93: Chapter 93 Mathematics Doesn't Require an ID Card

Professor Lu Zhixing's words made C216 quiet for a moment.

Professor Gu Nanzhou looked at him, and then glanced at Jiang Lin.

Only at this moment did his brain seem to be forcibly dragged back to the Real World from the frantic mathematical deduction, beginning to realize an absurd detail.

For the past full hour, the attention of everyone in the C216 discussion room had been pinned to those few pieces of draft paper as if adsorbed by a magnet.

They verified the edges, checked the vertices, exhausted the local topological structures, and even frantically constructed hierarchical structures on the two-dimensional plane in their minds.

That was a pure mathematical world; in that world, there was no age, no educational background, only logic.

Yet they had not truly confirmed Jiang Lin's identity.

"Professor Lu, let me confirm once more, is Jiang Lin currently a master's or doctoral student in your School of Physics?"

Professor Lu Zhixing fell silent for a fraction of a second.

He knew all too well what these words meant to Professor Gu Nanzhou and the entire mathematical community.

Looking straight into Professor Gu Nanzhou's eyes, he said, "Neither."

Professor Gu Nanzhou was momentarily stunned, then furrowed his brow tightly, a list of Jiangda University's Top-Notch Talent Training Program flashing through his mind: "An undergraduate, from the gifted youth class?"

"Neither."

Professor Lu Zhixing's reply remained brief, but every word felt like a heavy hammer.

Chen Yan subconsciously raised his head, his eyes filled with confusion and shock.

Professor Gu Nanzhou's gaze gradually fell back onto Jiang Lin.

Professor Lu Zhixing took a deep breath and finally spoke the answer that was enough to tear apart common sense: "He is a high school student at Jiangcheng No. 7 High School, and the Gaokao is a little over a month away."

As these words fell, C216 was as if someone had pressed the mute button.

Chen Yan's lips moved, seemingly wanting to smile, but his facial muscles completely refused to obey, and in the end, he only repeated in a low voice: "A high school student?"

What was this supposed to be?

A senior high school student who was reviewing Five Years of Gaokao and Three Years of Simulation ran into the professional reading class of the Department of Mathematics at Jiangda University, and casually drew an aperiodic monotile that might solve the sixty-year-long unresolved Einstein problem in the field of discrete geometry?

Professor Gu Nanzhou's first reaction was not questioning.

Because if it were questioning, he would have already questioned it when he saw that local neighborhood table.

His reaction was to look at Jiang Lin's face anew.

Just now he had been looking at the paper, the edges, the corners, and that table implicitly expressed in the language of algebraic topology.

Until this moment, he truly noticed that the student sitting in this discussion room auditing, who had used an ordinary Zhonghua pencil to draw that thirteen-sided polygon, indeed still carried the contours of a youth that had not fully faded from his face.

As expected, geniuses emerge in youth?

Or did the God of Mathematics decide to play an absurd joke at Jiangda University today?

Professor Gu Nanzhou took a deep breath, forcibly suppressing the churning huge waves in his chest, as reason told him that now was not the time to be shocked.

He turned around and said to the surrounding students who were still watching the commotion and whose brains had similarly crashed from the shock: "Classmates, today's reading class has already ended, everyone go back and digest the content yourselves.

As for this diagram and this polygon, do not mention it after stepping outside.

Before undergoing strict formal verification, this is still only a thought experiment."

The students looked at each other in dismay, but since the mentor had already dismissed them, and his tone carried unquestionable majesty, they could only pack their schoolbags one after another and leave.

Someone walked to the door, and could not help turning back to take a look at the youth who seemed indifferent to all the surrounding shock, hesitant to speak, but ultimately saying nothing.

Soon, only four people were left in the slightly spacious classroom: Professor Gu Nanzhou, Chen Yan, Jiang Lin, and Professor Lu Zhixing.

Professor Gu Nanzhou turned around, his gaze having recovered the calmness of a backbone of the Department of Mathematics at Jiangda University.

He said: "Put the matter of identity aside, verify the tile first."

After the two words "verify the tile" were spoken, that dizzying shock floating in mid-air in the room finally fell back to the ground, finding a fulcrum of gravity.

Right.

Verify the tile first.

Whether a senior high school student, an undergraduate, or a fields medal winner.

Mathematics does not check ID cards, nor does it check household registration books.

It only looks at whether your logical chain is closed, and only looks at whether you have completely solved the problem.

Professor Gu Nanzhou took out his mobile phone.

Messages from Nanjing University popped up one after another.

[Professor Gu Nanzhou, let me ask you seriously, did you see an unpublished rough construction on a preprint website and get excited without checking it yourself?]

In this circle, the more astonishing the news, the more one must first assume it is fake.

Ever since Hao Wang proposed Wang tiles in 1961 and conjectured that any tile set that can tile the plane can tile it periodically, this field had witnessed too many hopes and disillusionments.

Berger proved Hao Wang's conjecture wrong and provided the first set of aperiodic tiles, but that required 20,426 tiles.

Later, Robinson reduced it to 6 tiles.

And later, Roger Penrose used kites and darts to reduce it to the famous 2 tiles.

For decades, top brains all over the world had been searching for that one tile.

A singly connected tile that can and can only tile an infinite two-dimensional plane in an aperiodic manner.

Socolar and Taylor found one in 2010, but that tile was disconnected, or required complex matching rules, and was not perfect in pure geometric topology.

Now, someone said they found a pure single polygon without any additional matching rules?

The other party sending this message was not questioning Professor Gu Nanzhou's vision, but backing him up.

In case it was a misjudgment, stopping now was still in time, so as not to alarm the upper line and end up in an awkward finish after holding a formal seminar.

Professor Gu Nanzhou's fingers tapped rapidly on the screen, replying with four words.

[Drawn on the spot.]

The other end fell silent for a few seconds, as if one could feel the sound of the other party sucking in a breath of cold air along the radio waves.

[Your student?]

Professor Gu Nanzhou glanced at Jiang Lin.

[No, a nominal student brought by a young PI from the neighboring School of Physics.]

While speaking, Chen Yan had already connected the meeting room's projector and ran sweating profusely to the neighboring teaching and research office to borrow a high-precision scanner.

He carefully, as if treating a unique copy, scanned the few pieces of paper Jiang Lin had just drawn into the computer page by page.

Chen Yan looked at the screen, his hands shaking.

The file name was temporarily requested by Professor Gu Nanzhou to be named: pre_check_jiang_scan.pdf.

"Right now it is still only a candidate construction; before the complete proof and review are finished, do not call it a theorem, nor tell the outside world that it has been solved."

Professor Gu Nanzhou explained the temporary naming issue to Jiang Lin, his tone carrying a sense of protection.

Professor Lu Zhixing indicated no problem; he understood these rules.

Jiang Lin was even less likely to say anything.

He was not even a university student; he was not quite clear about the rules and unwritten rules of the academic circle regarding preprints and candidate constructions.

He simply stood there, watching the geometric structure he had deduced countless times in his mind magnified on the projection screen.

A dozen minutes later, the encrypted video conference was connected.

The projection screen was evenly divided into left and right halves.

On the left appeared a man of about fifty years old, with graying hair and thick rimless glasses, against the background of an office piled high with foreign journals, printouts, and various polyhedron models.

Professor Gu Nanzhou introduced in a deep voice: "Jiang Lin, Professor Lu, this is Professor Lin Zhaoye, Department of Mathematics at Nanjing University, an authority on discrete geometry and combinatorial topology."

On the right was a woman in her forties with capable short hair and black-framed glasses.

On the whiteboard behind her, several sets of quasicrystal diffraction patterns and dense projection relationships still remained.

"Professor Shao Mingtang, Institute of Theoretical Physics, Chinese Academy of Sciences, an expert on quasicrystals and aperiodic material structures."

These were the two absolute experts in the discrete field that Professor Gu Nanzhou had urgently invited via text message earlier.

Originally, such a level of informal meeting was extremely difficult to gather, but these two big shots, after preliminary communication and looking at an incomplete local diagram Professor Gu Nanzhou had photographed and sent over, could no longer wait for tonight's routine call.

They had turned down their evening meetings, wanting only to take a look at this legendary monotile as soon as possible.

After the two came online, Professor Gu Nanzhou extremely concisely introduced Jiang Lin and Professor Lu Zhixing.

Neither end of the video had any redundant pleasantries with each other.

At their level, nonsense was a desecration of time.

Professor Lin Zhaoye looked at the screen, and five minutes later, he slowly spoke: "Is this the tile you mentioned, which supposedly should not exist on a two-dimensional plane?"

Professor Gu Nanzhou nodded: "Candidate, currently only a candidate."

Professor Lin Zhaoye nodded: "Good, let's look at it as a candidate first; are the edge angles all integer multiples of π / 6?"

Jiang Lin stood by the table, his tone very calm: "Yes, all angles fall on the same set of 30° auxiliary grids, and the side lengths take only a limited number of categories of ratios. What truly needs to be verified is not whether it looks visually similar, but whether this set of finite angular shapes can be closed into a complete local state table."

"Whose hands is the original manuscript in right now?" Professor Shao Mingtang interjected.

Professor Gu Nanzhou pressed his hand on that hardcover folder: "Here, Professor Lu Zhixing is also present as a witness."

Professor Shao Mingtang's tone was unquestionable: "Do not make it public first, do not post it in WeChat groups, do not post it in Moments, do not leave traces on any social media, and even less let those students from just now take photos and spread them outside. If this thing is true, this is one of the greatest breakthroughs in geometry this century. If it is fake, leaking the news in advance will make the Department of Mathematics at Jiangda University become an international laughingstock."

Professor Gu Nanzhou glanced at Chen Yan.

Chen Yan immediately turned his phone face down on the table like he was electrocuted, screen facing down, and even raised his hands to signal: "Understood, absolute confidentiality."

Professor Lin Zhaoye pushed up his glasses, leaning forward as if about to drill out of the screen: "Let's begin, don't be in a hurry to discuss whether it can resist periodicity yet, let's go step by step. The first step, first prove that it is not a dead end you casually doodled."

Professor Gu Nanzhou tapped the keyboard, enlarging the few splicing diagrams composed of single tiles that Jiang Lin had just drawn.

Professor Lin Zhaoye stared at those gear-like interlocking edges and asked the most fundamental question: "Can it tile the plane?"

In mathematics, tiling means covering the entire R2 plane without any gaps and without any overlaps.

"Yes."

Jiang Lin stood by the table, his expression still without fluctuation.

"How to tile?" Professor Lin Zhaoye pressed on, his gaze extremely oppressive, "You cannot just show me a local pretty pattern and tell me that it can extend to infinity; I need a construction method."

Jiang Lin walked forward, pointing to the large-scale structures he had labeled as four categories on the diagram.

"These four categories of large blocks, I temporarily call them Type I (H), Type II (T), Type III (P), and Type IV (F). They are essentially super-tiles pieced together from different quantities and orientations of the same single tile. And the core of tiling the plane lies in the fact that each category of large block can be further composed of smaller Types I, II, III, and IV scaled according to strict proportions."

Professor Lin Zhaoye furrowed his brow slightly, quickly constructing algebraic relations in his mind: "Meaning, large tiles are pieced together from small tiles, and small tiles can form even larger tiles? A recursive substitution system?"

"Yes, its expansion coefficient is not a simple integer."

Jiang Lin replied, "This means that during high-order synthesis, the boundaries will produce extremely complex non-linear interlocking."

Professor Lin Zhaoye did not immediately say good.

He was too experienced, and quickly pointed out the trap within: "Layer nested within layer, capable of infinite magnification, holds true algebraically, but that does not equal being able to cover the entire infinite plane geometrically and topologically. You may have simply constructed an expanding isolated island, and outside the isolated island, there exists a vacuum it cannot reach."

Jiang Lin nodded, obviously having thought about this problem long ago: "Yes, infinite magnification only proves that for any given positive integer n, the nth-order super-tile exists and can be seamlessly spliced from basic single tiles. But in order to cover R2, it must also be guaranteed that the subsequent layer (the (n+1)th order) geometrically completely encompasses the previous layer (the nth order), and the scale measure of this containment diverges to infinity as n → ∞."

"So?"

Professor Shao Mingtang interjected on the other end of the video.

"Therefore, we can utilize the principle of compactness in topology."

Jiang Lin looked at the whiteboard diagram on the screen, his speaking pace neither hurried nor slow.

"Under this construction rule, if you randomly pick any coordinate point p(x, y) on a two-dimensional plane, as the hierarchy n increases, the boundary of the nth-order large block will expand outward exponentially. Sooner or later, this point p will fall into the interior of a certain nth-order large block. No point on the real number field can escape the coverage of this limit. Only in this way can the entire plane be considered thoroughly covered, without resulting in a morbid topological structure that drifts further and further off and misses distant areas."

Professor Lin Zhaoye stared at that intricate substitution diagram for a full ten minutes.

His brain raced frantically, trying to find loopholes in this set of recursive logic, but he found that, at least intuitively, the covering closed-loop given by this young man was self-consistent.

"Remember this step for now. I temporarily accept the existence of your coverage," Professor Lin Zhaoye said, his tone still stern. "However, being able to provide an infinite tiling method by no means equals that it cannot degenerate into a periodic tiling method. Many tiles can be tiled both aperiodically and periodically; what we want is forced aperiodicity."

Shao Mingtang took over the conversation, hitting the nail on the head: "Nor does it mean that all tiling methods will obediently grow according to your hierarchical structure. Perhaps in some corner of the universe, this tile can fill the plane periodically in a way you never expected, one that does not comply with these four types of large block rules."

Professor Gu Nanzhou stood aside, not speaking up for Jiang Lin, nor did Professor Lu Zhixing.

At this moment, no one should speak up for him, nor could they.

In the courtroom of pure reason, logic itself was the best defense attorney.

Jiang Lin nodded to himself, as if everything was within his calculations: "Therefore, the second step is forcing."

Professor Lin Zhaoye narrowed his eyes: "Speak plainly, don't pedantically quote books."

Jiang Lin thought for a moment, gave up using the complex Turing machine state transition metaphor, and switched to a more fundamental explanation.

"The first step is me telling you that this tile can be tiled this way. This is merely existence."

"The second step is for me to prove that, whether you like it or not, once you put this tile on the table and begin splicing outward, in the end, through the concave-convex interlocking of local boundaries, it will be forced—and can only be forced—to form the macroscopic structures of the four types of large blocks I provided."

Professor Lin Zhaoye finally began to nod: "Continue."

Jiang Lin pushed that densely packed local neighborhood table next to the scanner, and Chen Yan flurriedly cooperated, enlarging the page on the projector.

An extremely complex table appeared on the screen.

Every single row was a geometric configuration of local splicing.

Left side: The posture of other single tiles that might be connected around a specific notch or convex corner according to internal angle requirements.

Middle: Indicating whether the combination is purely geometrically valid (no overlaps, no gaps).

Right side: Indicating which type of large block the valid combination will eventually be forced into when extending outward, or at which step it completely hits a dead end.

Professor Lin Zhaoye glanced at the number of rows in the table and cut straight to the point: "How do you know you have listed them all?"

In proofs of aperiodic tiling, the most feared thing is omissions in exhaustive enumeration.

If you missed a valid tiling method, and that tiling method happened to lead to a trivial periodic structure, then all the preceding macroscopic edifices would collapse in an instant.

Jiang Lin answered without hesitation: "I didn't go about listing infinite planes."

"What do you mean?" Professor Lin Zhaoye's brow furrowed deeper.

"The tiling methods of an infinite plane are an uncountable set; they can't be fully listed," Jiang Lin said. "But what I listed are the local cases of that circle around the vertex."

He picked up a pencil and drew an extremely sharp notch on a blank piece of paper.

"The angles that this single tile can contribute are all integer multiples of 30°. When we make a full circle around any vertex on the plane, the sum must be 360°, which is exactly twelve 30° sectors."

Jiang Lin's pencil tip dotted twelve graduations on the paper.

"Therefore, surrounding a vertex, all possible angle combinations of single tiles that can be pieced together form a strict finite set. This is a simple combinatorial mathematics problem, not infinitely many types."

He quickly drew a few more connection methods, their gear-like edges interlocking on the paper.

"Some combinations can be completely connected locally."

"But some combinations look like they are connected at this vertex, yet as long as you push outward one more step and add a circle of tiles, an extremely bizarrely shaped gap will pop up in the secondary neighborhood—a fatal flaw whose area is not enough to cram even half a single tile."

He drew a heavy cross over that hypothetical gap.

"This kind is a dead end. It deceives you locally at the first layer, but at the second layer, it will collapse due to the exclusivity of geometric space."

While listening spellbound, Professor Lin Zhaoye mentally checked the number of combinations of the twelve sectors mentioned by Jiang Lin.

Jiang Lin continued to output his logic: "Therefore, what we need to verify is not the entire infinite plane, but all possible local vertex situations (finite sets). When each legal local situation grows outward by one step, either it will logically become the internal core of those four types of large blocks, or it will die immediately without any intermediate state."

He paused for a moment and added a computer science concept: "This is the same thing as the logic of pruning and cutting off dead ends in search algorithms."

Professor Lin Zhaoye had not heard of Jiang Lin's personal searcher or whatever, but he understood computational complexity theory.

"In other words, you utilized the discrete angle constraints of the vertices to compress the infinite tiling space into a finite, computable local situation state table?"

"Correct," Jiang Lin replied.

"A finite table is theoretically feasible, but it fears two things most. One is missing items due to human calculation errors, and the other is misjudging dead ends. You think it's dead, but actually, by flipping or translating another tile, it can come back to life."

Shao Mingtang spoke up regarding the technical details for the first time.

Jiang Lin nodded, agreeing with this rigorous skepticism: "Therefore, for every dead end marked in the table, I have written clearly which step it specifically died at and which angle conflict caused its death."

Chen Yan swallowed hard, moved the mouse, and zoomed in on the seventeenth row of the table.

Next to that row was drawn an extremely distorted notch, an abrupt convex edge, and a crimson red cross.

Professor Lin Zhaoye stared fixedly at the seventeenth row on the screen: "This row, where exactly does the problem lie?"

Jiang Lin took just one look at that familiar topological map: "It satisfies the internal angle sum equaling 360°. Locally speaking, it is indeed valid."

"Oh?" Professor Lin Zhaoye raised an eyebrow.

"It can indeed close perfectly on the first vertex, and all the polygon angles fit together." Jiang Lin redrew that connection method by hand on the blank paper, the lines uncommonly smooth. "So, if you only look at this central point, you can't pick out any faults; the algorithm will return True at this node."

As he spoke, Jiang Lin's pen stroke turned, and he began to supplement the second circle along the periphery.

"However, when we extend one step based on this legal core, the restraining force of the outer boundary begins to take effect. Here, a space boxed in by several borders will be forced out."

He heavily circled a sharp corner in the upper left, and then tapped that space with the tip of his pen.

"Due to the rigid structure of the surrounding tiles, this reserved angle is jammed at 150°, but its edge contour requires filling in a shape with 90° and 60° protrusions. No matter how we rotate this tile of ours, it is impossible to simultaneously satisfy these two constraints, and no other tile can be filled in."

Chen Yan listened with cold sweat pouring down beside them, and subconsciously blurted out: "So, it doesn't present a contradiction on the spot; it is only after extending one or two steps that its topological irreconcilability is exposed?"

"Correct." Jiang Lin glanced at Chen Yan. "This is also why many single tiles look like they can tile flat during preliminary testing, but collapse very quickly. Therefore, you cannot just verify an absolute isolated vertex; you must look at what mandatory conditions its local neighborhood as a whole will force outward when growing a circle outward."

On the other end of the video, Professor Lin Zhaoye's fingers tapped gently on the desktop, making clacking sounds.

Every single sound felt like it was tapping on the hearts of Chen Yan and Professor Gu Nanzhou.

After a full ten seconds of thought, Professor Lin Zhaoye slowly spoke: "This explanation is logically rigorous and can be written into the appendix of the paper as a core lemma."

As soon as these words came out, several people in C216 involuntarily let out a huge sigh of relief.

Chen Yan even felt his legs go weak for a moment.

Being able to be written into the appendix did not mean passing the final test.

But at least it showed that in the eyes of top experts, Jiang Lin was not talking nonsense whimsically. His proof framework was orthodox and met the standards of modern mathematical analysis.

However, Shao Mingtang did not let him off easily.

As a top scholar in the field of quasicrystals, she had seen too many models that were flawless on the microscopic level but ran into huge problems in macroscopic assembly.

"Vertex local forcing can indeed suppress the degrees of freedom; for now, I'll count it that you have a clear train of thought." Shao Mingtang's gaze pierced through the screen, nailing sharply onto Jiang Lin. "But I care about another more important matter."

"What?" Jiang Lin raised his head.

"You said all tiling methods will be forced to be divided into these four types of large blocks, but this is only the 'expansion' process. If we reverse it, pulling back a plane that has already been tiled to perform a reverse synthesis, is this division unique?"

As soon as these words came out, Professor Lin Zhaoye, who was just tapping the desk, changed his gaze slightly.

This truly was a major problem.

Pulling back the massive number of small tiles that had been tiled and merging them into large blocks of the previous layer—this action is called composition or inverse substitution in the mathematical language of aperiodic tiling.

If a few single tiles in the same area could sometimes be partitioned and synthesized as part of a Type I large block, and sometimes logically interpreted as part of a Type II large block, then the so-called strict hierarchy would become completely blurred.

Once the hierarchy is blurred, it means it loses its unique algebraic encoding.

The cornerstone of subsequently using hierarchical infinite expansion to proof by contradiction the existence of periodicity would collapse suddenly.

Yet Jiang Lin did not immediately follow Shao Mingtang's words to say unique.

He stated a fact extremely calmly: "Which type of large block each tile on the bottom-layer boundary belongs to is not completely unique."

Chen Yan abruptly raised his head, his heartbeat halting instantly.

Not unique?

Wouldn't that mean it's completely over?

Professor Lin Zhaoye also narrowed his eyes dangerously, as if evaluating whether this young man was playing with fire.

Shao Mingtang's voice rose by a notch: "Without uniqueness, your hierarchical tree will branch out, and proof by contradiction for periodicity is completely out of the question. How can your subsequent proofs proceed?"

"Because what I wanted in the first place was not for the attribution of every bottom-layer small tile to be absolutely unique."

Jiang Lin picked up a pen and drew two rows of staggered small tiles on one of the extremely winding large block boundaries projected out.

"Look here," he pointed at the screen. "In the marginal zone where two large blocks meet, there do exist a few tiles that can either be classified into the left large block or classified into the right large block, which creates ambiguity locally."

"But this is merely extremely limited wandering back and forth on the boundary line."

Jiang Lin used a thick black marker to brutally draw a macroscopic boundary line running through front and back right in the center of those two rows of small tiles—a skeletal line belonging to a higher-order hypergraph block.

"It can never penetrate into the core of the large block. It will absolutely not change the topological skeleton of the higher layer."

To make pure theory more concrete, Jiang Lin used a straightforward metaphor.

"Just like a dirt road between two villages, a few stones scattered by the roadside—you can say they belong to the left village, or argue for the sake of taking advantage that they belong to the right village. Locally, ownership is ambiguous."

He tapped the thick line with his pen.

"However, regardless of who these few broken stones belong to, the major road dividing the two villages still solidly lies right there. Where this macroscopic boundary line is will not undergo any displacement due to the attribution dispute of these few microscopic stones."

Although this metaphor was coarse, in the context of algebraic geometry, it was extremely precise.

Shao Mingtang understood immediately, and a bit of appreciation appeared in her eyes.

[part:gemini-3.6-flash]

She reframed Jiang Lin's colloquial explanation into rigorous academic terms: "So what you need to prove is that the action of synthesizing the topological skeleton itself is uniquely determined. Regardless of which local boundary tile you start pulling back from, even if there is ambiguity at the microscopic level, the resulting macroscopic hierarchical graph will be isomorphic. As for whether individual edge tiles belong to the left or the right, at the infinite limit scale, their measure is zero, so it doesn't matter."

Jiang Lin nodded. "Correct."

Shao Mingtang pressed further to confirm: "Rather than requiring the attribution mapping of every single tile to be an injection?"

"Right. Demanding an injection would be asking too much and would be an unnecessary over-proof," Jiang Lin answered decisively.

Shao Mingtang leaned back in her chair and let out a long breath. "This framing is far more important and insightful than your dry proof by contradiction earlier."

Professor Lin Zhaoye nodded heavily on the other side. "The uniqueness of the synthesized skeleton is the foundation of foundations. If the foundation is shaky, your entire proof by contradiction crumbles. But as long as the skeleton is unique, the periodic proof by contradiction has a solid footing. This step must be pulled out separately and clearly written as its own section."

Professor Gu Nanzhou immediately picked up his pen and quickly made a note on his memo pad.

[Core Key]: Prove the uniqueness of the synthesized skeleton, rather than the uniqueness of the attribution of underlying single tiles. Introduce the concept of equivalence classes for a rigorous formulation.

Standing to the side, Professor Lu Zhixing only half-understood it since it was outside his field, but he understood the philosophical implication for science behind those words.

Jiang Lin didn't stubbornly dig in his heels.

Facing the doubts of top experts, he didn't try to use ornate language to cover up the minor flaws in his model.

He openly admitted the presence of local ambiguity, then elevated the dimension of the discussion, reframing the core objective into a grander, more precise, and far more defensible skeleton isomorphism.

This was what doing real, top-tier research looked like.

Doing research isn't about packaging a problem to make it look flawless without a single defect.

It is about having the courage to lay the bloody flaws bare on the table, and then using impeccable logic to prove that they are not fatal.

Having resolved the crisis of synthesis uniqueness, the review entered deep waters.

Shao Mingtang pushed up her black-rimmed glasses and raised another question: "When tiling, is this single tile allowed to be flipped over?"

In aperiodic tiling, if reflection of tiles is necessary to complete a full tiling, it is called weakly aperiodic.

If it can tile space using only translations and rotations—like the legendary Spectre tile—then it is purely, strongly aperiodic.

Jiang Lin answered truthfully: "The current version I constructed allows flipping; it requires its reflection."

"Then write that in bold black text on the very first line of the first page," Shao Mingtang instructed solemnly. "Allowing reflection versus forbidding reflection are two completely different symmetry group propositions in group theory. The forbidden-reflection version is stronger and much harder. Whatever you do, don't let people misunderstand and think you were arrogant enough to casually solve the stronger proposition too, or you'll trigger an all-out academic siege."

"Understood," Jiang Lin nodded, taking the advice to heart.

"Also, given that reflections are allowed..." Shao Mingtang's speech slowed down suddenly, as if groping around for a hidden trap in the dark, "I'll grant for now that the overall tiling lacks global periodicity. But is there a pathological possibility: on an infinite plane, along a specific direction vector v, there exists a strip of finite width but infinite length, and the pattern inside this strip repeats periodically?"

As soon as she said this, the air in the conference room seemed to drop to the freezing point.

This instantly made Professor Gu Nanzhou think of the common hazard of long strips in aperiodic tilings.

Many aperiodic tilings do not repeat globally, but if you isolate a single narrow strip, you'll find that like an endless conveyor belt, it secretly repeats along a certain dimension.

Jiang Lin immediately asked back: "Do you mean that while the overall pattern is aperiodic, a semi-periodic translationally invariant strip might be hiding along some obscure direction?"

Shao Mingtang's brows twitched sharply.

She clearly hadn't expected Jiang Lin not only to instantly grasp her implicit point, but even to reply so quickly, summarizing her deepest concern right onto the topological point in a single sentence.

"Yes. Just because the full tiling lacks two-dimensional translational symmetry doesn't mean it's clean in one dimension," Shao Mingtang said gravely. "If such a periodic strip happens to be lurking inside, a fatal flaw will be carved into your overall proof by contradiction, and future applications in crystallography will carry a massive question mark."

Jiang Lin fell into deep thought.

This time, he stayed silent for a full ten-plus seconds.

Room C216 was dead silent; everyone was waiting for him.

Then, Jiang Lin raised his head and looked directly at the screen. "Regarding this part, I currently only have an intuitive framework in my mind; I haven't written it down in rigorous mathematical prose yet."

Professor Gu Nanzhou's heart tightened, and he shot a sharp look at him.

Jiang Lin remained unruffled and continued, "By proof by contradiction, suppose there really is such an infinite periodic strip of width W. Then the boundary contours on both sides of this strip must also repeat periodically; otherwise, it wouldn't be able to mesh with the surrounding tiling."

"However, please recall the substitution rules from earlier."

Jiang Lin pointed to the substitution diagram.

"For our four classes of macro-blocks, every time they synthesize up by one level, the string of geometric notch-and-protrusion markings recorded on the boundary gets stretched proportionally. Meanwhile, due to the combination rules of higher-order macro-blocks, new markings are forcibly inserted into the middle."

Jiang Lin paused, employing a subtle interdisciplinary metaphor.

"You can imagine the notch-and-protrusion markings on this boundary as a code sequence generated by a specific generative grammar."

"With each step upward, following the same set of nonlinear substitution rules, this code string expands, lengthens, and absorbs new symbols. This is extremely common in symbolic dynamics."

"For such a code sequence generated through recursion and non-integer inflation factors to maintain global periodic repetition would be equivalent to requiring it to align perfectly across a fixed translation interval T."

Jiang Lin's tone gradually sharpened.

"However, this part cannot rely merely on intuition. I need to isolate the boundary marking sequence and use substitution matrices alongside symbolic dynamical systems to prove it lacks eventual periodicity."

"Therefore, this code sequence cannot possess periodicity. By extension, that so-called semi-periodic strip simply has no room to exist within this framework."

Shao Mingtang looked at Jiang Lin on the screen in shock.

This kid was naturally drawing upon concepts from symbolic dynamics and one-dimensional substitution sequences to solve a two-dimensional topological problem.

But she quickly recovered and spoke bluntly: "Yet this is still just a very pretty verbal argument."

"Yes, this is physical intuition and an algebraic conjecture," Jiang Lin admitted without reservation. "I need to dedicate a separate section to strictly derive line by line the aperiodicity of that boundary code, using matrix eigenvalues and substitution rules."

Professor Lin Zhaoye clapped his hands hard on the left screen. "Excellent!"

Chen Yan was startled beside him.

This was considered excellent?

Jiang Lin clearly said he hadn't finished writing it, so how was it excellent?

Professor Lin Zhaoye glared at Chen Yan through the screen, as if reading his mind.

"Him admitting he hasn't finished writing it is ten thousand times better than blabbing 'it's trivially non-existent' just to save face. What mathematics dreads most is arrogant hand-waving. Knowing where the gaps are and knowing what tools to use to bridge them—that is top-tier research acumen."

Chen Yan immediately shut his mouth, breaking into a cold sweat.

Professor Gu Nanzhou firmly added a line to his memo: [Top Priority To-Do]: Exclude semi-periodic strip (worm) issue. Line of reasoning holds (based on symbolic dynamics and substitution matrix eigenvalues); urgently need to complete algebraic derivation text.

Now came the final stage of verification.

The ultimate crown of the entire aperiodic monotile theory—the periodic proof by contradiction.

Professor Lin Zhaoye took a deep breath, his tone becoming solemn like never before. "Jiang Lin, when you were discussing coverage earlier, you said that higher-level macro-blocks get larger in scale, approaching infinity. Thus, any fixed translation vector T cannot preserve the structure across all levels. Intuitively, yes, because the finite cannot contain the infinite. But in a rigorous proof, you cannot jump so quickly."

"Why?" Jiang Lin asked humbly. This was a relatively weaker link for him in pure mathematical proof.

"Because you are missing an extremely critical link in the logical chain," Professor Lin Zhaoye said, cutting precisely to the core like a scalpel. "You must first prove a lemma: assuming there exists a translation vector T that allows the entire tiled plane to overlap perfectly with itself, then this translation T must simultaneously and unconditionally preserve the macroscopic skeleton of every macro-block level."

Professor Lin Zhaoye's words rang out with absolute authority.

"Otherwise, a reviewer will sneer and ask you: why can't it preserve the translational symmetry of only the microscopic tiles at the bottom while shattering the macro-block boundaries into pieces at the macro scale? Who mandated that microscopic periodicity must imply macroscopic periodicity?"

Jiang Lin instantly understood this logical gap.

He nodded, a spark of clarity flashing in his eyes. "So the uniqueness of the synthesized skeleton that Professor Shao pushed me to admit and establish earlier is actually a necessary prerequisite here?"

"Yes, exactly," Professor Lin Zhaoye said, his eyes beaming with intense enthusiasm.

The fragments in Jiang Lin's mind quickly clicked together, and he articulated it in the simplest yet logically airtight manner: "If the entire infinite tiling, after being translated by vector T, remains identical to its original state..."

"...then due to the uniqueness of the synthesized skeleton, reading backward from this full tiling according to local forcing rules to reverse-generate the hierarchical skeleton graph must also yield an identical isomorphic graph after translation."

"Because this skeleton structure isn't some arbitrary auxiliary line I drew as a subjective observer."

"It is forced out by the intrinsic geometric properties of the tile itself; it is the unique, legitimately read intrinsic structure from the underlying picture."

"Since we proved synthesis uniqueness earlier, it mathematically guarantees this. If translation doesn't shift the base tiling, it inevitably cannot shift its skeleton tree either."

"Once the skeleton tree is forced to exhibit translational symmetry, and since the scale of the skeleton inflates to infinity with level n, a finite translation vector T attempting to translate a skeleton larger than its own scale will inevitably yield a misalignment contradiction. Q.E.D.!"

On the other end of the video call, Shao Mingtang couldn't help but lightly applaud. "The logic in that statement has come full circle."

Professor Lin Zhaoye took off his thick glasses and wiped away tears of excitement from the corners of his eyes. "It holds logically, but it still needs to be codified into an airtight written proof."

"I will formulate it in formal language," Jiang Lin solemnly promised.

Professor Gu Nanzhou noted down the final line on his memo pad: [Ultimate Theorem Dependency]: The foundation of the periodic proof by contradiction lies in -> The hierarchical skeleton can be uniquely read out (synthesis uniqueness). Logical chain closed.

Discussion Room C216 had entered a state of white-hot review.

No one cared about time; no one cared about fatigue.

No one asked anymore whether Jiang Lin was a high school senior.

Nor did anyone agonize over why a high school senior could construct something the world's top minds had spent sixty years searching for.

Because these mundane, sociological questions all had to give way in the face of this magnificent mathematical construction.

Now, the only true question remaining was whether this tile, temporarily dubbed Jiang's Tridecagon Monotile, could survive this brutal first round of interrogation.

After half an hour of dead silence and individual calculations...

Professor Lin Zhaoye put his glasses back on, rubbed his eyes which were aching from the screen's blue light, and declared in a heavy voice, "I still need to manually verify this exhaustive table of local cases line by line. Relying solely on human review, I cannot guarantee it one hundred percent. Standard practice for this kind of proof requires submitting the finite combinatorial cases to a computer for exhaustive verification—humans set the rules and machines run all possibilities—before one dares write in a paper that there are no omissions. But..."

He pivoted, his voice trembling with excitement he couldn't suppress.

"Judging from the logical review so far, even after racking my brain, I couldn't find a single counterexample that could shoot it down on the spot. At least in terms of logical intuition, it stands solid."

Following up on Professor Lin Zhaoye's words, Shao Mingtang added, "The uniqueness of synthesis and the exclusion of that semi-periodic belt both require a substantial amount of supplementary mathematical analysis lemmas, but this proof's roadmap is no longer a castle in the air."

[part:gemini-3.5-flash-lite]

Professor Gu Nanzhou pressed his hands against the desktop, staring at the two grandmasters in the video: "So, what is the final conclusion?"

Professor Lin Zhaoye was silent for a few seconds, as if handing down a historic verdict: "Let us first treat it as a candidate construction with extremely high potential."

Shao Mingtang added a sentence of immense weight: "At least in the field of aperiodic tiling, it is already the only pure monotile candidate in nearly thirty years worth initiating a large-scale official review for."

Professor Gu Nanzhou's tense shoulders finally relaxed.

"Next, all procedures must be nailed down completely." Professor Lu Zhixing suddenly interjected, his voice chillingly calm.

Everyone turned simultaneously to look at this professor from the School of Physics.

Although Professor Lu Zhixing did not understand the full proof, he knew all too well the cruelty of academia.

He looked at Professor Gu Nanzhou and stated point by point: "First, the original manuscript is to be physically sealed immediately, and no one is allowed to add a single stroke to it again."

Professor Gu Nanzhou nodded solemnly.

"Second, high-definition photos of all scanned copies must be backed up locally as read-only, tracked via internal campus emails, packed into encrypted archives, and timestamped across multiple mailboxes."

Chen Yan immediately reacted as if receiving a military order, frantically tapping on his computer keyboard to pack, organize, and encrypt the folder.

"Third, everyone in this room and in this video conference today acts as the first-round witnesses. Regardless of ultimate success or failure, today's conversation must be made into a memorandum."

"Fourth, and most importantly. The boundaries of contribution must be written out crystal clear. Who proposed this extremely counterintuitive construction? Who provided the core proof-by-contradiction route? Who is responsible if a computer team needs to verify the local configuration tables later, and who is responsible for turning rough ideas into formalized mathematical lemmas? There must not be the slightest ambiguity, and no possibility of anyone attempting to steal or dilute core contributions."

Professor Gu Nanzhou gave Professor Lu Zhixing a deep look.

As expected of a young Principal Investigator, he possessed an almost instinctive sensitivity toward the ownership of academic achievements, legitimate record-keeping of procedures, and potential risk control.

"I completely agree. This protects Jiang Lin, and it protects ourselves as well."

Professor Lin Zhaoye said loudly from the screen: "This must be done, and there must be no ambiguity."

Shao Mingtang's gaze softened somewhat as she looked at Jiang Lin: "Especially since Jiang Lin's identity is too special—a high school senior. If this thing really pans out, someone internationally will definitely suspect that some heavyweight published it under his name. Therefore, every minute and second of the original records must be written as clearly as court evidence."

The four words special identity made Chen Yan, who was packaging files, unable to help glancing at Jiang Lin again.

A high school senior.

With bloody well over a month left until the Gaokao.

Yet here, on the blackboard, he drew an aperiodic monotile that might be enough to make it into top journals and shock the entire discrete geometry circle.

Forcing these three extremely magical facts together still made his brainstem ache faintly.

Professor Gu Nanzhou put those few thin sheets of original manuscript back into the black hard-shell folder.

"From now on, the shape of this tile is legally fixed."

He turned his head to look at Jiang Lin, his tone as solemn as if holding some kind of ceremony.

"Jiang Lin, listen to me. When you go back tonight, do not change the length of even a single edge, and do not adjust the included angle by even a single degree."

Jiang Lin nodded calmly: "Understood."

"When doing mathematical proofs, the biggest fear is modifying the object while proving it," Professor Gu Nanzhou admonished earnestly. "Today's tile is today's tile. Regardless of whether it is eventually verified or falsified, it is set in stone. Even if you have a flash of inspiration tomorrow morning while washing your face and think of an even prettier hendecagonal tile, that is a different tile. You'll have to open a new file and calculate it separately."

Jiang Lin nodded.

He understood that this was fixing the object of proof.

Professor Lu Zhixing straightened his back and proclaimed like a notary public: "I, Professor Lu Zhixing from the School of Physics at Jiangda University, as an on-site witness, hereby confirm that the current candidate construction codenamed Jiang, along with all logical deduction explanations in this round tonight, was entirely proposed and completed independently by Jiang Lin himself right here, in front of us."

Professor Gu Nanzhou looked at Chen Yan.

Chen Yan snapped to attention: "I, Department of Mathematics PhD student Chen Yan, also testify as being present throughout."

Professor Lin Zhaoye pushed up his glasses in the video: "I will immediately send an encrypted email to you from the official Nanjing University mailbox, confirming that I received the first edition of scanned materials containing the original train of thought at this time."

Shao Mingtang nodded: "The Chinese Academy of Sciences will handle it the same way, with a timestamp as proof."

Professor Gu Nanzhou sat back down in front of the computer, placed his hands on the keyboard, and created a new folder.

He hesitated for a moment when naming it, ultimately choosing extreme restraint.

candidate_Jiang_2022_05_01

He pressed enter, paused for a second, and looked at Jiang Lin to explain: "For internal communication, let's call it this for now."

Jiang Lin had no objections; it was just a codename.

Professor Gu Nanzhou skillfully dragged the high-resolution scanned copies, the on-site record photos taken by Chen Yan per Professor Gu Nanzhou's request, the exhaustive table of local cases, the screenshots of the video conference, and a long list of participants along with a description of their division of labor all at once into the folder.

Then, he opened Jiangda University's campus email system and began drafting the internal memorandum email.

In the recipient list, he filled them in sequentially: Professor Gu Nanzhou, Professor Lu Zhixing, Professor Lin Zhaoye, Shao Mingtang, Chen Yan.

Finally, he asked Jiang Lin for his email address and added Jiang Lin.

Email Title: [Confidential/Tracked] Candidate Monotile "Jiang" —— First-Round Internal Verification Records and Consensus

For the body, Professor Gu Nanzhou wrote carefully, choosing his words with extreme brevity and efficiency.

Tonight (May 1, 2022), in Discussion Room C216 of the School of Mathematics at Jiangda University, a first-round small-scale internal verification meeting was held regarding the "Candidate Monotile Jiang" (tentative name) proposed by Jiang Lin.

Preliminary Conclusion: Upon preliminary review by Professor Lin Zhaoye from Nanjing University, Professor Shao Mingtang from the Chinese Academy of Sciences, and myself, no obvious obstacles that could immediately break down the construction topologically or algebraically were found. This construction and its accompanying forced substitution rules possess extremely high academic value for further formalized proofs and computer-aided verification.

Core work remaining to be supplemented ——

Uniqueness of the synthesis skeleton (depending on equivalence class partitioning) needs to be written as a formal mathematical lemma;

The exclusion argument for semi-periodic strips (Conway worms) (based on symbolic dynamics and eigenvalues) needs algebraic deduction completed;

The local case table must introduce a computer-aided exhaustive verification program as soon as possible to ensure no omissions.

Current Status Definition: Serious candidate construction.

In view of the potentially huge impact, the current stage is strictly confidential and will not be disclosed to the outside world or other non-core personnel for the time being.

When Professor Gu Nanzhou wrote the last line, his fingertips hovered over the keyboard for a moment.

Recalling Professor Lu Zhixing's warning, he struck the keyboard heavily and added the core disclaimer and attribution statement.

[Special Declaration]: The original polygon construction, monotile boundary shapes, recursive decomposition system of the four types of hypergraph blocks, and the core proof roadmap using local forcing and synthesis uniqueness for periodic proof-by-contradiction included in the attachments were all independently conceived by Jiang Lin himself and completely presented on-site today. All other participating personnel only provided review and formalization suggestions.

Professor Lu Zhixing had been standing behind Professor Gu Nanzhou staring at the screen.

When he saw this bolded last sentence, he finally nodded in reassurance.

Professor Gu Nanzhou took a deep breath, moved the mouse, and clicked send.

Whoosh ——

The progress bar flashed past.

A few seconds later.

Professor Lu Zhixing's phone placed on the desk vibrated once.

In the lower right corner of Chen Yan's computer screen, a green prompt box indicating the arrival of an email popped up with a beep.

Professor Lin Zhaoye in the video lowered his head to look at another screen, then looked up and said: "Nanjing University confirms receipt."

Shao Mingtang followed closely: "The Chinese Academy of Sciences confirms receipt, attachments intact."

The old phone in Jiang Lin's school uniform pocket also glowed faintly on the screen.

An email containing several high-definition attachments amounting to tens of megabytes, along with a network timestamp accurate to the second.

A bizarre tridecagonal tile that had drifted in the dreams of countless mathematicians for sixty years and should not have existed, from the very moment the mouse clicked send, finally drove down its first layer of traceable priority anchor in this material Real World.

It had become an existence jointly witnessed by Physics and mathematics, no longer just a lonely ghost in Jiang Lin's mind.

Professor Gu Nanzhou closed the computer screen and turned to look at Jiang Lin.

The fervor in his eyes had not completely faded yet, but his tone already carried the level gaze and consultation reserved for a scholar of the same rank.

"Jiang Lin, tomorrow, if you have time, come to Jiangda University once more. We need to strike while the iron is hot and pry out bit by bit the ideas in your head regarding the proof-by-contradiction of symbolic dynamics, putting them down in black-and-white LaTeX paper format."

Jiang Lin took out his phone, lit up the screen to check the time, and said: "I'll try my best."

As soon as these three words came out, Chen Yan on the side almost failed to hold it together on the spot.

Try your best?

Holy crap!

This was a candidate monotile with an extremely high probability of causing a major earthquake-level impact across the entire tiling field of global discrete geometry, crystallography, and condensed matter Physics!

If countless bigwigs were the discoverers, they would wish they didn't have to sleep tonight and would sleep right in the laboratory until the paper was published.

And now, whether tomorrow's review of this history-changing construction could continue depended on whether a high school senior had free time?

Absurd.

Simply absurd to the point of making one extremely want to laugh.

But looking across the entire C216, gazing at the geometric deductions covering the whiteboard, not a single person laughed.

Professor Lu Zhixing looked at his watch; it was almost ten o'clock.

He walked over to Jiang Lin and said in a gentle tone: "It's too late, I'll drive you back."

Jiang Lin packed his things into his schoolbag, put it on his back, shook his head, and said: "No need, Teacher Lu, I rode my bike here."

Watching this series of actions—putting away his pen, closing his notebook, putting on his backpack—so practiced just like an ordinary student who had just finished an evening self-study class, Chen Yan watched him closely.

The contrast between this scene and that tile from a few minutes ago finally made him unable to hold back the sentence he had been holding in all the way.

"You..." He opened his mouth, realizing he actually didn't know what he should ask, and finally squeezed out, "You don't have to go to class at Jiangcheng No. 7 High School tomorrow, do you?"

"Not really, I've taken an extended leave recently."

C216 once again fell into that familiar pin-drop silence.

Across the video, Professor Lin Zhaoye took off those thick glasses again, covered his face with his hands, his shoulders shaking uncontrollably, and finally let out a low laugh.

Mixed within that laughter were reflections on the passing years, envy of a genius, and infinite awe toward the marvelous field of mathematics.

Shao Mingtang looked at the slender youth with a schoolbag on his back on the screen, her eyes turning complex and gentle as she softly said: "Watch out for safety on the road, Jiang Lin."

"Thank you, Professor Shao, thank you everyone, all teachers, thank you."

"Be careful on the way, see you tomorrow."

"Alright, see you tomorrow."

Jiang Lin turned around, pushed open the door, and left C216.

The lights in the corridor were very ordinary, the common cold-white voice-activated lights, turning on one by one with his footsteps.

Looking out through the glass windows of the School of Mathematics building, the sky outside had completely darkened, and the city's neon lights reflected mottled halos on the river surface.

Jiang Lin walked down the steps, came to the bicycle shed, pushed out his bicycle, straddled the seat, pedaled with both feet, and rode out of the school gate along the tree-lined avenue formed by French plane trees.

The early summer evening breeze blew against his face with a hint of moisture, blowing across the hem of his school uniform and rustling loudly.