98: Chapter 98 A Spark of Hope

At zero hours Coordinated Universal Time, arXiv released the update list for the day.

For mathematicians around the world, this was a completely ordinary moment.

The massive machine with its server located at Cornell University, like a tireless academic throughput pump, indiscriminately poured the crystallizations of wisdom, heart's blood, or even wild fantasies submitted from all over the globe in the past twenty-four hours onto the internet.

At this time every day, hundreds of new preprints would be pushed online in batches, classified by discipline, and neatly arranged into their respective lists.

Algebraic geometry, number theory, partial differential equations, combinatorics...

Under each category was a long string of titles uploaded that day, the authors, and that string of numbers starting with the year and month.

The vast majority of papers would stay on this list for a day, have their abstracts glanced at by dozens of people around the world, perhaps two or three people would click to download the PDF, and then sink into obscurity.

What awaited them was a peer-review cycle with no foreseeable end, or simply being ignored in the vast ocean of academia.

This was the daily routine of the academic world.

Calm, massive, inconspicuous, yet with undercurrents surging.

Under the overall mathematics category, the number of newly added papers that day exceeded one hundred.

Under math.CO, the combinatorics category, there was also a long string of new titles.

Britain, Warwick.

At one-seven in the morning local time, Marcus Holt had originally closed his computer and prepared to sleep.

He was forty-six years old this year and a mathematics professor at the University of Warwick.

His research direction was discrete geometry; if defined in more precise academic language, what he did was plane tiling and aperiodic substitution systems.

This field was niche and profound, extremely testing a researcher's intuition for geometric topology.

Core researchers around the world who truly staked their lives and fortunes on this direction numbered no more than one hundred and fifty at most.

Most of them knew each other, and even knew the sub-topics each person was tackling at hand.

But glancing at the arXiv update list before bed was Holt's unshakable habit for over a decade.

Most of the time, the update list only featured minor improvements in familiar fields, generalizations of certain problems, or technical patches of certain models.

It did not bring him any surprises.

Holding a freshly brewed cup of chamomile tea, Holt leaned back in the somewhat faded leather armchair in his study, mechanically swiping down on the iPad screen with his right thumb.

Algorithmic boundaries of graph coloring problems on sparse graphs, skip.

Algebraic proofs regarding certain special hypergraph combinatorial identities, skip.

A new type of polyhedral Ramsey number estimation, glance at the abstract, save it to let the PhD students look at it later.

Next.

An Aperiodic Monotile via Local Forcing.

Holt's thumb hovered above the screen, and his first reaction was definitely not excitement.

Instead, what welled up in his heart was an occupational fatigue almost engraved into his DNA, accompanied even by slight irritation.

Because those five words "aperiodic monotile" were a special entity in his circle.

It was like the perpetual motion machine of the Physics world, or the Goldbach's conjecture of the number theory world, carrying a fatal magic of its own, attracting countless moths to a flame.

Precisely because it was famous, and precisely because it appeared so easily understandable in geometric intuition, every once in a while, in Holt's mailbox and on the arXiv list, a warrior would pop up claiming to have solved it.

The vast majority were folk mathematics enthusiasts.

Someone sent hand-drawn strange shapes they thought could tile the plane, accompanied by a letter dozens of pages long, filled with exclamation marks and fanatical emotions.

Someone modified the edge lines of Penrose tiling, changed the color, and announced that they had subverted condensed matter Physics and crystallography.

Someone's proof was riddled throughout with "obviously," "easily seen," and "not hard to think of," yet was riddled with loopholes on the most basic topological definitions.

Holt remembered that a few years ago, a person claiming to be a retired engineer even sent such a manuscript to several people in the circle simultaneously, and it was said that even Penrose himself, in his nineties, received a copy.

As a result, that thing didn't even survive the definition of the Euclidean space isometric transformation group on the first page.

He had seen too many.

So many that he had developed a conditioned reflex; seeing monotile plus aperiodic plus a name with no background, he could basically conclude that this was yet another piece of trash that could be scanned in three seconds and thrown into the trash can forever.

He sighed, picked up his teacup and took a sip, his thumb ready to swipe past.

But he didn't.

Because the last two words in the title, like an extremely fine needle, gently pricked his optic nerve.

Local Forcing

This word was too cold and too specific.

This didn't quite sound like a word that amateur scientists or enthusiasts liked to use.

The titles of papers by amateur scientists were often long and full, carrying an irrepressible excitement for fear that the world wouldn't know their greatness.

For example—《A Revolutionary Discovery of a New Class of Aperiodic Tilings and Its Profound Significance for Crystallography》, and the like.

Whereas this one only had six words.

An Aperiodic Monotile via Local Forcing.

As if possessed by a ghost, Holt gently touched the screen.

The tablet loaded the PDF.

Opening the details page, his gaze habitually swept towards the author and affiliation first.

Jiangcheng No.7 High School, China

Holt's brows immediately furrowed.

High school?

Chinese high school?

Single author?

These informational elements superimposed together, in the context of the mathematical world, were almost equivalent to typing the six words "do not waste time" onto the screen.

In highly specialized modern mathematics, an author whose affiliation was an ordinary Chinese high school single-handedly solved a geometric puzzle that had plagued the academic community for half a century?

The probability infinitely approached zero.

If not for having glanced at the abstract just now, Holt would definitely have closed this page immediately.

The abstract was also very short.

No "revolutionary", no "breakthrough", not a single adjective trying to persuade the reader how important it was.

It simply stated calmly: This paper constructs a simply connected polygon, proves that it can tile the plane, and any such tiling does not possess translational periodicity.

Restrained and precise, it was extremely standard modern mathematical academic language.

Holt flipped down, skipped the introduction directly, and came to the first figure.

Figure 1: Geometric construction of Tile J

A tridecagon appeared on the screen.

Holt's hand holding the teacup instantly hung suspended in mid-air, the tea rippling against the cup wall due to a slight tremor.

He had been doing tiling research for more than twenty years, and had an almost beast-like intuition and sense of smell for various geometric shapes.

The thing before his eyes had an extremely irregular outline, full of abrupt concave and convex corners, like a strange hat that had been crudely flattened and forcefully twisted by someone.

Too ugly.

But what made Holt's pupils slightly contract was that it wasn't the kind of ugliness that an amateur scientist would draw.

The ugliness drawn by amateur scientists was often random, lacking internal logic, and an ugliness of forcibly distorting edges just to piece together a puzzle.

Whereas the ugliness of this tridecagon before him carried a heart-palpitating inevitability.

It didn't look like a graphic drawn based on aesthetics, but rather looked like a solution tightly constrained after a set of extremely rigorous mathematical equations mutually crushed and compromised each other.

Every one of its corners seemed stuck at a specific angle.

Every one of its edges seemed to have an indescribable correspondence with other edges.

This indicated that the underlying layer of this shape was embedded in a very classic hexagonal/triangular grid.

Holt instinctively put down the teacup, sat up straight, zoomed in on the screen with two fingers, and carefully scrutinized that seemingly abrupt concave edge.

"Something's wrong."

He muttered under his breath, clicked open the PDF sidebar, and jumped to the third section.

Two lines of English were printed all alone at the beginning of the first page.

This section does not prove uniqueness of individual tile assignments.

It proves recognizability of the hierarchical skeleton under finite boundary ambiguity.

Holt stared at these two lines of English for a full minute.

The previous fatigue and casualness were swept away, replaced by the tremor of a hunter seeing the footprints of an unknown beast.

This sentence was too professional.

One could even say it knew the fatal weakness and pain point of the Einstein problem too well.

In the countless failed monotile candidate proofs over the past few decades, ninety-nine percent of people died in one place.

They naively thought that as long as they proved that every bottom-layer brick could be uniquely classified into a certain upper-layer large block, the proof would be complete.

However, on the infinitely extending plane, due to aperiodicity, there would always be certain ambiguous stitching methods on the boundary.

What truly needed to be proven was not that every bottom-layer brick could be absolutely uniquely classified into the upper-layer large block, but whether the hierarchical skeleton could be stably read out by local rules beyond finite boundary ambiguity.

This guy from a Chinese high school not only knew about this trap, but even actively pointed the blade right at it.

Pushing his already cooled herbal tea to the edge of the desk, Holt reopened the PDF sidebar, took a deep breath, and jumped straight to the most core and most easily crashing chapter of the entire proof.

04_no_periodic_strip.pdf

If this manuscript was some high-level prank, or contained some hidden fatal error, ninety-nine percent of it would give itself away here.

Holt first looked at the reading convention.

Then looked at the flip handling.

Then looked at the short repeating segments.

Soon, he saw a table.

S-3, S-4, S-5, S-6, S-7.

These were boundary segments that, locally speaking, seemed likely to undergo periodic extension along a certain axis.

The author didn't hide these dangerous local combinations at all, nor did they brush them off with a sentence that was easy to prove false.

Instead, like a cold surgeon, he cut open these tumors that could lead to theoretical collapse one by one and numbered them individually.

He listed the initial forms of these segments, the flip-reading constraints, and most importantly, he deduced the splitting situation of these segments after performing the third substitution.

Holt's finger stopped on the screen, and he sighed softly: "Interesting..."

Continuing to flip to the appendix.

An hour later, Holt leaned back against the chair back, only feeling an indescribable current rush from his tailbone to the back of his head.

He looked at the time in the lower right corner of his computer: 1:45 AM.

Pondering for a moment, he opened his email client and created a new email.

In the recipient field, he cautiously typed the first name: Craig Kaplan.

Pausing for a moment.

That bunch at the University of Waterloo were top experts in computational geometry, tiling, and visualization.

For such complex constructions based on substitution rules, Kaplan's team possessed the sharpest code intuition in the world.

If there were low-level errors, they could write a script to run it and find out.

Thus, he entered the second name: Chaim Goodman-Strauss.

The geometer at the University of Arkansas.

If there were any topological hallucinations in the hierarchical substitution structure of this manuscript, an old fox of that caliber would almost certainly have noticed it.

He originally wanted to add a few more names.

After thinking about it, he deleted half of them.

The discrete geometry circle is too small.

It is so small that once an email containing hints of a potential solution to the Einstein problem spreads, within half a day it could reach everyone globally who genuinely cares about the issue, and even attract unwarranted harassment from the media.

Before truly seeing clearly and without doing even a single basic review, Marcus Holt did not want to create unnecessary academic noise.

In the end, his email subject line was extremely short, without any punctuation.

"You need to look at this."

The body text was even more concise to the extreme.

"I am not saying it is correct."

"But it is not the usual nonsense. Look at Section 4, specifically the treatment of the S-5 ambiguity."

After clicking send, Marcus Holt did not go to sleep; instead, he returned to the first page of the PDF, picked up his stylus, and heavily drew a circle next to that ugly thirteen-gon.

This time, he began to read carefully, word by word, with the rigorous gaze of scrutinizing mathematical truth.

Outside the study window was the thick darkness of the British late night.

But within the neurons of the academic internet, a massive signal had already begun to transmit rapidly along the fiber optic cables.

A few hours later, Waterloo, Canada.

Craig Kaplan had just woken up.

As an expert long immersed in Penrose tiling, Islamic geometric patterns, and computer substitution systems, the first thing he did every day was brew coffee and check his inbox.

When he saw the email sent in the middle of the night by Marcus Holt, the coffee pot was making a gurgling boiling sound.

"You need to look at this."

Seeing this subject line, Kaplan raised an eyebrow.

He knew Marcus Holt too well; that typical old-school British scholar would never casually use such an imperative, urgent tone in an email.

Holding his coffee, he clicked on the arXiv link in the email.

Looking at the title, looking at the abstract, looking at the shape in Figure 1.

Then, his gaze fell on the author affiliation: Jiangcheng No.7 High School.

Kaplan stared at the screen in a daze for a full ten seconds, then burst out laughing with a snort.

"Was Marcus drunk last night?"

He instinctively wanted to close the page, but out of respect for Marcus Holt's academic reputation, he held back.

He scrolled the page down to Section 4, which Marcus Holt specifically pointed out in his email.

A few minutes later, the smile on Kaplan's face completely vanished.

He put down his mug and immediately woke up the two high-performance workstations next to him.

...

At the same time, Berlin, Germany.

In a computational geometry and combinatorial algorithms laboratory.

Young postdoctoral researcher Klaus was running an SMT solver.

He did not look at the proof text.

He had been @-mentioned in a group chat by his fellow senior lab mate.

"Use the De Bruijn grid method to test the local extensibility of this tile." Klaus bit into a cold sandwich while typing on the keyboard.

What he needed to do was have the supercomputer determine whether, given a disk region of radius R composed of Tile J, there must necessarily exist a way to extend it to infinity while forcibly breaking translational periodicity during the expansion process.

This was the ultimate computational touchstone of local forcing theory.

The terminal window on the screen was scrolling data frantically.

The state space exploded exponentially, but Klaus had written an excellent pruning algorithm to remove those branches that had already been proven to be dead ends in the paper's S table.

Half an hour later, the calculation finished.

The terminal output green characters: "NO PERIODIC PATCHES FOUND UP TO R = 10."

"SURVIVING BRANCHES MATCH SUBSTITUTION DATA."

The sandwich in Klaus's hand dropped onto the desk.

He posted a screenshot in that private little group chat consisting of only seven or eight top algorithm researchers.

Attached below were a few sentences.

"I expected nonsense."

"This is not nonsense."

"At least the local forcing table is not collapsing."

"This deserves a real proof check."

...

Europe, North America, Japan...

During these seemingly calm few hours, the flame began to spread rapidly in the extremely hidden depths of the academic circle, tracing along a few of the top brains.

No one was shouting on social media; those who truly knew the ropes were silently downloading the PDF, silently writing code to verify, and silently deducing the topological dead ends of the S-5 table on scratch paper.

Two in the afternoon.

On a hardcore academic forum named Tiling & Tessellation, someone finally could not help posting the image of Tile J.

The post title was extremely cautious: "Another Einstein claim? (Tile J, on arXiv today)"

The first dozen replies were filled with the sarcasm and mockery common in academia.

"High school? Seriously?"

"Wake me up when someone verifies the substitution system."

"The outline is too casual; it does not look like something with deep algebraic properties."

However, after just forty minutes, the wind direction changed.

An account with a University of Waterloo IP (many guessed it was someone from Kaplan's lab) replied.

"Wait, this substitution structure looks real; I just used their coordinates to regenerate the first four layers of super-large blocks without any overlap."

After a while, a scholar working at CNRS in France posted a matrix screenshot of the four categories of super tile blocks in the paper.

He accompanied it with the caption:

"The local rules are extremely ugly. It is constrained to an unbelievable degree. This is usually a good sign. Fake proofs often look too perfect and too symmetric. But this one looks like reality."

The once-noisy sarcasm and mockery in the forum began to quiet down rapidly.

Those who understood a bit had already shut up to chew on the main document, supplementary proofs, and code verification package.

Those who did not understand were still staring at that bizarre thirteen-gon pattern, arguing about what on earth it looked like.

Some said it looked like a stepped-on kite.

Some said it looked like a broken arrow.

Some said it looked like a crab that had been severely tortured by mathematicians.

The first batch of nicknames for Tile J was born haphazardly amidst this atmosphere of both astonishment and absurdity.

Jiangcheng University, School of Mathematical Sciences, Room C216.

Three in the afternoon.

Professor Gu Nanzhou's email began to pop up new email notifications continuously, like an air raid siren being sounded.

As one of the leading figures of the School of Mathematics at Jiangcheng University, he clearly knew what everything happening right now meant.

The first came from Rutgers University in the United States.

The tone was polite, but the questions were as sharp as a scalpel.

"Professor Gu, is the author Jiang Lin really a high school teacher?"

"Has this construction been independently verified by your team, and are there implicit assumptions in Lemma 3.2?"

The second came from the Collège de France in Europe.

It was sent by an eccentric yet highly authoritative old geometer from the Bourbaki school.

His email was shorter than the code.

"Are there supplementary codes? We want to reproduce the local state table. I suspect the completeness of the S-4 case."

The third came from the Research Institute for Mathematical Sciences at Kyoto University in Japan.

The sender asked extremely detailed questions, clearly having devoured the paper inside and out.

"In Section 4, the short repetitions of S-5 seem to rely on specific reading conventions. If chiral reflection is applied to the right boundary, does topological homeomorphism still hold, and is there a detailed elaboration?"

The fourth came from an unfamiliar Gmail address without any institutional suffix, containing only a dry sentence.

"Please confirm this is not a prank."

...

Two o'clock in the afternoon London time, ten o'clock in the evening Beijing time.

On Marcus Holt's personal Twitter account, which had over twenty thousand academic followers, an extremely brief update appeared.

"I am not yet ready to endorse the complete topological proof, but the Tile J preprint deserves serious and rigorous attention, and its substitution rules are computationally robust."

There were no exclamation marks, no exaggerated meme images, no emotional adjectives, and it did not even mention the conjecture that it might have solved the Einstein problem.

But for those who knew the ropes, the weight of this sentence was heavier than Mount Tai.

Because it was an extremely clear signal.

It meant: the bigshots in academia had personally stepped into the field to run the data.

This paper from a Chinese high school had officially broken away from the category of amateur crackpot nonsense, obtaining the admission ticket to enter the sacred halls of academia for final judgment.

Everyone could start spending time pondering it.

Half an hour later, a screenshot of this tweet was forwarded into several core pure mathematics WeChat groups in China.

After another dozen minutes, an anonymous question appeared on Zhihu and was quickly tagged on the trending list.

"Question: How to evaluate Marcus Holt saying that the newly launched Tile J paper on arXiv deserves serious treatment?"

The first top-voted reply underneath carried an irrepressible sense of doubt and astonishment.

"Thanks for the invite, just saw the screenshot in the math group, and my whole mind is blank. Looked up the author's affiliation: Jiangcheng No.7 High School. Is this real or fake? Isn't this a prank by some bigshot with the same name?"

The second reply was more professional and cautious.

"Do not rush to hype it up. Professor Marcus Holt's words are very rigorous; he only said the calculation is robust, not that the topological proof is correct. If it really is an Einstein, then it is not just big news, but the most conclusive achievement in the direction of aperiodic tiling, something that can be directly written into the textbooks of the history of mathematics. If it is fake news or marketing hype, the backlash could ruin a person. So my suggestion is to wait for peer review first."

But no matter how cautious everyone was, the flame had unstoppably begun to spread from the edges of the academic circle into the public eye.