91: Chapter 91 A Brick That Shouldn't Be There

The next day, Jiang Lin finally gave a proper name to that clumsy thing that made programs find programs.

Microprogram Searcher (MPS).

It sounded like a product of the last century, but to Jiang Lin, it was a cyber drudge willing to run into every foolish method and never get tired of it.

On the quantitative side, it could already help Jiang Lin do quite a bit of the dirty work.

It helped him ferret out those hot paths hidden in the system that were unremarkable on a single run but looped tens of millions of times every day.

It was also a ruthless quality inspector; every newly found set of practices had to be compared item by item with the old workflow, and as long as one output did not match, it was immediately sentenced to death.

Today it ranked five numbers, and tomorrow it might bucket sixteen numbers.

Moreover, Jiang Lin soon realized that the most valuable part of this gadget was not just being a little faster.

Rather, it was forcing himself to become more rigorous.

Because when using this searcher, if he wanted to make it work more smoothly, he had to clearly tell it:

What counted as right, what counted as wrong, what boundaries must never be touched, and what actions looked like shortcuts but were actually shifting future debts to the present?

This was simply the most hardcore self-learning process.

In the past, whenever he learned a new concept, he always had to engage in mental combat with himself.

Would a state compression lose information?

Had a recurrence secretly mixed in future data?

Would a seemingly exquisite proof slip and fall on some extreme boundary?

Now, it was equivalent to having an unintelligent but infinitely energetic, stubborn teaching assistant by his side.

Jiang Lin typed a line of test in the terminal: "Are there any counterexamples to this statement within ten elements?"

MPS did not understand why; it would only dump all situations out like flipping open a drawer.

If there were none, it would tell you that none were found in this small range.

If there were, it would slap that bloody counterexample right in your face.

This was enough to save Jiang Lin a lot of time to verify whether his intuition was wrong.

But MPS v0.1 was too straightforward.

For sorting five numbers, the code ran merrily.

For six numbers, the background printing traces began to become messy.

When it came to eight numbers and even larger numbers, in order to find the shortest, most stable, and least error-prone action sequence, it had to probe among massive comparison orders.

The fan in the chassis suddenly began to spin wildly, emitting a dull buzzing sound.

On the Task Manager curve, the CPU usage instantly spiked to 100%.

There were too many paths.

All possible action sequences overflowed like a breached flood, directly drowning this infinitely energetic worker in the memory.

The vitality supported by pruning was negligible in front of the exponential explosion.

Hearing the roar of the chassis, Jiang Lin pinched the space between his eyebrows.

This was a structural terminal illness.

It was no longer a problem that could be solved by idling for a few more hours.

To make this toolchain the core, he had to solve more underlying logic.

Which states could be merged?

Which dead ends could be cut off in advance?

And after cutting them off, how to prove mathematically that he did not mistakenly kill the true optimal solution?

Jiang Lin switched back to the log document and typed a few lines of words.

MPS v0.2 goal: No longer just trying, learn to merge states, eliminate dead ends, and prove the legality of pruning.

After writing, he paused for a moment.

The cursor flashed rhythmically at the end.

He added another line.

Mathematical language needed: local rules, state transitions, global constraints.

Somehow, these words unexpectedly connected to another line in his mind.

Reminding him of an older problem.

What if the physical boundary of a shape was a natural local rule?

What if two tight splices of graphics were a state transition?

Incorrect connections would automatically walk into dead ends, while correct connections would be forced to spread into increasingly grand structures like growing crystals.

This was essentially no different from letting MPS make frantic trial-and-error runs in the chassis.

Jiang Lin put the cursor back into the search box.

This time, he did not search for code, but instead typed a line of English.

local rules force global structure

Press enter.

A few seconds later, a few words popped up on the page: tiling, substitution, aperiodic.

He clicked all the way through the links.

Penrose tiling, Wang tiles, quasicrystals.

Finally, at the corner of an academic review, he saw that word as sharp as a needle tip.

monotile.

Jiang Lin stared at this word for a full half minute, his eyes shining.

Then he re-entered in the search box: aperiodic tiling monotile.

...

In the afternoon, Jiang Lin went to the Jiangda University library and borrowed four books.

One on finite automata, one on combinatorial optimization, and one on symbolic dynamical systems.

As well as a lecture notes on discrete geometry with somewhat curled covers.

He wanted to find a way of proof, a mathematical language that could explain how finite local constraints could still be effective in infinite expansion.

When flipping to the middle part of the discrete geometry lecture notes, his gaze stopped on a chapter name.

Aperiodic Tilings.

He scanned through quickly.

First Wang tiles, then Penrose tilings, followed by substitution tilings.

Finally, near the end of this chapter, the lecture notes mentioned a concept in an extremely short paragraph:

Einstein problem: whether a single tile can force aperiodicity.

(The Einstein Problem: whether there exists a single-shaped tile that can force the tiling to exhibit aperiodicity.)

A single brick.

Relying solely on its own geometric shape, it could tile an infinite plane, but all tiling methods could never produce periodic repetition.

This was simply the perfect geometric embodiment of the MPS idea.

In the program, local actions eliminated error paths, forcing the searcher into the unique feasible domain.

In geometry, the local boundary of a single brick eliminated all incorrect spellings, forcing the infinite plane into a profound hierarchical structure.

One virtual and one real, fitting together seamlessly.

Closing that curled-cover lecture notes on discrete geometry and coming out of the library, Jiang Lin still had the three words—local rules, state transitions, and global constraints—swirling in his mind.

The state space explosion problem originally encountered in the Microprogram Searcher (MPS) seemed at this moment to be projected into a concrete entity under the context of geometry.

Walking through the shaded avenue, he originally intended to go straight to the bike shed to get his bicycle back to No. 7 High School, when his peripheral vision caught a poster on the roadside bulletin board.

Discrete Geometry Reading Group

Theme: Aperiodic Tiling, Quasicrystals, and the Monotile Problem —— How Local Rules Force Global Structure.

Location: Building C, School of Mathematics, Room 216

Host: Professor Gu Nanzhou (Associate Professor)

Time: 15:30

Jiang Lin's steps suddenly stopped.

Local rules, how to force out global structure?

Thinking of the core dilemma in MPS, he pulled out his phone to check the time; there were still ten minutes before the class started.

He immediately fetched his bicycle and rode toward the School of Mathematics.

Building C of the School of Mathematics was much quieter than the Physics building.

The walls were pasted with posters of various academic lectures, mostly filled with vocabulary like tensors, manifolds, and homology groups.

The door of C216 was half-open.

Jiang Lin pushed the door open and went in, finding a window-side seat in the last row to sit down.

The classroom was not large, with several long tables pieced together in the middle, and a dozen or so students sitting scattered around.

Some were flipping through printed papers, while others were typing on their laptops.

The atmosphere was not serious, but it did not seem very relaxed either.

By the whiteboard at the front stood a man in his thirties.

He was wearing a light gray shirt with his cuffs casually rolled up to his elbows, spinning a whiteboard marker in his hand.

He must be Professor Gu Nanzhou.

The projector was on, projecting an intricate geometric figure onto the screen.

It was precisely the classical Penrose tiling.

There were only two basic shapes in the diagram.

One was a kite, and the other was a dart.

Both edges bore specific arrow markers.

"We are discussing aperiodic tiling today. Before that, I need to clarify a concept that is often misunderstood by the outside world, and even by some beginners."

Professor Gu Nanzhou turned around and wrote a line of words on the whiteboard.

non-periodic ≠ random

"Many people feel that aperiodic means messy, meaning without regularity." Professor Gu Nanzhou pointed to the equation on the blackboard, "Randomness certainly does not possess periodicity, but the beauty of mathematics has never lain in lawless chaos."

"The reason Penrose tiling is fascinating is that its local rules are very limited, one could even say very simple, yet in the process of infinite extension, it can irresistibly force out a global aperiodic structure."

He quickly sketched the outlines of the kite and dart on the whiteboard, and heavily dotted a few arrows on the edges.

"For two bricks, as long as these matching rules are added, dictating that the arrows must meet in the same direction, you will never be able to piece together a periodic pattern that can be translated and overlapped, because it is locked down."

A PhD student sitting in the front row raised his hand, pushed up his glasses, and asked, "Professor, Penrose tiling relies on edge-matching rules. If we realize it physically, such as making these arrows into real convex-concave buckles and directly internalizing the matching rules into the geometric shape, would it be equivalent?"

"Your intuition is very accurate."

Professor Gu Nanzhou nodded appreciatively.

"In fact, this is precisely what many construction methods later on were doing. Using pure geometric boundaries to replace artificially attached symbolic rules. As long as the shape is properly designed, two bricks can indeed achieve this."

Speaking of this, his tone suddenly shifted, becoming heavy.

"But what if we push the conditions to the limit?"

He picked up the blackboard eraser, erased the kite and dart he had just drawn, and rewrote three short lines of English.

one tile

tiles the plane

only non-periodically

"The Monotile Problem (The Einstein Problem). This name does not refer to the physicist Einstein, but rather ein stein in German—a single stone, a single brick."

Professor Gu Nanzhou's gaze swept across the students by the long table, as if casting a heavy anchor.

"Using only a single brick, it can seamlessly and non-overlappingly tile the entire two-dimensional plane. However, all tiling methods cannot possibly produce periodic repetition."

A brief silence fell in the classroom.

"A single brick can tile a plane, which has zero difficulty; squares and regular hexagons can both do it." Professor Gu Nanzhou tapped the whiteboard, "The difficulty lies in the fact that it must not have any possibility of a periodic tiling method. You cannot say that I happened to arrange a flashy aperiodic spelling for it and consider it a success; that is far from enough."

He emphasized his tone: "You must prove mathematically that anyone, even if he racks his brain trying to piece together a periodic pattern, as long as he holds this brick, he cannot escape the destiny of aperiodicity. The geometric shape of this brick itself must become a tyranny, a compulsion."

Sitting in the back row, Jiang Lin's breathing hitched slightly.

He immediately opened the black notebook he carried with him and wrote a line of words on a blank page.

Instead of providing one aperiodic tiling method, it was forcing all tiling methods to enter an aperiodic hierarchy.

This was a binding force akin to a dimensional-reduction strike.

And it was precisely the feature that his MPS system currently lacked the most.

MPS was merely running around frantically among massive possibilities to find feasible solutions, whereas the single brick in Professor Gu Nanzhou's words used its own geometric rules to directly cut off all error paths in their bud.

The previous discussion was still ongoing.

Professor Gu Nanzhou began to sort out the historical context.

From the undecidability of Wang tiles to substitution tilings, and then to how to use metatiles to build larger hierarchical structures.

As the lecture went deeper, the blackboard was already crowded with substitution diagrams, local patches, angle constraints, and a few lines of simple combinatorial counting.

"Why does the seemingly simple condition of a single brick make the problem so tricky?"

Professor Gu Nanzhou pointed to the complex derivation on the blackboard.

"Because without a second shape to act as a buffer and constraint, you cannot easily cut off periodicity. A single brick alone is too easy to connect head-to-tail to form translational symmetry."

A second-year graduate student smiled bitterly and chimed in: "Professor, this sounds like asking us to forge a key that can open all the doors in the world, yet is destined by physical laws not to be copied by any door. This requirement is too counter-intuitive."

Professor Gu Nanzhou smiled rarely.

"That is a good metaphor. Its most profound part lies in the fact that you must fold massive, even approaching infinite, global information without omission into a tiny local geometric boundary."

Jiang Lin stared at the sentence in his notebook: hide global information into local geometry.

This was essentially the same thing as the search_sort_network he had deduced earlier.

In the underlying quantitative system, he was trying to hide the global sorting correctness into a string of local compare-swap operation constraints.

In the tiling problem, it was to hide the aperiodicity of the infinite plane into several convex-concave boundaries of a single brick.

The underlying logic was interconnected.

Over an hour's reading group ended in the blink of an eye.

Professor Gu Nanzhou announced the end of the meeting, but no one left immediately.

The lingering warmth of the academic discussion was still there.

Several students gathered around the long table, studying some high-definition printed drawings brought by Professor Gu Nanzhou.

Some were arguing about the symmetry of the quasicrystal diffraction pattern, while others were discussing the stability of tenfold axes of symmetry in actual materials.

The second-year graduate student who had just asked the question picked up a pencil and casually drew various distorted polygons on the scratch paper, trying to piece together an irregular shape.

"If there really is such a monotile meeting the conditions, what on earth will it look like?"

He shook his head while drawing, his tone relaxed, treating it entirely as a recreational mathematical joke.

And what about Jiang Lin?

Of course, this was not his first time seeing such a problem.

The state space explosion of MPS forced him to think repeatedly about one thing.

How to compress seemingly infinite paths into a limited few categories of states.

The magnetic geometric Rubik's cube in the Alibaba preliminary contest had also forced him to compress chaotic spatial configurations into connection graphs and upper bounds.

The blind box unpacking problem made him confirm once more that as long as the symmetry was strong enough, states should not be named after specific objects, but rather by how many such equivalence classes were still missing.

In his mind, those three questions for MPS v0.2.

Which states can be merged?

Which dead ends can be cut off in advance?

After cutting them off, how to prove that he did not mistakenly kill the true solution?

The metatile Professor Gu Nanzhou had just talked about seemed to suddenly change these three questions into a geometric shell.

Transition was no longer compare-swap, but one brick sticking onto another brick.

Dead ends were no longer unrunnable candidate programs, but an unfilled corner gap.

He had originally drawn many state transition sketches for MPS.

Some sketches, in order to see the boundaries clearly, had already been drawn by him as geometric splices.

Now, these sketches suddenly had new names.

metatile.

substitution.

forced hierarchy.

Therefore, his first thought was not whether the pattern looked good or not.

But the three questions.

Could it generate an entire plane?

Could it be proven that all legal spellings were forced into the same set of hierarchies?

If the hierarchy was established, could periodicity be disproven?

At this thought, Jiang Lin's wrist began to move.

The tip of his pen fell onto the paper without hesitation.

He first drew a tiny kite shape.

Those edges all fell on the same auxiliary grid, and several types of angles appeared repeatedly, just allowing the notches and convex edges to form a finite number of adjacency relationships.

Then, with a specific topological structure, he spliced eight such small kite shapes together.

A tridecagon suddenly appeared on the paper.

It was not regular.

The edges were filled with abrupt concavities and convexities, resembling a weird hat that had been roughly flattened and forcefully twisted by someone.

The outline lacked any symmetrical beauty of conventional geometric figures, exuding a cold and hard mechanical feel.

The PhD student drawing next to him caught Jiang Lin's notebook out of the corner of his eye, stopped his pen, and leaned over with keen interest.

"Classmate, what are you drawing, a broken kite?"

He smiled without malice, just finding the shape weirdly comical.

Jiang Lin did not raise his head, nor did he answer.

His wrist continued to move, and next to this weird hat, tightly clinging to its groove, he drew a second identical brick, only with a rotation.

Seamless and snug.

Then came the third and fourth blocks.

These bizarrely shaped bricks intertwined in a way that seemed extremely awkward and visually dislocating.

But their edges meshed together like precision industrial gears, leaving not a single gap.

Professor Gu Nanzhou, having answered the student's question and about to leave, happened to see this scene as he passed by Jiang Lin.

He could not help but stop.

Looking down at those interlocked tridecagons, he fell silent for a few seconds and asked, "This student, did you draw this yourself?"

"Yes."

Professor Gu Nanzhou asked thoughtfully, "Can this shape tile the plane?"

"Yes."

Hearing Jiang Lin's affirmative answer, the PhD student nearby immediately perked up, put down his draft paper in hand, and turned around.

"Being able to tile is nothing rare; many shapes can be periodically tiled as long as they can form a periodic unit cell."

Several nearby students also noticed this side, thinking Jiang Lin was just bragging casually.

Whether it can be periodically tiled is not something that can be asserted just by looking at a few hand-drawn diagrams.

Professor Gu Nanzhou did not take the PhD student's words to heart; he was staring blankly at the spliced figures on the paper.

The combinatorial relationships of those concave angles, convex angles, and a few slanted edges were rapidly reorganizing in Professor Gu Nanzhou's mind.

After a long while, he pulled open a nearby chair and sat down, looking at Jiang Lin, and pondered, "Would you like to try proving it?"

Jiang Lin nodded.

Because this was not a flash of inspiration doodle he had just made in the classroom.

Over the past few days, he had repeatedly drawn many local state diagrams for MPSv0.2.

Some were states in the program.

Some were casually drawn by him as boundaries on a plane.

Originally, he just wanted to find a visualization method to help himself understand which paths would die and which paths would be forced into the same category.

Until Professor Gu Nanzhou talked about metatiles, he realized that those boundary drafts he regarded as auxiliary diagrams could themselves become a geometric problem.

He just lacked an orthodox set of mathematical language to wrap it.

And this afternoon, Professor Gu Nanzhou's reading class happened to hand this set of language into his hands.

substitution.

forced hierarchy.

metatile.

These words, like solid molds, compressed his boundary sketches—which were originally only used to aid understanding of dead ends, categorization, and inheritance—into a framework that could be scrutinized by the contemporary discrete geometry community.

"Step one, I will not directly prove aperiodicity, but first prove that it can tile the entire plane."

When Jiang Lin spoke, on the left side of the paper, he drew several single bricks and pieced them together to form a larger block whose outline still bore specific concave-convex characteristics.

Then he marked a letter next to this block: H (Hex).

Immediately afterward, using different splicing methods, he drew three other composite blocks composed of single bricks, labeled respectively: T (Turtle), P (Propeller), F (Fang).

Without exception, their edges all bore highly characteristic geometric teeth.

That PhD student furrowed his brows slightly and muttered in a low voice, "Is he making oversized bricks?"

Professor Gu Nanzhou raised his hand, signaling him to be quiet.

Jiang Lin ignored the outside noises and continued to deduce on the blank paper.

"These four types of blocks belong to the first-level metatile."

Beside the H block, he began to draw the next-level substitution map.

"According to the geometric characteristics of this single brick, the H block itself can be pieced together from smaller H, T, P, and F according to an unalterable topological structure."

"T as well."

"P and F are the same."

The pen tip moved rapidly across the paper, and the proportions of every edge and every corner were exceptionally precise.

"Every higher-level block can be rigorously combined from these four types of metatiles of the upper layer."

When drawing the third-level substitution, the structure on the paper had become quite complex, but no matter how the interior intersected, the overall boundary still maintained the topological characteristics initially set.

At this moment, no one casually questioned anymore.

Because from this moment on, the question had changed from whether this was a doodle to whether this local neighborhood table had any missing items.

The PhD student just now was even in a state of holding his breath and concentrating, staring fixedly at those meshed gaps.

"Is this a strict substitution rule?" Professor Gu Nanzhou asked softly.

"Yes." Jiang Lin stopped his pen, "As long as this substitution rule can be iterated infinitely in mathematics, it means this plane can be completely covered by metatiles whose scale grows larger and larger until approaching infinity."

He wrote on the edge of the paper:

level 0 : tile

level 1 : H / T / P / F

level 2 : substituted H / T / P / F

...

level n → ∞ : tiles the plane

"Able to tile, the first step is closed." Jiang Lin said calmly.

"You have only proved that, relying on this substitution rule, this brick has a tiling method with a strong aperiodic flavor." Professor Gu Nanzhou stared into Jiang Lin's eyes, "I said in class that this is far from enough."

Jiang Lin nodded: "I know."

"So you must prove that it is not you who chose this hierarchical tiling method from God's perspective, but that the geometric boundaries of this brick itself are forced to and can only grow into this hierarchy."

"Therefore, the second step is not tiling."

Jiang Lin pushed the A4 paper full of substitution rules aside and pulled out a new sheet of paper.

Alone in the center of the paper, he drew the tridecagon single brick.

"It is Forcing."

He circled the three key concave angles on the boundary of the single brick with his pen tip, and then circled several corresponding convex angles.

"We do not look at the global picture, only the local. Under the constraint of rigorous tiling, the plane is not allowed to leave any gaps, nor is any overlap allowed. This means that every concave angle on the single brick faces a choice of life and death; it must be bitten by a certain kind of convex angle of another brick."

Around the concave angle, he drew the first tiling method of adjacent bricks.

"The first filling method, legal."

Then came the second.

"The second, legal."

The third.

"The third, legal."

Then, he drew the fourth tiling method.

This brick fit perfectly at the concave angle, but at the other end, a section abruptly protruded.

Jiang Lin unhesitatingly drew a huge cross next to it.

"This type of local filling method looks connected at a vertex on the surface, but as long as it extends outward for another circle, its other end will form a 30° dead corner. And in our single brick system, there is no protrusion that can fill a 30° gap; this is a dead end."

The PhD student couldn't help chiming in: "Wait a minute, how can you be sure that there are only these few splicing possibilities for the local neighborhood? Planar combinations explode exponentially."

"Because angle resources are limited." Jiang Lin didn't even raise his head, "In a plane, the sum of angles around any vertex must strictly equal 2π. The interior and exterior angles that this brick can provide are all within the integer multiple system of 30° and 90°."

"The combination of angles it can contribute is a finite set; therefore, the types of local neighborhoods around any vertex are also a finite set."

"The only ones that can close and extend outward are the categories I listed."

"Those that cannot close will quickly expose dead corners in the first or second circle, making it impossible to continue tiling."

Professor Gu Nanzhou couldn't help nodding.

He knew very well the weight of this derivation.

This was the hardest part of the single brick problem: not artificial design, but the spontaneous pruning of geometric forms.

Jiang Lin continued to expand the first legal local tiling method on the paper.

After the first circle was filled, several bricks were combined together, inevitably forming a larger groove area with a specific shape around the center.

"Do you see this new boundary?" Jiang Lin tapped the large groove with his pen tip, "The shape of this groove cannot be arbitrarily filled by a single brick. Through the angle exclusion method just now, the only legal way to fill it is to combine it into the H block or T block I defined in the first step."

While quickly sketching, he stated the geometric rules: "The local compatibility constraints of the single brick force out the first-level metatile."

"And the new boundary formed after the combination of the first-level metatile still carries the same type of concave-convex constraints, and the scale is enlarged, which will continue to force out the second-level metatile."

"Once this process starts, it cannot be stopped. The underlying geometric rules force the superstructure to evolve along a uniquely designated route."

"How do you guarantee that all legal local tiling methods will eventually fall without omission into the structures of those four types of metatiles?" Professor Gu Nanzhou pursued.

Jiang Lin put down his pen, pulled a folded squared paper from behind the black notebook, unfolded it, and pushed it in front of Professor Gu Nanzhou.

That was an extremely detailed table.

The handwriting was neat and the logic rigorous.

The table header clearly read:

Local Patches (Local Neighborhood Types) | Valid (Legality) | Forced Metatile (Attribution Forcing Metatile) | Dead End (Dead-End Termination Condition)

In the table, dozens of splicing combinations around vertices were densely listed.

Each row was like a judgment on a certain possibility.

Seeing this table, the muscles in the PhD student's eye corners couldn't help twitching.

He had studied tiling for three years, the second-figure in his advisor's research group, and he could deduce the undecidability of the Wang tiles from beginning to end.

So he subconsciously wanted to pick fault and find loopholes.

How could local neighborhoods be exhaustively enumerated; combinations exploded exponentially.

But this folded squared paper.

Dozens of neighborhoods, the legality, attribution, and death conditions of each were neatly listed.

The loophole he wanted to ask about had already been crossed out in the seventeenth row of the table.

The full belly of words reaching his mouth turned into a sentence: "Classmate, did you calculate this in advance?"

"Part of it." Jiang Lin answered.

"No, this is not a part." Professor Gu Nanzhou interrupted the PhD student, his gaze rapidly scanning the data on the table like a scanner, "All the key neighborhoods leading to topological divergence are right here."

No one chimed in casually anymore.

When the constraints of local boundaries, the exhaustive exclusion of finite neighborhoods, and legal forced categorization were laid bare on the table, any emotional doubt appeared pale and weak.

The geometric forcing of the first layer.

The structural inheritance of the second layer.

On this piece of paper, it had already presented a structural closure that was hard to casually break through.

This was definitely no casual classroom doodle.

In classroom C216, the remaining few students all gathered around, but no one spoke.

They looked at the weird tridecagon on that paper, just like looking at a creation fallen from outer space.

Professor Gu Nanzhou stared at Jiang Lin with burning eyes and asked, "What about the third step?"

"Proof by contradiction for periodicity."

Jiang Lin took back the paper and pen, writing a concise formula in the blank space on the far right.

T + v = T

"Suppose there exists a non-zero translation vector v such that this entire infinite tiling pattern, after being translated along v, can closely coincide with the original pattern. This is the definition of periodicity."

After speaking, he drew a schematic diagram of the boundary of the first-level metatile on the left. Then he outlined the second layer outside with a dashed line.

Then came the third layer.

One layer nested within another, and the scale of each layer was magnified at a fixed ratio.

"The second step just now has proved that any legal tiling method will not be chaotic; it will definitely be strictly decomposed into the first-level metatiles by the forcing power of the local boundaries."

"The first layer will in turn be forced to combine into the second layer."

"The second layer combines into the third layer; this hierarchy with a fixed structure will extend infinitely outward."

Professor Gu Nanzhou looked at the nested figure, his palms sweating slightly.

He had completely anticipated Jiang Lin's subsequent logical trajectory.

That would be the final decisive kick.

Jiang Lin continued to push forward the logical lock: "That being the case, if this periodic vector v exists, then what it translates is not only the underlying single bricks."

"It must also keep the decomposition boundaries of each level of metatile unchanged."

"Because once changed, the translated pattern will cause the original hierarchical boundaries to misalign, which contradicts the conclusion that all legal tiling methods must conform to this forcing hierarchy."

He wrote down the strict derivation conditions on the paper:

v preserves level 1 boundaries

v preserves level 2 boundaries

...

v preserves level n boundaries

Jiang Lin drew a huge metatile outline occupying half a sheet of paper.

"However, the spatial scale of the n-th level metatile increases unboundedly as n increases."

"For any fixed non-zero vector v, we can always find a sufficiently large n in the infinite hierarchy such that the length of vector v, |v|, is much smaller than the characteristic scale of the n-th level metatile."

"Under such a disparate scale difference, translating a tiny distance v can never send the huge boundary of the n-th layer as a whole back to the position of another n-th layer boundary; it will definitely experience irreconcilable misalignment."

"Therefore, the only translation vector that can satisfy maintaining the coincidence of all infinite-level boundaries is one."

At the end of the derivation, Jiang Lin wrote the final conclusion.

v = 0

"Therefore, there is no non-zero periodic translation."

"This brick can only tile the plane aperiodically."

End of proof.

Jiang Lin put down the pencil.

You could hear a pin drop around the long table.

Sunlight beat in from outside the window, shining on those crude hand-drawn figures and formulas on the paper.

Able to tile.

Local boundaries generate a forcing hierarchy.

Scale differences of infinite hierarchies prove periodicity by contradiction.

Three steps of logic, interlocked, leaving no obvious breaks.

On this humble desk, a closed loop was completed.

Everyone knew in their hearts that this was not a complete paper that could be directly published.

It lacked a lot of formal language embellishment.

It also needed to convert those hand-drawn charts into rigorous computer-aided vector graphics.

It needed to complete the lengthy exclusion process of the remaining obviously invalid local neighborhoods in the table.

It also needed to clarify whether this brick was allowed to be flipped when tiling (the chirality problem).

But at least it already had a skeleton that could withstand the first round of review.

The second-year graduate student swallowed a mouthful of saliva, his voice trembling slightly: "Professor Gu, what on earth is this?"

Professor Gu Nanzhou did not answer him immediately.

[part:gemini-3.6-flash]

He took a deep breath, pulled out his phone from his pocket, and opened the camera.

His hands were somewhat stiff as he took a dozen photos in succession of the papers Jiang Lin had drawn on and the folded table.

After photographing the whole set, he focused the lens on the inconspicuous, hat-like 13-gon single tile on the first sheet, taking a standalone close-up.

Having done all this, Professor Gu Nanzhou placed his phone face down on the table, his eyes fixed on Jiang Lin.

"I need to confirm a few boundary conditions." Professor Gu Nanzhou's voice was serious as never before. "Does this tile require artificial color markings when actually tiling?"

"No."

"Does it require arrows drawn on the edges to indicate orientation?"

"No."

"All forcing properties rely solely on its own geometric polygonal boundary?"

"Rely solely on geometric boundaries," Jiang Lin answered.

"During the tiling process, is flipping the tile allowed?" Professor Gu Nanzhou asked one of the most crucial technical details.

Jiang Lin nodded. "The current proof version allows flipping; it includes a mix of left-handedness and right-handedness."

Professor Gu Nanzhou pondered for a moment. "Allowing flips is already astonishing enough for a single tile problem. As for the chiral version that doesn't allow flips, that's another, stronger requirement, which shouldn't be mixed into today's proof."

Speaking up to this point, he paused for a few seconds, his gaze becoming extremely complex.

To understand Professor Gu Nanzhou's sudden loss of composure right now, one must first know what kind of problem lay behind this tile.

It all started with something as ordinary as tiling a floor.

Taking a single shape of tile to cover a whole wall or floor is something everyone has seen.

Squares work, rectangles work, regular hexagons work—that's how honeycombs come to be.

What they have in common is that the tiled patterns repeat.

The pattern you see on this side of the room is identical to the one on the other side.

If you shift the entire pattern over by one unit, it fits perfectly back over itself.

This property, where shifting a certain distance allows it to coincide with itself, is called periodicity in mathematics.

For thousands of years, almost all floor tiles, wall tiles, and mosaics laid by humans have been periodic.

Repetition is the most natural, least effort-requiring state of this matter.

Until someone posed an odd question.

Can one find a way of tiling that can cover an entire infinite plane, yet never repeat?

Not by deliberately making a small irregular patch with extra effort, but such that even if you extend this pattern infinitely to the ends of the earth, you can absolutely never find two regions that align perfectly.

The entire pattern is similar everywhere, yet different everywhere, never yielding that neat overlap from a simple shift.

This kind of tiling is called aperiodic tiling.

The mere question of whether it existed or not was agonizing enough.

But what truly kept generations of mathematicians awake at night was not this problem itself, but a trickier variant of it.

You might think: what's so hard about not repeating?

I could just take a heap of broken tiles of various shapes, toss them randomly on the ground, one here, one there, topsy-turvy, and doesn't that lay out an endless, non-repeating messy pattern?

True, that indeed doesn't repeat.

But that is because you don't let it repeat.

It relies on your hands constantly watching over it, intentionally avoiding repetition, barely keeping it non-repeating.

The moment you let go, or if a careless person comes to lay them, it could collapse back into that neat, repeating form at any moment.

This non-repetition is fragile, requiring human supervision.

What mathematicians wanted was something else.

They wanted to find a few specific shapes of tiles whose very shapes forbade repetition.

With these tiles, even if you intentionally tried to tile a repeating, neat pattern, you couldn't do it.

Every time you tried to align them and make them repeat, the tile edges wouldn't interlock, leaving gaps that couldn't be filled.

The more you wanted it orderly, the more it forced you back onto that non-repeating path.

It didn't need you watching, nor anyone watching.

The rules weren't written on paper; they were grown into the edges of the tiles.

Shape is the command.

This was the true core of the aperiodic tiling problem.

And its most extreme, scalp-numbing version was that unavoidable question everyone circled back to:

Exactly what is the absolute minimum number of tile shapes needed to achieve an aperiodic tiling?

From the 1960s, when Wang tiles were proposed, generations circled around this question.

At first, to force a tiling pattern that could never periodically repeat required over twenty thousand different tile shapes.

Later, generation after generation pressed the number down—hundreds, then dozens.

Until 1974, when Penrose made an earth-shattering breakthrough by achieving aperiodic tiling with just two shapes of tiles.

Two.

A thick rhombus and a thin rhombus.

By relying solely on matching rules on their edges, they could cover the entire plane yet never assemble into a repeating pattern.

This was a monumental milestone already written into every textbook.

From then on, everyone knew where the next step lay.

Wasn't it just going from two tiles down to one tile?

Laymen hearing this would mostly think it was a logical, minor step forward.

Since it was reduced to two tiles, how hard could saving one more tile be?

However, the truth was quite the opposite.

A most counterintuitive catch was hidden here.

The fewer tile types there were, the harder it became—so hard that half a century passed without anyone finding it.

The logic wasn't actually convoluted.

The reason two tile shapes could lock down periodicity was that they constrained each other.

The thick rhombus and thin rhombus had to alternate and interlock according to rules; neither could leave the other. It was precisely this tension of mutual control that forced the entire pattern into non-repetition.

But what if only one tile shape remained?

It would have to control itself.

What a single shape feared most was end-to-end connection and self-replication.

If you duplicated it, translated it by a distance, and fitted it seamlessly, periodicity appeared.

Squares did this, regular hexagons did this, floor tiles and honeycombs all did this.

The vast majority of shapes you could think of, as long as they could cover a plane, could definitely tile periodically.

So the requirement of a single tile was essentially asking an almost paradoxical question:

Could one design a shape that could cover an entire infinite plane, yet be so tightly jammed by its own geometric boundaries that no matter how or who tiled it, it could absolutely never yield even a single bit of repetition?

It had to be sociable enough to fill every inch of space, yet solitary enough to forever refuse aligning with its own copy.

This was like crafting a key that could open every door in the world, yet was physically fated never to be duplicated by any door.

It had to be universal and unique at the exact same time.

Stuffing these two contradictory properties simultaneously into the few concave and convex boundaries of a single tile was where the difficulty lay.

Without a second tile to help constrain it, all the binding force had to be borne by this single tile's own shape.

Precisely because of this, this single step was delayed for half a century.

Professor Gu Nanzhou knew very well what half a century meant.

When he was doing his PhD, his advisor mentioned this problem, calling it a pit that could consume a person's entire life.

Several peers he knew had circled around it for twenty years, publishing a heap of peripheral papers, before finally daring to admit over drinks:

"I bet it exists, but I probably won't see it in my lifetime."

Even more people didn't even dare to bet that it existed.

Because even the direction of this problem was blurry.

Nobody knew if that tile actually existed.

If you spent ten years proving it didn't exist, that would still be an answer.

But if it clearly existed, yet you spent ten years concluding it didn't, that would be a catastrophe for an academic career.

Thus, most smart people chose to steer clear.

Yet the student before him used a pencil, a few sheets of A4 paper, and one afternoon...

...to draw it out.

And in passing, he deduced the three-step framework of why it was necessarily aperiodic right in front of a dozen people.

Professor Gu Nanzhou felt a surge of absurd dizziness.

He even subconsciously wondered if this kid had seen a similar construction in some preprint somewhere and recited it from memory.

But as soon as this thought surfaced, he snuffed it out himself.

If there really were such a preprint, given the scale of this field, it would have spread everywhere long ago; it was impossible he hadn't heard of it.

What's more, memory could recite diagrams, but not that table of local neighborhood forcings.

That table required working through every combination of angles one by one in one's head, categorizing and filing them before listing them out.

That wasn't memorized; that was calculated.

Looking at Jiang Lin, he asked the question that he himself felt was redundant.

"Student, do you realize what it is you just drew?"

Jiang Lin lowered his head, looked at the 13-gon with rigid lines, and answered softly, "A tile that cannot repeat."

Professor Gu Nanzhou almost choked on this answer.

A tile that cannot repeat.

A featherlight phrase of a few words.

But these words were a sentence that generations of mathematicians spent their entire careers without daring to easily write down. Yet from Jiang Lin's mouth, it sounded like describing a random pebble picked up by the side of the road.

"From the moment you step out of this classroom door, do not say this phrase casually to anyone outside."

Professor Gu Nanzhou gave him a deep look, a touch of warning in his tone.

Beside them, the PhD student was still staring intently at the folded grid paper on the table.

Having researched tiling for three years, he was the acknowledged number two person in Professor Gu Nanzhou's research group, able to deduce the undecidability of Wang tiles from start to finish.

So from the moment Jiang Lin drew the first 13-gon, his immediate instinct wasn't amazement, but nitpicking.

This was the most instinctual action of a well-trained researcher.

Local neighborhoods cannot be fully enumerated; the combinations explode exponentially, he thought.

How can infinite iterations of substitution rules guarantee convergence? he thought.

Could that scale argument in the third step quietly collapse on some boundary case? he thought.

He had even organized his first question into words, preparing to throw it out as soon as Jiang Lin finished speaking.

This was his strength; he had demolished many junior fellow students' reports with this trick during group meetings.

Then he saw that table.

Dozens of vertex neighborhoods, the validity, forcing classification, and death conditions of every single one were listed neatly.

The exponential explosion loophole he was about to ask about was casually plugged by a line reading 'finite set of angle resources'.

The boundary case he was preparing to press on already had a clear cross drawn on it in the lower rows of the table.

He opened his mouth.

Ultimately, he swallowed his words back down.

And realized an even more terrifying thing.

Before putting pencil to paper, this person had probably already anticipated every objection that someone like him, who had studied this for three years, could think of, and had resolved them all in advance.

He wasn't debating with you; before you even asked a question, he was already waiting for you at the finish line of the debate.

Hearing his advisor's warning words, the PhD student drew a sharp breath.

He understood the true weight of his advisor's sentence.

This wasn't out of fear that Jiang Lin would embarrass himself.

It was fear of trouble.

The academic community had always possessed an extremely brutal sense of priority for drafts of this caliber.

Whoever fixed the shape first, left the timestamp first, and wrote out the proof chain clearly first would hold the hard evidence in subsequent disputes.

A tile that shouldn't exist, once leaked, would instantly become prey targeted by countless eyes.

Before it was formally anchored and stamped with an indisputable name, any small movement could trigger uncontrollable consequences.

Someone might happen to be working on a similar construction; someone might coincidentally post a preprint first.

"This original manuscript must not be lost."

Professor Gu Nanzhou took a hard-shell folder from the side, carefully gathered the sheets of paper Jiang Lin had drawn on into it, and flattened them.

"I need to send a message to two colleagues right now."

Professor Gu Nanzhou didn't post on his Moments, nor in any academic exchange group with hundreds of members.

Instead, he opened WeChat and selected two contacts with extreme caution.

One was an old classmate from Nanjing University working on discrete geometry, and the other was a researcher at the Chinese Academy of Sciences studying quasicrystal and aperiodic tiling theory.

These were the two pairs of eyes with real weight in this field domestically that he could find in his urgency.

His finger hovered above the screen, pausing for two seconds.

He knew what sending this message meant.

He had always been known for his steadiness before those two, never exaggerating, let alone disturbing others over unconfirmed matters.

For these two contacts, he usually felt even forwarding conference announcements was bothersome and hadn't initiated contact with them for over six months.

Because of this, what he was about to send had to match the weight of this interruption.

He deleted his first draft of wording, then deleted the second draft.

What remained at last was so brief it looked like an encrypted telegram.

[Are you free tonight? Let's have a call. Something that shouldn't exist—a tile—might have appeared in my hands.]

Sent.

Setting down his phone, Professor Gu Nanzhou's fingertips trembled slightly.

Having done research for twenty years, for the first time, he felt the tangible sensation of personally pushing open a door.

Even though the person pushing open the door was actually not him.

Soon, the screen lit up on the desk.

The classmate from Nanjing University replied extremely quickly with just three words.

[Which tile?]

Professor Gu Nanzhou stared at those three words for a few seconds.

He could imagine the other party's expression at this moment.

In this circle, the phrase "a tile that shouldn't exist" required no explanation; everyone knew which problem it referred to.

The other party's "Which tile?" wasn't asking what tile it was, but rather, "Are you crazy? Do you know what you're saying?"

Almost at the same time, the one from the academician's team replied as well, even shorter than the previous message.

[Where is the person?]

Professor Gu Nanzhou naturally understood the meaning behind those words.

In the face of a breakthrough of this caliber, the first thing to confirm was never authenticity.

Authenticity could be verified slowly.

It was the person.

Who was it, where were they, were they safe, and would someone else reach them first?

He placed his phone face down on the table, turned his head, and looked at the student by the long table.

At this moment, no one had the mood to discuss Penrose's two rhombuses anymore.

Those two rhombuses that had once been revered as classic entryways and filled half the whiteboard were still quietly remaining on the projection screen.

But no one spared them another glance.

They were like a monument of an era, quietly retreating into the background in front of this sudden 13-gon, becoming a former answer.

The classroom was so quiet that one could hear the hum of the projector fan.

Among the dozen or so people, not a single one spoke.

They looked at that bizarre polygon with rigid lines on the paper, feeling a sensation hard to describe.

It was like watching with their own eyes an object that should have lain in textbooks decades in the future fall ahead of time onto this paint-chipped long table before them.

And the person who dropped it was currently putting his pencil back into his pencil case quietly, his actions as calm as if he had just finished an after-class exercise.

The PhD student's gaze fell on Jiang Lin's face.

That face was excessively young, its jawline still carrying the soft roundness of youth, and his thick black hair hardly looked like that of a researcher who regularly stayed up late.

No matter how one looked, he didn't seem like someone who had spent even a year in this pit.

He opened his mouth and asked the question everyone wanted to know most at this moment:

"Student... which department are you from?"