The Spire of Abstraction

Tokyo, Japan. The National Astronomical Observatory of Japan, First Research Lab.

The ‍‍‌‌⁠​​⁠‌​​‍​​‍​‌​⁠‍‌‍⁠⁠‌⁠‌‍‌‍‌‌‌‍⁠​‌​‌​‌‍‌‌‌⁠‍​​‌‍⁠‌‌⁠⁠silence, which was usually filled only by the low hum of running computers, was shattered to pieces.

It started with an urgent phone call from Watanabe Hiroshi, the chief professor of mathematics at Tokyo University.

—Sato, you're in your lab, right? Open arXiv right now! Not the astronomy section—there's a paper cross-listed in theoretical physics and mathematics that's causing an absolute uproar! No, wait. Just go to MathOverflow! Every single page is talking about it.

Professor Sato Kenichi pushed up his reading glasses on the bridge of his nose and frowned.

He was the godfather of observational astronomy, a man who despised pure theory or abstract equations as mere armchair philosophy. To him, who believed only in vivid, high-resolution images and raw data, the noisy clamor of mathematicians was a distant affair.

"Ha, what on earth is it, Watanabe? We're busy enough just filtering data here."

—That data is the problem! Reference number one in that paper! The paper you published in Nature last year is plastered right there!

Sato's eyebrow twitched.

And the corner of his thin lips curled up slightly.

His paper was reference number one?

And in a paper that was the talk of the mathematical community?

That was quite enticing to hear.

It meant his research had influenced the mathematical world to that extent.

Sato Kenichi moved his mouse and entered the mathematics community forum, a place he usually wouldn't even glance at.

He easily found the thread about the sensational paper.

Hmm... Seohan University?

Ironically, the astronomy research team that had been constantly nitpicking his papers was also from Seohan University.

And yet, a mathematician ‍‍‌‌⁠​​⁠‌​​‍​​‍​‌​⁠‍‌‍⁠⁠‌⁠‌‍‌‍‌‌‌‍⁠​‌​‌​‌‍‌‌‌⁠‍​​‌‍⁠‌‌⁠⁠from that very university had cited his paper.

As Sato scanned the contents, his brow furrowed deeper.

"...Computational limitation of a machine?"

Sato Kenichi's face flushed red.

The simulation they had spent their entire lives building, utilizing the finest equipment and algorithms, was being dismissed by some mathematicians scribbling equations on scraps of paper as a mere calculator error. They were coming at him in broad daylight.

—Are you looking at the paper?

"Ha... arrogant bastards..."

—No! No! You can't think like that right now!

"What nonsense are you talking about? Surely this paper isn't..."

Sato Kenichi trailed off as a cold shiver ran down his spine.

Indeed, if it were just a paper spouting garbage claims, it wouldn't have caused such a stir.

His friend wouldn't have called him in such a panic, either.

—Because it's such a closed theory, the math community might not accept the paper's citations, but the logic of the equations is flawless.

"Wh-what? Flawless?"

—In other words, your data is wrong!

"Phew, that's impossible. How..."

Sensing the grim atmosphere, the researchers in the lab cast anxious glances at Sato Kenichi.

He was usually very quiet, but once he lost his temper, all hell broke loose.

And right now, he was on the verge of losing it.

—It's already spread too far. It's too late to pull strings in the math community to overturn it. Fortunately, the math community is hesitant to recognize this paper, but Edward Kang just made things worse. He proposed an online open debate to the math community. It's tomorrow.

"..."

—For now, looking at the atmosphere in the math community, they can't touch those equations. There are absolutely no holes. So they'll probably ‍‍‌‌⁠​​⁠‌​​‍​​‍​‌​⁠‍‌‍⁠⁠‌⁠‌‍‌‍‌‌‌‍⁠​‌​‌​‌‍‌‌‌⁠‍​​‌‍⁠‌‌⁠⁠debate the fundamental principles. There's no stopping your paper from being refuted.

Sato Kenichi didn't care about the fundamental principles of mathematics.

What he needed most was a flaw in this paper.

"Phew... Look, Hiroshi. Is there no way?"

—Your only option is to dig through the Subaru Level 0 data yourself and find something. Though time is short.

"..."

—Or, you'll have to find some fault during the open debate tomorrow.

"Ha..."

Sato Kenichi dropped his phone.

"What the hell is this!"

Professor Sato Kenichi, the chief research director, screamed in a fit of rage.

—So, you're finally stepping up.

Renoir chuckled at Edward Kang's words, which sounded as if he had expected this.

"Well, isn't this what you wanted?"

—Exactly. I wanted you to have a conversation with this mathematician.

"Hmm. Is he a disciple you're raising yourself?"

—No, he's already complete.

"Ha... There was someone like that in Korea? I haven't seen anyone of that caliber in the IMO over the last few years to warrant you saying that."

—Haha. That's right. He's never participated in the IMO.

Renoir's eyebrow twitched.

A mathematician who only began to shine late in life?

—Anyway, brace yourself. You're in for quite a shock.

"..."

—Well then, see you on video tomorrow.

"...Yeah."

Renoir let out a soft sigh.

Hearing his old friend's voice made his mind even more complicated.

Has he met an incredible successor...

That must be why his voice sounded so excited.

Renoir pressed his thumbs against his temples, thinking of his own researchers.

Ha, I'm jealous...

He still didn't have anyone who completely satisfied him...

At best, Jacques was barely keeping up.

By the way, in for a shock?

It was just a video debate between mathematicians. What could possibly be so shocking...

And the next day.

Connecting to the video conference, Renoir found out the reason.

"Hello! I'm Lee Cheol-jun, fifth grade, class five, Han-young Elementary School!"

On the screen, a very small child smiled brightly and greeted them in English.

Renoir's mouth fell wide open.

The moment the screen of Gather Town, a video conferencing platform created during the pandemic, turned on, countless mathematicians staring at their monitors blinked in stunned silence, their mouths agape.

Even the massive online communities, which had been flooded with comments and chats before the screen even turned on, froze completely the moment the tiny child appeared on screen.

—Oh, do I just look over there and say hello?

The small child asked Edward Kang. When Edward Kang smiled warmly and nodded, the child looked at the camera, smiled brightly, and bowed his head.

—Hello! I'm Lee Cheol-jun, fifth grade, class five, Han-young Elementary School!

It was an English greeting.

Everyone heard it loud and clear.

The name of the first author of that paper.

He wasn't just a guest appearing briefly for the opening.

In an instant, the frozen comment sections and chat rooms exploded.

A chaotic tangle of numerous languages poured out.

Most of the people watching this screen were mathematicians.

Therefore, they knew exactly what kind of person Edward Kang was.

He was absolutely not someone who would play such an absurd joke with mathematics.

Thus, there were far more reactions of sheer astonishment than skepticism.

The reactions, which had been close to panic, gradually regained rationality and soon began to shift toward interest in the tiny child on the screen.

The format of this online academic conference was set up so that Cheol-jun would explain his paper, and once the explanation was over, only the mathematicians in the video conference room—who had clearly verified their identities—would take turns asking questions.

Renoir was already in the meeting room.

For now, only Renoir's face was visible, but Edward Kang was chuckling to himself, imagining that the reactions of mathematicians all over the world wouldn't be much different.

Turning his head, he saw Cheol-jun hesitate just as he was about to write something on the whiteboard.

Edward Kang tilted his head and looked at Cheol-jun.

Was he nervous?

According to the presentation order, he was supposed to start by expanding the Fourier series.

With that thought, he peeked at Cheol-jun's face, only to see Cheol-jun twitching one eyebrow with a mischievous expression.

Suddenly, Cheol-jun turned around to face the camera.

A highly meaningful smile was plastered all over his face.

Sensing that he was about to speak, Edward Kang leaned in, ready to translate.

Cheol-jun opened his mouth.

"So, I'm sure you've all had a chance to look at the paper I wrote with the teachers from Seohan University."

Edward Kang's eyes went wide.

...Wait, what?

His English was remarkably fluent.

In fact, regarding language, there were automatic translation subtitle programs, and he himself could translate, so he hadn't worried much.

Furthermore, he hadn't expected anything beyond a simple English greeting from an eleven-year-old, so he hadn't even asked about his language skills.

That was why he had always provided translated versions of books and papers.

And yet,

He was this good at English?

No wonder his pronunciation during the greeting had been unusual.

Edward Kang was just as shocked as the mathematicians watching the screen.

Cheol-jun spoke again.

"Anyone who has taken even a brief look at the paper will know that no errors have been found in it."

Edward Kang shook his head.

His speaking style was quite something.

He had just made even those who hadn't personally verified the paper nod in agreement.

"So, rather than explaining that, I'd like to talk about something else."

Cheol-jun smiled brightly at the camera.

"Well, first of all, I haven't been learning math for very long, so I'm not that good at it. I still can't properly see the things that Professor Edward Kang here takes for granted."

This was a statement that would shock the countless mathematicians watching the screen.

His point of comparison was a Fields Medalist.

By Cheol-jun's logic, everyone watching this broadcast was in the same boat.

"So I'm not here to stubbornly insist that I'm right. I'm not trying to beat anyone, either."

Cheol-jun used a dry-erase marker to write Euler's formula in large letters in the center of the whiteboard.

Below it, he wrote out one of the most notoriously, monstrously long sigma sequences from his paper.

"This Euler's formula is truly wonderful. They call it art that summarizes the universe in a single line. I only found out about it recently. To be honest, because I didn't know about it until not too long ago, I ended up with an incredibly long equation like the one below."

Cheol-jun scratched his head as he compared the two equations.

Placing them side by side like this, they really were completely different.

It was a difference he hadn't realized before.

But that didn't mean the path he had found wasn't beautiful.

It wasn't a matter of right or wrong, but merely a difference in perspective.

Of course, these were equations meant for extreme comparison and couldn't be connected directly.

Cheol-jun wiped both equations clean with an eraser.

Then, he wrote another equation in the middle of the whiteboard.

ind(D) = ∫_M ch(E) · td(TM)

"Um, so... I discovered this index theorem not too long ago."

Cheol-jun blinked his round eyes, then gave up trying to explain it in words.

This was something he just had to write out and show them.

If I can convey this properly, everyone will find it incredibly fun.

The corners of Cheol-jun's mouth curled up slightly, and the hand holding the marker began to dance across the whiteboard.

And Edward Kang, who had been sitting in his chair, stood up abruptly, his eyes wide with shock.

"The Atiyah-Singer Index Theorem..."

Renoir, staring at the screen, muttered blankly.

The researchers beside him also gulped.

Jacques, one of the researchers, glanced at his tablet screen.

The community's reactions were updating in real-time.

It was a community page where only those who knew a thing or two about math gathered.

Even that space was in utter chaos.

Jacques quickly turned his head back to the monitor.

The equations he had been struggling with for days were being listed one after another, centering around the fundamental formula of the Atiyah-Singer Index Theorem.

The Atiyah-Singer Index Theorem is written in an extremely abstract language.

One can learn ‍‍‌‌⁠​​⁠‌​​‍​​‍​‌​⁠‍‌‍⁠⁠‌⁠‌‍‌‍‌‌‌‍⁠​‌​‌​‌‍‌‌‌⁠‍​​‌‍⁠‌‌⁠⁠the formula itself and follow the proof.

Of course, one can even use it on an exam.

However, when faced with a practical problem that needs to be solved, those who can actually pull it out and apply it are exceedingly rare.

Because to actually apply it, one must be able to perfectly translate the problem at hand into the language of this theorem.

In other words, one had to possess the intuition to see exactly how the problem they were trying to solve connected to a specific part of the index theorem.

Without that intuition, it was a theorem that could never be pulled out and used, even if one knew it.

And yet, that child was currently applying that notoriously difficult index theorem to his own equations.

The very theorem that his revered Professor Renoir had praised endlessly, calling it the highest spire in the temple of abstraction.

This is insane...

To bind the messiest, microscopic chaos and the most elegant, clean macroscopic order into one...

Jacques gulped.

For a long time, there was only the sight of the child silently filling the whiteboard, but to those who could recognize how wondrous the unfolding equations were, it felt like watching the creation of a new world.

The tiny creator wrote the final equation at the very edge of the whiteboard.

ind(D_H) = ⟨c₁(L_ω), [T²]⟩ ≠ 0

Then, after slowly and meticulously reviewing the equations he had written, he turned back to the screen.

—Hehe. That took much longer than I thought. Thank you for waiting!

The incredibly cute child smiled brightly and scratched his head.

"...Mon Dieu."

Someone muttered the very words that had instinctively surfaced in Jacques's mind.

Turning his head, he saw it was Professor Renoir.