92: Chapter 92 Measuring Mountains with Vernier Calipers

"Classmate, which department are you from?" As soon as those words left his mouth, the gaze of everyone in C216 landed on Jiang Lin.

Jiang Lin froze for a moment and replied, "I'm not from the School of Mathematics."

"Then who is your advisor?" Professor Gu Nanzhou blurted out almost instantly, his voice half an octave higher than usual, carrying a hint of eagerness.

"Professor Lu Zhixing," Jiang Lin mused, "School of Physics."

Professor Gu Nanzhou's brows twitched slightly, as if he were searching for this name within the massive database in his mind.

He had heard this name before.

Over at the Physics building, there was a young PI who specialized in high-stability measurements. Not long ago, his research group seemed to have produced a moderately sized achievement that had circulated for a while within the Science Faculty building.

Whether it was optical path ultra-low frequency drift suppression or some kind of quantum precision measurement, he couldn't remember clearly.

His name was Professor Lu Zhixing.

But that was the extent of it.

An associate professor from the School of Mathematics working on discrete geometry and aperiodic tiling, and a young researcher from the School of Physics doing precision measurement experiments—they normally had nothing to do with each other.

Being in the same university and in two buildings separated by an artificial lawn and a few rows of roadside trees, they had never spoken a single word to each other.

Within the vast, precisely operating machine of academia, they belonged to completely different gear assemblies, and even their lubricating oil wasn't interchangeable.

Professor Gu Nanzhou didn't even have his phone number.

"Can you get in touch with Professor Lu right now?" Professor Gu Nanzhou looked at Jiang Lin.

Jiang Lin understood what he meant.

He fished out his phone, flipped to Professor Lu Zhixing's WeChat, and without sending any text, directly dialed a voice call.

Three rings sounded in the earpiece.

"Hello, Jiang Lin?"

The call connected, and Professor Lu Zhixing's voice carried a hint of clear surprise, with the faint, low roar of a vacuum pump coming from the background noise.

This student rarely took the initiative to call him.

In Professor Lu Zhixing's impression, Jiang Lin was a person whose efficiency was almost mechanical. When he had matters to discuss, he usually finished them in a few sentences on WeChat, and even his punctuation marks exuded a precise restraint.

"Teacher Lu, I'm over at Building C of the School of Mathematics, in the C216 discussion room," Jiang Lin's voice lacked a single ripple. "There's something going on, and I might need to trouble you to come over. If it's convenient, the sooner the better."

The other end of the phone fell silent for a second.

Professor Lu Zhixing believed he understood this student fairly well.

Ever since solving the interferometer low-frequency drift last month, he knew that Jiang Lin was someone whose choice of words always carried considerations of uncertainty, always keeping his words within bounds, never crossing the line, and never exaggerating.

For such a person to let slip the four words "the sooner the better" was in itself a highly unusual signal.

It was probably even more urgent than an ordinary student shouting that the laboratory was on fire.

"What is it?"

Professor Lu Zhixing's voice immediately sank, revealing an advisor's instinctive vigilance.

Jiang Lin didn't answer, instead handing his phone to Professor Gu Nanzhou.

"Professor Gu, please explain it to Teacher Lu."

When Professor Gu Nanzhou took the phone, he deliberated for three seconds. The words he finally spoke were as brief as an encrypted telegram, just like the message he had sent to those two insiders.

"Teacher Lu, I am Professor Gu Nanzhou from the School of Mathematics, working on discrete geometry," he said. "You have a student here who, this afternoon in my reading seminar, drew something that I find inconvenient to fully describe over the phone. You need to come over. For this matter, you as his advisor probably need to be present at the very first moment."

Another period of silence came from the other end of the earpiece.

Then came a very light intake of breath, as if someone had been choked.

"I'll be there in ten minutes."

Returning the phone to Jiang Lin, Professor Gu Nanzhou looked at him and asked a question he had been holding back all along.

"Can you tell me where on earth this brick of yours came from?"

This was the most unreasonable link in the entire affair.

Why would a student from the School of Physics plunge headfirst into aperiodic tiling?

This was one of the most obscure, coldest, and most agonizing corners in mathematics.

From Wang Hao's Wang tiles to Roger Penrose's world-shaking two pieces of kites and darts, and then to the later complex substitution tiling systems.

This field was like a labyrinth open only to madmen.

Without one or two years of training in discrete geometry, symbolic dynamical systems, and tiling theory, ordinary people couldn't even touch the threshold.

Yet this brick had somehow been drawn out.

Even its internal forcing mechanisms and skeleton had been fully deduced.

There must be a path in the middle—a path that did not come in from traditional algebraic topology or geometric group theory.

"Sorting," Jiang Lin said.

After this word dropped, C216 fell quiet for a moment.

It wasn't shock.

Rather, they hadn't understood it at all.

Standing off to the side, PhD student Chen Yan instinctively frowned.

As a senior student who had endured three years in tiling theory and whose hair had thinned quite a bit, he had thought of countless answers.

For instance, Jiang Lin happened to have seen some extremely obscure and unpublished arXiv preprint.

For instance, some top research group abroad, such as a certain geometric group at Cambridge or Princeton, had leaked unpublished internal lecture notes.

For instance, Jiang Lin actually had a retired old academician specializing in discrete geometry guiding him from behind the scenes.

To be even more absurd, perhaps he was exceptionally gifted as a child and had come into contact with Penrose tiling puzzle toys, quietly playing with them in his mind for over a dozen years and accidentally stumbling upon a solution by sheer luck.

But sorting?

Between a sorting algorithm and a single tile capable of tiling the entire Euclidean plane in an aperiodic manner, there lay a mountain.

A mountain—the kind where different professions are worlds apart.

Professor Gu Nanzhou didn't speak immediately either.

He looked at Jiang Lin as if confirming whether he was suffering from auditory hallucinations due to excessive excitement.

"Sorting?" Professor Gu Nanzhou confirmed once more.

"Mhm."

Jiang Lin didn't offer much verbal defense.

He turned sideways, unzipped his backpack, and pulled out a thick black hardcover notebook from inside.

Flipping to the first few pages, he pushed it flat across the table in front of Professor Gu Nanzhou.

"To be precise, it's a small sorting program," Jiang Lin said, pointing at the dense symbols on the paper.

Chen Yan finally couldn't help speaking up, his voice carrying a trace of a tremor: "Sorting networks?"

Jiang Lin turned his head to look at him and nodded slightly: "In the beginning, it was sorting networks."

As soon as those four words came out, Chen Yan instead heaved a sigh of relief, his tense nerves relaxing slightly.

He knew about sorting networks.

In the cross-field of computer science and discrete mathematics, this wasn't unfamiliar.

Fixed-size inputs, fixed-order comparators, without the data-dependent dynamic branch prediction found in ordinary sorting algorithms.

It was like a hardwired circuit where out-of-order data was thrown in, and through layers of comparison and exchange, the final output was inevitably ordered.

But he still didn't understand.

No matter how much a sorting network tested the extremums of combinatorial mathematics, it was still just a small tool in computer science used for underlying hardware optimization.

It was still far too distant from the problem of a single tile required to tile an infinite two-dimensional plane without any translational symmetry appearing.

One was a linear combination of finite states, and the other was an infinitely extending geometric topology.

Professor Gu Nanzhou lowered his head to look at the notebook.

He had originally thought he would see a few pages of obscure code written in C++ or Python, or some calculation formulas regarding time complexity.

Yet he found that what he was looking at wasn't code.

At least, not just code.

On the first page of the notebook, Jiang Lin had used a black gel pen to draw five circles extremely neatly.

Inside them were labeled: 0, 1, 2, 3, 4.

Between the circles were connected various intersecting solid and dashed lines.

Annotated on the side in small script were—

Compare 0 and 1.

Compare 3 and 4.

Compare 2 and 4.

Swap if they should be swapped.

Continue if they shouldn't.

Professor Gu Nanzhou didn't understand the performance optimization of low-level computer programs, but he possessed top-tier mathematical intuition.

He could see that this was not sorting in the ordinary sense at all.

It looked more like a set of extremely compressed, fixed topological actions.

Five positions, a limited number of comparators. All full permutation states (5! = 120 possibilities) as inputs had to be forcibly compressed and collapsed into the same unique ordered result by this inflexible set of actions.

Flipping another page, the diagrams began to grow complex.

No longer just five simple circles.

Instead, it was a small, rigorous state transition diagram.

Dozens of nodes were densely distributed across the paper.

Directional arrows connected the nodes.

Several branches had huge red Xs drawn over them by Jiang Lin.

A few lines of text were written beside them: State merging, early pruning, do not mistakenly kill the optimal path.

Professor Gu Nanzhou's page-turning action slowed down.

By the third page, there were very few traces of code or algorithmic logic left in the notebook.

Some charts still retained the shadow of program state trees, but some nodes and connecting lines had begun to undergo mutations.

They twisted and folded, turning into boundaries.

A circle of inward-concave line segments.

An outward-convex tooth-like structure.

A small notch crossed out in red pen and marked as unmatchable.

At the very bottom of this page, a line of English was written with extremely heavy strokes.

Local rules force global structure.

(Local rules force global structure.)

Professor Gu Nanzhou's fingers paused abruptly on that line of text, as if struck by lightning.

"I'm writing a tool, temporarily calling it the Micro-Program Searcher (MPS)."

Jiang Lin chimed in to explain at the appropriate moment.

"It's still very foolish right now, and its computing power is extremely limited. It will only brute-force test all possible low-level approaches within a strictly defined, very small range."

"For instance, if we want to find the optimal network for sorting five numbers, it can test all candidate comparison orders one by one, leaving behind the set that is correct, stable in state, does not violate missing value rules, and runs faster on low-level instruction cycles."

Hearing this, Chen Yan instinctively took half a step forward and interrupted, "Isn't this just exhaustive search in the field of algorithms? What on earth does this have to do with tiling? You surely can't just arrange numbers and turn them into a brick, right?"

Jiang Lin raised his head and glanced at him indifferently.

"The connection is not in sorting itself."

"The connection lies in why the searcher blows up."

Jiang Lin reached out and flipped the notebook to another page.

Written on it was—MPS v0.2 core bottleneck: Which states can be merged?

Which dead ends can be pruned in advance?

After pruning them, how to rigorously prove mathematically that I haven't mistakenly killed the real solution?

Jiang Lin's slender fingers hovered over the paper, pointing at the first line of text.

"If two algorithmic states, no matter what kind of input they face in the future or how they proceed downward, will face the exact same subsequent choice space, then they should be classified into the same equivalence class."

His fingers moved downward, pointing to the second line.

"If a certain path starts from the current node and, no matter how it is repaired or how comparators are added in the future, it cannot possibly exceed the currently known best result, then this path is invalid. It should be pruned in advance at the very first step so it no longer consumes computing power."

Finally, his finger tapped heavily on the third line.

"But there is a problem here: when you decide to use an algorithm to prune a category of states, you must provide a global proof proving that it is absolutely impossible to participate in any legal final sorting network."

He spoke very calmly.

Yet it caused Professor Gu Nanzhou's face, which had originally only shown shock, to slowly creep with a layer of near-terrifying realization.

Because if the subjects of these three sentences Jiang Lin just spoke were changed and replaced with the exclusive language of discrete geometry and tiling theory, it would be another set of things that made all mathematicians tremble all over.

If the future extension methods and constraint conditions of two local boundaries are equivalent, they can be classified into the same type of super-tile.

If a certain local patching method, when extending outward by one circle, inevitably produces dead corners at the edge that cannot be closed, then it is a dead end.

You must prove in the local neighborhood and boundary extension—that it cannot possibly participate in any legal tiling covering the entire plane.

This wasn't the relationship between sorting and bricks.

These were two seemingly unrelated problems that shared the exact same skeleton at the most fundamental abstract logic level.

Chen Yan also slowly understood.

He stared at those few pages of notes, his scorching gaze seemingly wanting to burn a hole through the paper.

"So from beginning to end, you weren't looking for bricks at all."

"I wasn't at the beginning," Jiang Lin readily admitted.

"Then what on earth were you looking for?" Chen Yan's voice was close to a moan.

"Find a mathematical language." Jiang Lin lifted his eyes, his gaze as profound as a cold, emotionless precision instrument, "I need a language that can clearly explain why those limited, localized micro-constraints still retain absolute compulsiveness when expanding outward to very large or even infinite scales."

Professor Gu Nanzhou took over the conversation, his voice much deeper than before, as if a huge boulder were pressing down on his chest.

"So, following this logic, you looked into Tiling."

"Mhm."

"To solve the problems of state substitution and scale amplification, you then looked into Substitution."

"Mhm."

"Finally, you inevitably hit the ultimate problem in this field, and you looked into the Monotile."

"Right." Jiang Lin nodded.

At this moment, the logical loop was closed.

Professor Gu Nanzhou did not ask any further. With trembling hands, he flipped the notebook back.

He began to examine the contents of the notebook from a completely new perspective.

On the surface, what he saw were pages of sketches; in reality, this belonged to an empirical record of human wisdom crossing disciplinary chasms.

At the very beginning were the circles and directed arrows of computer science.

In the middle were the equivalent state merging tables of algebraic structures.

Further down, the lines began to twist, turning into geometric outlines drawn with extremely irregular, jagged edges.

Those outlines were still very rough, carrying a strong sense of crude engineering aesthetics.

Beside some polygons, it was written in red pen: DEAD.

Beside some blocks of similar shapes, they were circled together and labeled: SAME CLASS.

There were also chunks where several shapes interlocked, labeled with exclamation marks beside them: FORCED LARGER BLOCK.

These were definitely not formal mathematical symbols that would be used in orthodox textbooks of the Department of Mathematics.

There were no homology groups, no manifolds, and no algebraic number theory.

But Professor Gu Nanzhou completely understood their meaning.

Dead end, same class, forced into a larger block.

How could this be a mathematics beginner drawing tiles?

This was an engineer, a physicist, trying to forcefully draw out the massive and abstract state-pruning logic within the search engine he wrote, using the physical boundaries of a two-dimensional plane.

Professor Gu Nanzhou abruptly raised his head.

"The tridecagon on the draft paper grew out of these algorithmic boundary state diagrams by itself?"

"Yes." Jiang Lin answered with certainty.

"It was definitely not deduced or modified by you from Penrose's kite and dart tiling, or any existing Aperiodic set?"

"No." Jiang Lin shook his head, a hint of rejection toward such inefficient methods in his tone, "Deducing from existing sets will trap you in preconceptions of geometric intuition. And..."

He paused for a moment.

"This afternoon was the first time I systematically looked at this problem."

Room C216 fell into a dead silence once again.

This time, even the sound of the wind blowing through the leaves outside the window seemed to have been sucked away.

An afternoon.

Looking at it systematically for the first time.

Then, using the concretization of the algorithmic state space, he drew a highly potential candidate monotile.

If this sentence were taken alone to any international discrete geometry academic conference, it would be treated as hopeless mad talk.

Security guards would directly escort you out of the venue.

Yet now, the black notebook on the table made this mad talk incomparably real, real to the point of being hair-raising.

Because Professor Gu Nanzhou and Chen Yan had seen that path with their own eyes.

It did not start from the monotile problem itself.

It started from the sorting of five numbers.

It started from a trivial operation called tens of millions of times per second in the underlying hardware.

It started from the exponential explosion of algorithmic states.

It started from how to cut off dead ends in advance and use absolute rigor to prove that the cuts were not wrong.

Charging all the way forward, it forcefully smashed into the pure mathematical southern wall of how local rules compel global structures, and punched a big hole in the wall.

Professor Gu Nanzhou flipped back to the large grid paper tucked in the middle of the notebook.

That was the local neighborhood table made by Jiang Lin.

The columns in the table formed a perfect mirror symmetry with the objectives of the MPSv0.2 algorithm Jiang Lin had just mentioned.

Local Patch — corresponding to the current state in the algorithm.

Valid — corresponding to validity check.

Forced Metatile — corresponding to state merging.

Dead End — corresponding to algorithm pruning.

Professor Gu Nanzhou suddenly felt that the significance behind this was even more hair-raising than an old mathematics expert who had studied tiling for thirty years finally puzzling out the answer in a flash of inspiration.

This student was not aiming to solve Einstein's problem at all.

He did not have that morbid pursuit of geometric symmetry and aesthetic beauty characteristic of pure mathematicians.

He had been driven to desperation by the engineering bottlenecks of a completely different field; to break out, he was forced before the same mathematical structure, and ultimately used brutal yet exquisite means to tightly fold and compress the infinite information of the global scale into a small local rule.

In sorting networks, this approach cleverly hid the global sorting correctness within a series of local pairwise comparison and exchange operations.

In discrete tiling, this approach forcefully engraved the aperiodicity of the infinite plane into the few uneven geometric boundaries of a single tile.

The underlying logic was entirely the same thing.

Instead, a pair of eyes that had never been trained by traditional tiling theory—and therefore had never been bound by its prejudices, its axioms, or its historical failure cases—crashed in through the side door of the discipline in the most unreasonable way.

This evoked an even more soul-striking exclamation in Professor Gu Nanzhou than if an expert who had steeped in this field for twenty years had solved it.

After a long while, he asked on a whim, "Can you bring out your search engine and let me see it?"

As soon as these words were uttered, several students nearby all raised their heads.

Jiang Lin nodded, "Yes, it's all on my computer. However, the current version is still very weak; it hasn't undergone large-scale cluster parallel processing and can only handle very small-scale problems."

"Whether it's weak or not is a secondary discussion; right now I want to know how on earth you compressed a small problem into a transferable set of methods."

...

Nine minutes later, the door of Room C216 was shoved open violently.

Due to the excessive force, the door panel slammed heavily against the wall stopper, emitting a muffled thud.

Professor Lu Zhixing practically rushed in.

He was still wearing the gray anti-static overalls from the Physics laboratory, and didn't even have time to take out the laser goggles in his pocket.

A thin layer of sweat hung on his temples, and his breathing was heavy.

The first thing he did upon entering was look for Jiang Lin.

Confirming that the student was standing there safe and sound, his tense expression relaxed slightly.

Only then did he quickly adjust his state and let his gaze fall upon the middle-aged man with a complex expression on his face.

He stepped forward quickly and extended his hand.

"Hello, Professor Gu, I am Professor Lu Zhixing from the School of Physics."

"Hello, Professor Lu." Professor Gu Nanzhou grasped his hand with great force, so much so that Professor Lu Zhixing was surprised.

After greeting, Professor Gu Nanzhou skipped any pleasantries or polite remarks.

He went straight to the point, without hesitation or ambiguity, spilling the entire sequence of events that had occurred in the past half hour like beans spilling from a bag.

From Jiang Lin correcting the topological errors on the blackboard to tossing out the concept of sorting networks, and then to deriving the boundaries using algorithm pruning logic.

With every sentence spoken, Professor Lu Zhixing's brows twitched.

After finishing, Professor Gu Nanzhou pulled out the A4 sheet of paper that he had just stored in his folder like a top-secret document, carefully spread it out, and placed it in front of Professor Lu Zhixing.

Drawn on the paper was a weird tridecagon with stiff lines and no aesthetic appeal whatsoever.

It looked like the product of someone brutally flattening it and then twisting it hard.

Beside it was the folded grid paper.

Dozens of vertex neighborhood splicing possibilities, validity, forced rules, and dead-end conditions were listed neatly by Jiang Lin with an extremely rational brushstroke.

Professor Lu Zhixing could not understand this tile.

At least at first glance, from a professional mathematics perspective, he completely failed to understand it.

High-stability optical measurements, optical path drift of the Fabry-Pérot interferometer, mechanical stress hysteresis in high vacuum environments, thermal cycle noise under extremely low temperatures...

These things were far too familiar to him; he could write out the error differential equations with his eyes closed.

However, discrete geometry, substitution systems, local matching rules, aperiodic tiling...

This was not his field; even the formula symbols looked like heavenly scripts.

Yet, while he couldn't understand the tile, he could understand Professor Gu Nanzhou's expression.

The man in front of him was an associate professor of mathematics who had secured a tenured faculty position at a top-tier university.

People who had ground their way out in the field of pure mathematics carried a bone-deep, ultimate arrogance toward logic.

Such a person would absolutely not summon someone in a rush from the Physics building just because a student from another department casually drew a bizarre polygon on paper.

Nor would they use phrasing on the phone as if dealing with some extremely dangerous explosive, solemnly saying that as an advisor, you had better be present at the first instance.

That was the look of someone who had discovered an unmined gold mine, terrified that someone else would snatch it away, yet equally terrified that the gold mine itself would collapse.

Professor Lu Zhixing opened his mouth.

Of course he knew that this nominal disciple of his was formidable.

Yet he never expected that this student, who was formidable in the field of engineering Physics, could also transplant his methodology intact into a purely fundamental theoretical field that was completely unfamiliar to him—and one that even many mathematicians shrank back from.

To cut out a tile.

And it was highly likely that very tile everyone had been looking for, yet no one had ever truly been able to find.

If confirmed, it would be that single tile the mathematical community had been searching for for half a century.

Professor Lu Zhixing looked again at Jiang Lin beside him, whose expression was as calm as if he had just finished a post-class exercise.

All his previous evaluations and judgments of this student were still underestimations.

He had likely only seen the local projection of his massive system along a certain specific dimension.

His Physics intuition was exceptionally good.

His engineering fundamentals were absurdly solid.

His sense of smell for error chains and system redundancy was acute to the extreme.

All these words themselves were perfectly verified in this student.

But looking at it now, they were too small.

The framework was too small.

This was like someone acquiring the sharpest cutting tool in the world, and the School of Physics thought he was merely a top-tier machinist capable of cutting the flattest parts.

But in reality, when this tool cut physical parameters, there was fundamentally no difference when it cut computer algorithm states or pure mathematical topological spaces.

In his eyes, everything was a geometric structure that could be compressed by finite states and approximated by error chains.

Professor Lu Zhixing let out a long breath.

He used to think he was helping to cultivate a future experimental physicist.

But now he realized this was simply like taking a vernier caliper to try and measure the shadow of an endless, colossal mountain range.

"Professor Gu." Professor Lu Zhixing raised his head, his eyes having recovered the decisiveness and protectiveness a Principal Investigator (PI) should have, "Since I am here, we can now discuss the verification progress of this tile, as well as the issues of confidentiality and authorship."