192: Chapter 192 Why is it so sure it's true?

In late October, the afternoon sun was no longer as scorching as it had been in midsummer, slanting diagonally through the row of ginkgo trees in front of Zijing Building No. 17 and sifting a layer of shimmering broken gold onto the concrete ground.

The wind was a few degrees cooler than yesterday, swirling a few scattered leaves that had turned yellow ahead of time, spinning against the base of the wall.

An orange-yellow shared bicycle leaned crookedly against the ginkgo tree pit.

The mechanical lock on the rear wheel was stuck in a half-open position, with half of the latch protruding. It neither fell back into the slot nor pushed into the lock body, resembling a stiff gesture paused halfway.

Jiang Lin pressed his fingertips against his phone screen. The moment the green unlocking checkmark popped out, the phone vibrated with a hum, but the rear wheel remained completely motionless.

He tapped it again, and the checkmark on the screen lit up once more. The latch remained stuck in its original place, refusing to move forward or retract.

He squatted down and flicked that cold, hard piece of metal latch with his fingertips.

The latch was jammed right there, neither up nor down.

Confirming that it wasn't a minor problem like an angular deviation that could be solved with brute force, he didn't waste any more time wrestling with it.

His fingertips opened the malfunction reporting column, checked the option, and he stood up and walked towards another bicycle nearby.

The faint sound of tires crushing fallen leaves came from behind. Gu Mingche held the handlebars and stopped beside him, teasing with a smile: "I thought you could fix it with a couple of moves as soon as you squatted down."

"It's not my bike, how could I mess with it without authorization." Jiang Lin replied with a smile.

"With your hands-on ability, if Low Entropy Workshop entered the shared bicycle market, you could save half on maintenance costs alone." Gu Mingche pedaled half a circle and casually joked, "By the way, you could also replace all these broken locks stuck halfway."

"Where would Low Entropy Workshop find the time to make shared bicycles now."

Accompanied by a crisp click, the latch of the second bicycle unlocked by Jiang Lin popped open smoothly, and the rear wheel turned very smoothly, as if intentionally used to set off the uncooperativeness of the first bike.

"Indeed, your robots are overwhelmingly popular right now, there's completely no need to make this simple gadget."

"Where are you going?"

"Sixth Teaching Building, Discrete Geometry Seminar, starts at three-twenty." Gu Mingche looked at his empty bicycle basket, "And you?"

Jiang Lin wanted to know too.

This was one of the rare afternoons in the past half year that had not been scheduled in advance.

In the morning at the R&D center, he re-sorted through the matters before the end of the month that required his personal review and signature.

Low Entropy Workshop had just finalized three delivery sequences.

The first mining order had entered the engineering evaluation stage, and the public-facing experiential activities had to wait for a testing window to reduce risks to an acceptable range.

The production line for the second delivery unit was still being set up, and the schedule for equipment entry had been changed several times.

Messages popped out one after another as usual, and the title bars lit up with unread red dots one by one, but they all had their own places to go and didn't need to pass through his hands for everything.

The scenario data was handed over to Liang Zhixia, the engineering boundaries belonged to Xu Man, and the money belonged to Chen Zhi.

He re-confirmed this three-line division of labor and clicked handover item by item.

The top column of the work terminal originally read urgent matters, and at this moment, it was empty for the first time.

On the way back to school from the R&D center, Jiang Lin deliberately reserved three hours for this afternoon. As for what exactly to do with these three hours, he still hadn't figured it out when he rode his bike downstairs to Zijing Building No. 17.

For a long time in the past, what he always lacked was time.

Now that he had managed to squeeze out such a blank space from a tightly packed spreadsheet, he instead felt a bit of strangeness.

Not knowing what to fill into it.

"Before five o'clock, just walk around casually." he said.

"That's rare, as busy as a bee like you, you actually have an undetermined afternoon?" Gu Mingche sighed, with a tone of watching the fun, and said, "Then come with me to sit in on a class. Today we're talking about the unit distance problem, and the problem statement is very simple. The teacher said last week that as long as you know points and lines, you can understand the first ten minutes."

"You chose this course?"

"Auditing. I went once last week and halfway through found out I was missing two prerequisite courses, so I crammed desperately for six days." Gu Mingche's tone carried a hint of self-deprecation, "Today's problem statement looks simple, and now I'm ready to go confirm again whether my remedial studies missed the mark."

"What did you make up?"

"Graph theory, incidence geometry, and a part of analytic number theory."

...

The two bicycles rode westward along the campus road.

The shallow layer of fallen leaves on the campus road was crushed by passing wheels, making a fine, dry rustling sound.

Gu Mingche habitually stood up when crossing speed bumps, letting the bike body cushion slightly in the bumps, and habitually turned back to look at the intersection, as if to confirm that the person behind him was still following.

The first time he turned back, Jiang Lin lagged behind by a little less than half a bike length.

The second time, the two were side by side.

The third time he turned back, Jiang Lin had already passed him from his left.

"On your left."

More than half of the amphitheater in the Sixth Teaching Building was already seated, with tiered seats spreading backward from the podium, and a few empty seats remaining in the rows near the window.

When Jiang Lin entered the door, several students near the aisle recognized him first.

Someone in the front row raised a finger to point at the empty seat in the back, and someone simply hugged the schoolbag originally used to save a seat back onto their thighs, making room for the person next to them with movements as practiced as a rehearsal.

No one felt abrupt because of his arrival.

Whichever classroom Jiang Lin appeared in no longer counted as an abnormal event worth verifying.

The teacher on the podium was erasing the blackboard. Turning around and seeing Jiang Lin, he patted the chalk dust off his hands and joked, "Oh, the Math God is here. Today's problem is not far from the direction you used to work on, so let's hear what opinions you have."

The air in the classroom suddenly became cheerful.

Gu Mingche picked a seat on the edge of the third row from the bottom. After sitting down, he pushed half of the notes he had organized last week in front of Jiang Lin.

The first page of the notes had only three words, the handwriting traced repeatedly with varying strengths.

Point, distance, edge.

A string of paper years began to appear on the second page, and the third page had already reached the incidence theorem, with dense small-print annotations interspersed beside the formulas.

Gu Mingche opened another small space in the corner of the page, listing the remedial materials from the six days, with two question marks discounting the remedial results.

The problem statement of this course was so simple that it only required a ruler, but the prerequisite knowledge was bottomless.

Jiang Lin took a pen and circled a spot for him: "This won't be used here."

"I read it for three hours for nothing?"

"It might be used in the future."

"These words of comfort for a Department of Mathematics student are about the same as drinking more hot water."

At three-twenty in the afternoon, the class bell rang on time, and the lingering voices in the classroom quickly quieted down.

The lecturer said a few brief words, picked up a piece of chalk, and directly drew a dozen scattered points in the very center of the blackboard, casually connecting two of them.

"Today's problem only requires a ruler. Place n points on a plane. If the distance between two points is exactly equal to 1, we draw an edge between them. Now the question is, at most how many such unit edges can be connected?"

He drew a row of equidistant points beside it with neat strokes, and then drew a square with neatly labeled side lengths.

"If arranged in a straight line, it's obvious that we can get n-1 lines, with each point connecting only to its left and right adjacent points." He tapped that row of points with the chalk, "If arranged in a square grid array, the horizontally and vertically adjacent edges add up to about 2n. Continuing to increase it is also possible; the method is to make more neighbors at a distance of one appear around the same point."

The chalk drew a larger two-dimensional grid array on the blackboard, densely spreading across half of the blackboard.

Starting from a central point, the teacher drew a few lines in different directions.

"Find an integer m that can be written as a squared plus b squared in many ways, and then shrink the grid points as a whole to 1 / √m of the original. The line segments originally of length square root of m now all become unit lengths." While speaking, he supplemented a small calculation formula beside it, "The more representations of m as a sum of two squares, the more unit edges can extend around each point."

Someone in the audience lowered their head and took notes quickly, and someone frowned and stared at the blackboard, obviously still digesting how the scaling step turned the length into unit one.

This problem could indeed be explained clearly with just a ruler, and the problem statement was so clean that even an elementary school student could understand it.

The problem was that once the ruler landed on the grid points, geometry quickly veered into the deep waters of integer factorization.

Gu Mingche followed along very smoothly in the first ten minutes, and even had the spare capacity to add a note next to the representation formula.

After entering Gaussian integers, his page-turning speed slowed down, and his pen tip stopped behind an incomplete equation.

Jiang Lin turned his draft paper sideways and added a line for him.

Gu Mingche took a look and lowered his voice to ask: "How much was omitted in this step?"

"About two pages."

"No wonder I started feeling last week that the teacher was erasing the blackboard faster and faster from here on."

Jiang Lin took back the draft paper and continued looking toward the podium.

The teacher turned around and wrote the currently best upper and lower bounds on the right side of the blackboard. The two rows of numbers were arranged side by side, with a large blank space left in the middle, as if deliberately placing the gap before everyone's eyes.

The upper bound is on the order of n to the power of four-thirds.

Although the existing lower bound is higher than linear, that extra piece in the exponent tends to zero as n increases, and it has never been able to leave a fixed positive surplus.

"For decades, the strongest constructions have grown out of this type of grid point." The teacher tapped beside the lower bound, "Erdős guessed based on this that the number of unit distances on a plane is at most n to the power of one plus o of one. In plain English, no matter how you arrange the points, it is impossible for it to maintain n to the power of 1.01 for long, let alone 1.1."

A rustling sound of page-turning and note-taking rang out in the classroom, and someone whispered one plus o of one, as if confirming they hadn't misheard.

Gu Mingche's pen tip swept across a line, and he stealthily pushed his notebook in front of Jiang Lin.

[The upper and lower bounds are so far apart, how dare they guess off the top of their head?]

Jiang Lin glanced at it, picked up his pen, added eight characters, and pushed it back.

[Conjecturing is fine, proving is not.]

Gu Mingche pressed the pen cap against his lips, his shoulders shaking for a long time, finally managing to hold back his laughter.

On the podium, the teacher had already thrown the question to the entire classroom.

"Let's talk first, why are so many people willing to believe it?"

A boy in the back row replied: "From 1946 to now, basically all imaginable regular point sets have been tried. Grids, circles, point deletion, combinatorial constructions—the best results have always been stuck right here."

"What can this prove?"

"At least it shows that fixed routines find it hard to squeeze out another constant exponent."

"Much more rigorous." The teacher wrote the four words fixed routines on the blackboard, "And what else?"

Someone in the front row mentioned that two circles have at most two intersection points, and the unit distance graph cannot arbitrarily become dense; another person explained from the incidence upper bound how geometric structures cause a large number of candidate edges to conflict with one another. The teacher asked about several boundaries and weakened one of the conclusions that had been stated too strongly.

While listening, Gu Mingche added a second half behind the initial sentence [The upper and lower bounds are so far apart, how dare they guess off the top of their head?].

[Turns out it wasn't a wild guess.]

Seeing this, Jiang Lin reached out to pull his notebook a little closer and added two words behind it.

[Still is.]

"Then why are you listening so attentively?"

"Wild guesses also come in many varieties."

On the podium, someone began trying to improve the construction.

A Ph.D. student proposed keeping only the local parts in the grid points with exceptionally many number representations.

The teacher asked him to estimate how many translation positions remained after deleting the points.

Another person wanted to stack multiple sets of circles together, but as soon as he reached the third sentence, a classmate in the front row pointed out that different radii could not directly contribute to unit edges. The two argued for a few sentences across three rows of seats, then lowered their heads together to modify the formula from just now.

Most of these paths had already been walked by someone.

Every few steps the blackboard writing advanced, the teacher forced it down to a specific enumeration.

A few minutes later, the gains just obtained leaked away from elsewhere.

At first, Jiang Lin was still supplementing conditions for Gu Mingche.

Later, his pen stopped beside the Gaussian integers and didn't write any further down.

Fixed grid points, fixed quadratic number fields, fixed decomposition rules.

The attempts on the blackboard had many arrangements, but the underlying arithmetic structure remained unchanged.

They could show that this set of structures could not support a fixed positive exponent, but they could not vouch for all point sets on a plane.

Jiang Lin's pen stopped beside the Gaussian integers and wrote a K.

The phone vibrated gently on the desk, and the screen lit up with the icon of the work terminal.

Jiang Lin lowered his eyes to take a look, his thumb already touching the edge of the screen in an almost subconscious movement.

Right at that moment, a student in the front row happened to ask, "For the restriction on numbers represented by the sum of two squares, is the key the Gaussian integers themselves, or does it happen in any fixed number field too?"

Jiang Lin's hand paused over his phone, not pressing down.

The professor replied, "In a fixed number field, similar counts usually cannot grow the exponents we want. Of course, people have thought about changing the number field, but the troubles will change along with it."

The screen went dark.

Jiang Lin flipped his phone face down under his scratch paper, suppressing the temptation of it lighting up again, and wrote a K beside the Gaussian integers.

Originally, he had only come to sit in on a lecture.

Whether he understood it or left halfway through, he didn't need to give anyone a result before five o'clock.

This should have been the only burden-free arrangement for this afternoon.

But after that K landed on the paper, the lower bound on the blackboard suddenly seemed too familiar—familiar enough to make a certain corner of his heart prickle with alertness.

Everyone repeatedly pressed on how to squeeze a little more out of the old construction.

He wanted to know even more why they still had to stay within the old construction.

Jiang Lin wrote d in the upper right corner of the K, and drew an arrow pointing toward the complex plane. The end of the arrow did not stop, as if it still wanted to keep extending outward, but was blocked by the edge of the paper.

Gu Mingche leaned over and looked at that K: "Making the degree larger, can that really be useful?"

"I don't know yet."

Gu Mingche's gaze lingered on Jiang Lin's face for a moment.

This expression of focus bordering on stagnation was not the first time he had seen it.

Then he withdrew his gaze, letting it fall back onto the K, and asked no more.

The professor began to wrap up this section of the discussion, preparing to move on to the subsequent incidence upper bound.

Jiang Lin stared at the arrow for a few seconds and still raised his hand.

In the classroom, everyone's gaze lifted from their notebooks.

"Professor, if the degree of the number field grows along with the scale of the point set, how many of the restrictions in the fixed number field remain?"

The professor put down his chalk and pondered for a moment: "You want to keep the points that finally land on the complex plane, and let the degree of the underlying number field grow with the scale?"

"Yes. That way, the counting restrictions in a fixed number field can no longer be directly applied."

That PhD student in the middle turned back to look at Jiang Lin, a hint of surprise in his eyes, and then said, "For degrees greater than two, the ring of integers is usually not discrete under a single complex embedding. How many points can be squeezed into a disc is unclear, and no matter how many unit edges there are, they cannot be compared."

He spoke very quickly, as if this idea had already crashed against the walls of his own mind long ago.

Jiang Lin asked, "What if the point set doesn't take the entire ring of integers?"

"Then what should it take?"

"A finite window."

The professor added a few strokes to the lattice points that had already been half-erased: "Where is the window opened?"

"Put all conjugate embeddings together, restrict their sizes, and then carve out a finite point set from within."

Someone in the front row flipped back their notes, the chalk fell onto the blackboard again, and the rustling sound of writing filled the classroom once more. The professor drew several columns of coordinates side by side, then circled three places among them, the force of the circles growing heavier each time.

"You owe at least three accounts." The professor turned to face him, his tone turning serious, "How many points can remain in the window; how many difference vectors with a modulus length of one can be found; how many times these difference vectors can repeat within the window. The discriminant and class number are still waiting outside to collect payment."

"Mm."

"And," the professor added a horizontal line across the first column of coordinates, "under one embedding the modulus length is one, but the other embeddings might not be so well-behaved. If you want to control them simultaneously, what conditions must the number field satisfy?"

Jiang Lin looked at that horizontal line.

For the moment, he could not answer.

Jiang Lin marked one, two, and three in turn beside the three obstacles, leaving half a page of blank space for the third item.

"I'll have to go back and look," Jiang Lin said.

The professor nodded, "Then find it first before we talk. Today's class can't be completely borrowed away by you; we still owe the second half."

A sound of page-turning rang out from the front row. Someone tucked the newly opened scratch paper back into their notes, but the PhD student was still leaning against the back of his chair, looking at Jiang Lin as he asked, "Even if all these can be controlled, in the end, the number of unit edges must have an extra fixed exponent. The gains brought by high degrees might all be eaten up by the window volume."

"That's why we have to calculate it."

"I calculated the simplest cubic field, and lost thoroughly."

"Cubic might not be enough."

"Then how high are you planning to increase the degree?"

Jiang Lin looked at the d in the upper right corner and replied calmly, "I don't know yet."

The professor tapped the podium with the eraser: "You two can exchange your 'I don't knows' after class. Now, turn to the incidence upper bound."

This time the laughter was a bit louder than before.

Gu Mingche also smiled, but when turning his book, he pressed down a corner of Jiang Lin's scratch paper with the K on it to keep it from being blown to the floor by the wind stirred up.

The second half returned to the originally scheduled content.

Jiang Lin listened with the same focus, but his notes were different from the first half.

Every time the professor mentioned a fixed condition, he would leave a narrow blank space beside it.

Two places were unrelated to unit distance; he only crossed them out after finishing writing.

Another derivation was originally very familiar to him, yet he had missed copying down the middle line.

Gu Mingche pushed his own notes over for him to fill it in.

Jiang Lin looked at it for a long time and asked, "Fill in where?"

Gu Mingche tapped that line of blank space with the tip of his pen: "Before class, you even said you could save two pages here."

Jiang Lin filled in the missing equation.

"I was thinking about something else just now."

"I could tell."

At four-fifty-five, the class bell rang.

The professor finished writing the final step and left the students with two papers to read.

People in the classroom began packing up their books. The PhD student from earlier squeezed out from the middle row and handed his draft to Jiang Lin.

"That counterexample of mine for the cubic field was calculated very roughly. Take it and have a look; at least it can save you from stepping into a trap."

The paper was densely covered with writing for two pages, with the final ratio circled in red and a downward arrow drawn beside it.

Jiang Lin took it: "I'll return it after I finish reading."

"No need to return it; it has already died completely." The PhD student slung his backpack over his shoulder, "If you really manage to add the degree up, remember to come back and tell me which layer it died on, just let me die knowing the truth."

After saying that, he chased after his companion out of the classroom.

Just as Jiang Lin stood up, ready to leave, the professor on the podium called out to him.

"That path you pointed to just now—actually, many people have thought about it in fragments. The difficulties are no secret either; high-degree number fields easily generate a bunch of things that one can't control oneself." At this point, a touch of casual amusement crept into the professor's tone, "Still, I really didn't expect that one day in my own class, someone would drag this moss-grown old train of thought back out."

He looked at Jiang Lin and added, as if remembering something: "Jiang Lin, back when you were auditing graduate courses at Jiangcheng University, didn't you casually churn out Jiangs Brick? At that time, our research group meeting even used it as material."

Among the students who hadn't yet left the classroom, someone chuckled softly.

Without waiting for Jiang Lin's response, the professor continued on his own: "So, can today's class be considered somewhat interesting? With you coming to listen to my lecture, I'm actually looking forward to whether you'll listen out something new that can flip the table again."

Speaking up to this point, his tone returned to the rhythm of discussing the problem itself.

"It's just that I hope you don't aim too high right off the bat. Perhaps you could consider lowering the requirements a bit first, as long as you construct a family of point sets so that the number of unit edges can stably exceed an arbitrarily close-to-linear bound."

In the end, he dropped a hint half-jokingly and half as a kind reminder.

"If you really manage to do this step, Erdős's quote won't hold up anymore."

"Mm, I'll first see if it has an entrance worth entering," Jiang Lin replied.

Walking out the gates of the Sixth Teaching Building, the sky was already starting to turn dark, and the streetlights lit up one after another. Gu Mingche pushed his bicycle and caught up: "I roughly understand now. Everyone has been farming in the same plot of land, finding that no matter how much fertilizer they use, it won't grow fast, so they guessed that the land all over the world is like this. You want to change plots."

"Pretty much."

"How close is pretty much?"

"The new land might be entirely saline-alkali soil, or it might be impossible to measure its area at all."

"Then why did you get more and more enthusiastic asking questions in class?"

Jiang Lin's action of pushing his bicycle slowed down a bit.

The professor's three accounts, the downward red arrow on the PhD student's draft, the ring of integers crammed together in a single complex embedding—they lined up in his mind from the beginning all over again.

If all three accounts are calculated badly, this path could die the very same day.

Jiang Lin had already started thinking about where exactly that cubic field had lost its gains.

"That old plot of land has been farmed for seventy-six years," Jiang Lin said.

"So you're really going to do it?" Gu Mingche asked.

"We'll see in November."

Gu Mingche pulled out his phone and deliberately glanced at the date: "Today is the twenty-first. Do you call eleven days later 'we'll see'?"

"At least not today."

"Sounds like a deferred execution."

"It originally is."

Gu Mingche steadied his bicycle and suddenly smiled: "You weren't saying that when you first walked into the classroom."

"When I first walked into the classroom, I didn't know it was here."

When they returned to Zijing Building No. 17, the sky had already grown dark. The voice-activated lights at the stairwell entrance sensed their footsteps and lit up one after another.

Jiang Lin noticed that the shared bicycle that had been reported broken that afternoon had already been moved to the maintenance area by the staff, parked side by side with a few other bikes similarly tagged with malfunction labels.

He glanced at its rear wheel once more; it was still tightly fastened in the half-open position, the latch completely unmoving.

Returning to the dorm at night, Jiang Lin sat at the desk in Room 402, the preparation checklist before the end of the month spread out across the screen.

There were nine days left until the thirteenth transmission. The final trajectory of the northern maintenance branch had not yet been finalized, the supplementary plan for the fifth relay, and the mechanism behind the amber reflection once every forty-seven seconds behind the door were all waiting for him to complete the next round of plans.

Placed beside the desk was also the engineering summary of the Low Entropy Workshop. Tomorrow morning, the freezing range of the first mining verification machine still needed to be confirmed.

All these matters had dates, and someone was waiting for the results for each of them.

The two pages of drafts brought back from the Sixth Teaching Building were pressed at the very bottom.

Jiang Lin worked along the checklist for over an hour.

When the roadmap zoomed into the northern fork, a set of distance markers popped up on the screen.

Staring at two of the points, he casually wrote down a difference vector on the paper.

After writing it, he saw that K beside it.

Jiang Lin put down his pen and got up to pour a glass of water.

Returning to the desk, he reached out and pulled those two pages of failed drafts from beneath the documents.

The ratio inside the red circle had indeed lost very thoroughly.

The direction increase given by the cubic field was limited, but the price paid by the window to accommodate the conjugate embeddings was much greater. Continuing to calculate along this set of numbers would only yield a few more pages of more precise failure.

He wrote beside the red circle——

[The degree cannot increase sporadically.]

The pen tip moved down half a line and then stopped again.

If he wrote any further down, tonight's route plan would have to take a back seat.

Jiang Lin raised his head to look at the bottom right corner of the screen.

Ten-fifty-eight.

Jiang Lin created a new folder on the desktop, and the naming box lit up immediately.

Then he typed six characters——

[Unit Distance Counterexample]

There were only three materials in the folder.

One set of lecture notes, one failed draft from someone else, and a sheet of paper rewritten by Jiang Lin.

Three questions were retained on the paper.

[How to keep the degree of the number field growing continuously?]

[How to simultaneously control all conjugate embeddings?]

[How to make a sufficient number of unit difference vectors appear repeatedly in a finite point set?]

He wrote the date at the very top.

[After November 1st]

After writing it, Jiang Lin closed the folder and brought up the roadmap of the northern maintenance branch again.

At eleven-seventeen, the work terminal popped up an ordinary reminder.

Liang Zhixia inquired who should sign the delivery instructions for the first mining verification machine.

Jiang Lin replied with the person in charge's name and added another sentence, telling her to follow the established procedure and not wait for him.

After the message was sent out, a small space cleared up on the right side of the screen.

That newly closed folder still remained on the desktop.

Jiang Lin opened the to-do list, created a new item for November 1st, and hung that folder under it.

The roadmap of the northern maintenance branch refilled the screen.

In the to-do list at the bottom right corner, an extra line for an item ten days later appeared.

[Unit Distance Counterexample]