjusted his grip on the chalk.

The world he had seen only through intuition.

Turning it into language, then into symbols, had never been easy. It was the agony of trying to contain something perfect within an imperfect vessel.

His yearning for a world beyond the Turing Machine had been an inevitable consequence.

‘Humanity’s digital civilization is nothing more than a tower of approximations.’

Then did the truth have to remain a shadow forever?

Countless unsolved problems, including the Yang-Mills Hypothesis, had become trapped at this point, unable to advance even a single step.

No matter how faithfully one imitated a shadow, one could never reach the essence itself.

Seo-ha could no longer tolerate that.

‘Let it all out. Everything, onto this blackboard.’

He did have concerns.

It was impossible even to predict how society might change in the distant future. But that could never be grounds for turning away from the truth.

scratch-scratch.

The Cantor Set, Sigma Algebra. Structures humanity had trusted for the past hundred years.

Seo-ha took direct aim at that seemingly absolute proposition.

‘The branch points that emerge at the intersection of Number Theory and geometry can never be confined within a countable sum.’

Irregular curves appeared before Seo-ha’s eyes. It was a violent and intricate structure, like a storm raging across the sea.

His gaze shifted toward their points of intersection.

The place left behind after the storm had passed.

The patterns left in the sand along the shore were different every time, and no two points traced the same curvature. Yet there was unmistakably an order within them.

‘I probably can’t prove this yet.’

Seo-ha clenched his fist.

The day all thirteen steps of the hypothesis were proven and the secrets of Complex Systems were unraveled. Until then, he would simply have to be patient.

But this alone was enough to explain the contradictions within modern mathematics.

The culmination of the proof.

At the center of the blackboard, he wrote the core formula of Measure Theory, one of the foundations of modern analysis. It was the standard axiom of Countable Additivity.

「All complexity is the sum of sufficiently small pieces.」

‘I’ll attack this part.’

Until now, humanity had believed that infinitely divided fragments could be arranged in sequence and added together one by one.

It was the world of countable infinity permitted by Cantor, and the foundation that allowed a Turing Machine to continue computing forever along its tape.

scratch-scratch.

Seo-ha began writing a new equation beside that axiom. At once, the lemmas he had proven so far seemed to swirl together and converge upon it.

Seo-ha’s hand raced back and forth across the blackboard.

‘Ah! So that was it!’

Whitman’s eyes shook. Without realizing it, he leaned forward.

‘He wasn’t trying to deny Countable Additivity.’

Looking beside him, he saw Cronen completely absorbed. His gray eyes relentlessly followed the tip of Seo-ha’s chalk.

The branching structures that appeared in Number Theory, the singular points of intersection found in geometry, the irreversible flows discovered in Complex Systems.

Until now, the results had seemed entirely unrelated. Yet Seo-ha was gathering them together and drawing them into the world of Yu-Riemann Geometry.

A quiet exclamation escaped Whitman’s lips.

“He was looking at all of them as a single thing from the very beginning.”

‘The reason Measure Theory established Countable Additivity as an axiom is obvious. It makes integration possible.’

But Seo-ha was telling all of humanity something here. Nature and the universe did not exist for the convenience of human calculation.

He attempted to divide space into countable pieces and confine it within them. It was a feat possible only within Yu-Riemann Geometry. But the two sides could not withstand it and diverged.

Cronen could see it.

The discrete fragments of the branch points losing equilibrium and scattering in every direction.

“Ah....”

It meant that countable infinity and uncountable infinity would have to be the same size for the space to be completed without contradiction.

A topological contradiction that was mathematically impossible.

With an exhausted hand, Seo-ha carefully placed the final period at the bottom of the blackboard.

「...Therefore, the density of branch points in a continuous topological space cannot be measured countably.」

「Q.E.D.」

It was the moment when the fortress humanity had trusted for the past hundred years could no longer withstand the contradiction it contained and collapsed before a single equation.

As the tension left his fingertips, the piece of chalk he had been holding slipped from his hand and fell to the floor with a soft tap.

Stagger-

His body looked ready to collapse.

Just as Theo, standing below the podium, was about to rush forward, Cronen reached Seo-ha first and caught him by the shoulder. Looking Seo-ha directly in the eyes, he said,

“Pay your respects.”

Seo-ha slowly turned toward the audience. At some point, the seats had filled with countless people. Question marks floated through their minds.

‘A bow? This isn’t even a formal presentation.’

‘What does he mean?’

But Theo understood.

And so did giants of academia like Whitman, Leonard, and Ramanathan.

Cantor’s infinity, Henri Lebesgue’s Measure Theory, Andrei Kolmogorov’s axiomatic probability theory, and the Turing Machine.

These were the theories he had brought down. And before him sat the modern scholars who had spent their lives developing them further. Seo-ha bowed deeply toward them.

They were the crystallized blood and sweat mathematicians had left behind in their struggle to grasp even a fragment of the universe’s order from within the darkness. It was because of that legacy that they could stand here today.

It was both a farewell announcing the end of an era and a jubilant declaration that a new continent had been discovered. Cronen spoke quietly.

“Congratulations. Humanity has finally gained a milestone pointing toward the next era.”

Clap, clap, clap!

Applause began from one side of the lecture hall.

It was Professor Leonard.

Once the turmoil in his heart had settled, he could see the future. Quantum mechanics was filled with unanswered questions. Countless experiments said it was correct, yet no one had ever managed to explain why the world behaved that way.

Seo-ha’s proof offered hope that they might finally see what lay beyond it.

Before he knew it, Leonard’s anxiety had transformed into a desire to join this great journey.

Clap, clap, clap!

Professor Ramanathan felt the same.

The applause gradually spread. The undergraduates joined in without even understanding exactly what had happened, but they could vaguely tell from the atmosphere.

They were witnessing a moment in history.

The applause swelled throughout the room.

It was only the first step of the 13-Step Hypothesis.

But the repercussions of this proof would be anything but small.

***

The proof written on the blackboard was quickly refined into the form of a paper at Team Apex’s lab and uploaded to arXiv.

“They say Yu Seo-ha’s new paper is out?”

The clinical effectiveness of the Life Matrix had already been demonstrated in medical settings. Countless scholars had been waiting for Seo-ha’s next paper.

Stanford, Cambridge, Oxford, the University of Tokyo, Tsinghua University, KAIST, and Seoul National University.

Researchers at leading universities around the world began dissecting his paper all at once. They gathered in small groups and exchanged their impressions.

“This is way too difficult.”

“Ugh! I spent all day yesterday reading it, and it was still hard.”

“Is he rejecting Turing’s Halting Problem? I thought that had already been proven.”

“He must have changed the premises and the scope. And the paper mathematically proves that. Judging from what Professor Cronen posted on social media, I think that’s right.”

Click.

With a click of the mouse, Professor Cronen’s social media page appeared on the screen.

-Let us celebrate! Today, one of the walls that has held humanity back has fallen.

We have built the systems of modern mathematics and computer science upon the absolute foundation of the Turing Machine. But Seo-ha has proven that this foundation falls into contradiction under certain topologies within Yu-Riemann Geometry.

Do not misunderstand. This does not mean that everything you have learned until now was meaningless.

Neither Turing nor Lebesgue was wrong.

Seo-ha has merely revealed that their theorems are valid only within a far more limited domain than we had believed.

“....”

“We have to admit there’s something to it. The survival rate of pancreatic cancer patients actually went up, after all. At the very least, he really has presented a new methodology for handling Complex Systems.”

Their tone was calm, but the atmosphere in the laboratory was strangely excited.

The dark age of science.

Technology visibly advanced every year, yet there were no theories capable of properly explaining it.

Artificial intelligence discovered patterns humans could not understand, and new materials displayed properties that had never been designed into them.

The same was true in practical engineering. Engineers could not clearly explain, in theoretical terms, why certain algorithms performed far beyond expectations.

It was an age in which phenomena were discovered first and their causes sought afterward. The world was changing rapidly, yet no one knew why.

Modern science had endured this contradiction for far too long.

At last, they had seen the possibility that this stagnation could be broken.

***

A few weeks later.

The paper was approved by the editors at Science without a single recommendation for revision. Then, the very next week, journals bearing the same cover appeared in bookstores and on newsstands around the world.

Science magazine very rarely used photographs of people on its cover. Traditionally, it favored graphics visualizing research results or abstract scientific illustrations.

But this time was different.

Seo-ha’s back as he stood before a blackboard.

The cover design created the illusion that he stood there with a single piece of chalk in hand, gazing out over a vast sea of equations.

The title was simple.

[Beyond Turing]

With every paper he published, Seo-ha continued the remarkable feat of gracing the covers of major journals.

*****

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