205: Chapter 205 Five Hundred Dollars, Seventy-Six Years

November 6th, 11:00 PM.

Gu Mingche refreshed Jiang Lin's personal homepage for the third time.

The counter in the upper right corner of the browser flickered slightly, and the preprint downloads jumped straight from 284 to 631.

The disposable chopsticks in his hand still held half a mouthful of rice noodles, and the soup had dripped down the noodles back into the paper bowl, splattering a few light brown dots on the plastic lid.

"It doubled in three minutes."

"That's because the first person shared the link."

Jiang Lin leaned back in his chair, casually opening the backend traffic source analysis. At the very top was a European combinatorial geometry academic mailing list.

[A claimed counterexample to Erdős's unit distance conjecture]

The body of the email completely reposted the word "Yes" that Jiang Lin had replied on the forum half an hour ago, followed by a direct link to the ArXiv preprint and a bolded reminder from the sender.

[The author claims that ε here is a fixed positive constant. If the quantifiers are correct, the conclusion is precisely the one we understand.]

Only then did Gu Mingche put that half-mouthful of rice noodles into his mouth, his eyes never leaving the pop-up new email notification box in the lower right corner of the screen.

The reaction speed of the academic community was even faster than anticipated.

The first email came from Frankfurt, directly questioning whether ε depends on the number of points n.

The second came from Paris, asking whether the degree of the CM field can grow infinitely without destroying the boundary conditions.

The third was even more direct. The senders were two German number theory experts, who got straight to the point and requested two independent code verification records of the pro-2 tower.

Shortly after, a fourth person downloaded the algebraic structure module Q-001 from the main attachment.

Three minutes later, he left the first question in the public comment section of the preprint page.

Since the prime number 101 will split into more and more prime places in subsequent tower layers, why doesn't that single splitting condition at the bottom layer need to repeatedly pay the order of the relation module layer by layer?

Gu Mingche's chopsticks paused as he read this.

"The first cut hits the exact same spot as Professor Zhou," he said, turning his head sideways.

"That spot has the narrowest margin," Jiang Lin answered calmly.

Instead of typing an explanation directly in the comment section, he opened the webpage backend editor and added a new entry with a highlighted border in the most prominent position on the homepage.

[Key Inquiries and Independent Verification]

Below the entry, two lines of monospace font were placed side by side.

[Q-001: The tower infinitude lemma has been reconstructed via two different routes.]

[Verification records are used to expose attack entry points and cannot replace independent checks by readers.]

That questioner refreshed the page eight minutes later.

Leaving a message that he had read the relation module expansion and was ready to bypass the two verification records and recalculate everything himself starting from the local decomposition group, his avatar then went dark.

Gu Mingche finally swallowed the mouthful of rice noodles in his mouth and let out a long breath.

"By putting the part people should be most suspicious of right on the homepage, someone is bound to dig deeper into it."

"It's also to save them time in finding the entry point," Jiang Lin said, typing a few lines of traffic-splitting scripts on the keyboard.

"The entry point is saved," Gu Mingche glanced at those sender addresses, "but those combinatorial geometry folks in Europe probably won't be sleeping tonight."

Just as he finished speaking, the phone Gu Mingche placed beside the keyboard suddenly vibrated violently.

"Can you guys give a sentence that ordinary people can understand within ten minutes?" Liang Zhixia got straight to the point.

The media inbox of the Low Entropy Workshop was originally stuffed with interview applications for Qingya Tunnel rescue robots, year-end production scheduling plans, and offline experience stores.

However, just half an hour ago, more than twenty all-English technical inquiries about mathematics suddenly flooded into the inbox.

The latest one came from a national comprehensive media outlet in China, and the reporter had even drafted the news headline for them.

[Eighteen-year-old Tsinghua University freshman overturns the Erdős conjecture?]

"The question mark is still there for now."

Liang Zhixia's voice was efficient and prompt.

"But if you guys don't give a unified official stance soon, clickbait accounts will help us delete that question mark before dawn."

Gu Mingche casually switched the phone to speakerphone and placed it between the two of them.

"6 times 10 to the negative 38th power can be written in the description, but don't simplify the entire achievement into a string of zeros," Gu Mingche reminded.

"Isn't this number the most intuitive one?" Liang Zhixia asked on the other end of the phone.

"Although it's small, it has passed two sets of strict interval algorithm certifications," Gu Mingche explained. "What's truly important is that it is fixedly greater than zero and will never shrink back into zero as the number of points grows."

"Ordinary readers seeing this number will only count how many zeros there are first, and then ask what on earth such a small number is good for," Liang Zhixia said.

"That's why the little-o notation must be explained at the same time," Jiang Lin interjected.

Gu Mingche smiled: "This sentence can finally be released on its own."

The crisp sound of rapid keyboard tapping quickly came from the other end of the phone.

"You guys are responsible for making the mathematical facts clear," Liang Zhixia said while typing, "and I'm responsible for blocking clickbaiters on the outside to prevent them from inventing a new theorem for Jiang Lin tonight."

Jiang Lin opened a blank document and dragged the top two lines of core formulas from the theorem page of the paper into it.

The old conjecture held that, no matter how the point set is constructed, the maximum number of unit distances can only reach n^(1+o(1)), and the extra part in the exponent will gradually shrink back to zero.

As for the new conclusion given by Jiang Lin, it directly gives n to the power of 1 plus δ, where δ is fixed to the right of zero right from the construction, and even if the point set expands infinitely, it will never retreat.

"In the past, people believed that the denser the point set, the more unit distance edges each point can connect on average, but it is impossible for them to continue growing according to any fixed power."

Jiang Lin spoke into the phone.

"This paper constructed a family of special geometric point sets and proved that they can indeed continue to grow according to a certain fixed power."

The sound of keyboard tapping on the other end of the phone stopped abruptly.

"Add a sentence at the end: Therefore, if this conclusion holds, does it constitute a counterexample to the Erdős conjecture?"

"Sure."

"Add another sentence about the current status?"

"Controlled verification has been completed, and it is currently undergoing broader public review."

"When has any of your achievements ever failed review?"

That being said, Liang Zhixia hung up the phone and immediately went to update the official website description.

Gu Mingche used his disposable chopsticks to fish out the soggy rice noodles soaked in the paper bowl, shook his head with a smile: "The Low Entropy Workshop is probably the only company in all of China that needs to popularize the hardcore mathematical meaning of the little-o symbol for readers under its robot delivery announcements."

Almost at the same time, the first authoritative scholar in the tower direction outside the review chain took away Attack Package One.

Professor Lin Zhaoye also seized the opportunity to forward the preprint link into the core discussion group of international discrete geometry.

At 12:41 AM, an email from Terence Tao quietly appeared at the top of the inbox.

In the email, he explicitly stated that he had completed the preliminary verification of the main theorem's logical quantifiers, geometric interfaces, and public verification descriptions. He confirmed that what the paper claimed was indeed a fixed positive constant lower bound, and also confirmed that once proven, this claim would officially overturn the usual form of the Erdős unit distance conjecture.

At the end of the email, Terence Tao wrote.

He would start directly from the relation bound corresponding to Q-001, completely bypassing the argument order of existing verifiers, and re-conduct an independent derivation of the entire pro-2 tower's construction.

Gu Mingche read this email twice and asked: "Why did he also choose to start from Q-001?"

"Because there's only a quarter of the relation margin left there."

"Turns out all the big shots in the world like to hit the thinnest spot first?"

"It saves effort."

A few minutes later, Terence Tao reposted this preprint on his personal academic homepage.

His recommendation blurb introduced the core claims of the paper, the current verification status, and the three interfaces that required key review. At the end of the entire passage was that far-reaching phrase—"if correct".

It was precisely this "if" that attracted more researchers hidden behind the scenes around the globe to flock into the paper's attachments.

They wanted to use their own pens to deduce firsthand whether the subsequent "correct" could hold true.

At 1:03 AM, the paper download count on Jiang Lin's personal homepage officially crossed the 10,000 mark.

The preprint link was already hung at the top of the homepage of the world's top three mathematical academic forums, and a new email would pop up in the public inbox almost every ten seconds or so.

Some people requested source files, some applied to join the second round of cross-verification, and several German researchers directly compressed and sorted the more than 600-page attachment into a clear node dependency graph, using red circles to mark the nodes they planned to focus on conquering tonight.

Sandwiched between these emails were inquiry letters from three top journal editors-in-chief asking whether he had chosen a submission target, as well as seven interview applications from mainstream international tech media.

Gu Mingche glanced at the time in the lower right corner of the computer, stood up, and cleared away the empty paper bowl on the table.

"I have class during the first period tomorrow morning. If I go back now, I can still sleep for five hours," Gu Mingche yawned. "What about you?"

"Sorting the public inquiries into three modules: algebra, geometry, and computation," Jiang Lin's hands didn't stop moving. "As for the emails sent before 12 o'clock, I'll sleep after I finish processing them."

Gu Mingche pointed at the unread email count that had piled up to 472 in the lower right corner of the screen and raised an eyebrow: "Big Brother, it's already 1:06 AM."

"So the ones before 12 o'clock are already fixed and won't increase anymore," Jiang Lin's tone was casual and matter-of-fact.

Gu Mingche couldn't help laughing: "You top mathematics people all have your own unique understanding of deadlines, right?"

Jiang Lin casually activated the automatic response system for the attack package.

The system would strictly filter all emails.

It only received letters that explicitly included page numbers, proposition numbers, and specific deductive inquiries.

As for those emails simply asking whether the conclusion was true or false, they would all be automatically replied to and directed to the public status description page.

The growth rate of the unread count finally slowed down noticeably.

Gu Mingche put on his backpack and walked to the door of 402. Before going out, he turned back, pulled open the printed paper, and glanced at the verification scope below his own name.

[Definitions, Citations, and Version Consistency.]

Last night, that line of small text still looked like a work-related injury certificate for doing hard labor, but now it had quietly laid in the computer hard drives of tens of thousands of top mathematicians and enthusiasts around the world alongside the over-600-page paper.

"Hey, Jiang Lin," Gu Mingche patted the door frame, "if someone later discovers deep inside the attachment that a certain symbol D was defined twice, is that on you or on me?"

Jiang Lin looked up at him: "Discovered before, it's mine; discovered after, it's yours."

"This division of labor is truly suitable for cultivating friendship."

Gu Mingche smiled and pulled open the dormitory door.

In the quiet midnight corridor, two students from the same building were walking over from the elevator direction.

One of them recognized Gu Mingche at a glance, his pace suddenly slowing down by half a beat at the door of 402, a trace of disbelief in his eyes.

"Gu Mingche?"

"It's me. What's up?"

"Are you that Gu Mingche in the paper attachment verification list?"

Gu Mingche turned his head back and looked at the spare folding chair he had just been sitting on.

An hour ago, he was still an ordinary freshman who had run over to 402 in the middle of the night to help check Greek letters and cross-references.

But now, someone had already started holding that 600-page proof attachment that was creating a sensation in the academic community, matching names to recognize him.

Gu Mingche flashed an utterly harmless smile at the other person: "Same name."

With that, he casually closed the door of 402, leaving the two students standing blankly in the midnight corridor, trying their best to judge whether that sentence could be believed.

The next morning.

One of the soybean milk machines on the second floor of the Zijingyuan Cafeteria unfortunately broke down.

The queue for buying breakfast wrapped around from the window all the way behind a thick pillar.

Gu Mingche queued for six minutes, and the two Ph.D. students from the School of Mathematical Sciences ahead of him had also been debating Jiang Lin's paper for a full six minutes.

The two of them engaged in a fierce clash surrounding the relation budget of the pro-2 tower.

One heavily doubted whether prime number 101, after completely splitting, would produce redundant relational constraints in high-level recursion.

The other held a pencil, deducing along the lower bound of the class group rank, attempting to find implicit constants that might decay as the number of layers increased.

Gu Mingche stood in the back holding an empty plate and listening, until he heard one of them casually remark that the review group had already proven the tower lemma for the whole world. He finally couldn't help but reach out from behind with his phone, showing the verification scope on the official page to the two of them.

"These two verification records can only prove that the two groups have successfully walked through their respective routes; they cannot save everyone from making independent logical judgments."

The two doctoral students turned around simultaneously, their gazes falling on Gu Mingche's phone interface.

One of them was stunned for a moment: "You finished reading it?"

"I helped him check the consistency of version citations and definitions." Gu Mingche said truthfully.

The other person suddenly raised his head: "You are Gu Mingche?"

Gu Mingche sighed inwardly, thinking that his excuse of having the same name last night would probably only work once.

Hearing this name, the people waiting in line around them cast curious gazes over.

The two doctoral students simply followed him and moved next to a pillar, and one of them hurriedly opened the main theorem page on his phone.

"The internet has been spreading six times ten to the negative thirty-eighth power. This number is really too small; just how big of a difference can there be between it and zero?"

Gu Mingche looked at the breakfast window and explained earnestly: "The strict lower bound given by the theorem is this explicit numerical value. No matter how small it is, it is an unequivocal constant, firmly nailed to the right of zero."

"So the current δ has an explicit value?"

"Yes, both sets of independent interval certificates have confirmed this lower bound."

Hearing this, a bystander listening nearby finally asked the question that had been frantically discussed on the Chinese internet since last night.

"Is a positive number so small that writing it out requires dozens of zeros really that different from zero?"

"For this problem, it is different by a whole conjecture."

As he spoke, Gu Mingche circled the traditional o(1) on his phone screen: "The value here will shrink all the way toward zero as the number of points n increases infinitely."

Immediately after, he circled δ.

"And this number, although small, is nailed in place, so n to the power of δ will eventually exceed the logarithm of any fixed power."

The bystander gazed at the two mathematical formulas on the screen and nodded thoughtfully: "In other words, everyone thought that part of the growth would become thinner and thinner and eventually go to zero, but Jiang Lin proved that it can at least retain a layer of fixed thickness?"

"That's pretty much what it means."

"How thick, we don't know, but it will definitely not disappear."

"Right."

The bystander breathed a sigh of relief, took out his phone, and typed quickly, sending this popular explanation he had just understood into the class group chat.

Just two minutes later, these three lines of dialogue were screenshotted and forwarded to the Tsinghua University internal BBS forum, with the post title bolded in bright red.

[Why as long as it is greater than zero, the conjecture has already lost]

When Gu Mingche finally bought his tofu pudding and found an empty seat, the view count of that post had quietly surpassed the total download count of his preprint paper when he left Dorm 402 last night.

Eight fifteen in the morning.

The public briefing session originally scheduled to hold thirty people was forced to urgently change locations three times within a short hour.

The Mathematical Sciences Center initially decided to move to a lecture hall that could hold one hundred and twenty people, but the registration system announced it was full just four minutes after opening.

After emergency coordination, the university authorities finally borrowed the Tsinghua Grand Auditorium, which could hold six hundred people, and handed over the on-site online live-streaming work to the university news center.

When the Instructor called Jiang Lin, Jiang Lin was sitting in front of the computer in 402, assigning numbers to new doubts in the public attack catalog.

"You have to sit in the front row today, right?" The Instructor exhorted over the phone.

"Of course, but the explanation will still be handled by the independent verification group." Jiang Lin replied.

"Right, they are responsible." A trace of helplessness tinged the Instructor's tone: "However, among the six hundred people coming today, at least five hundred and ninety of them are not here to listen to them talk about class field towers."

"We can hand out attack packages to them on the spot."

"If you brat dare to hand them out, they'll be able to bury you in the Grand Auditorium with print paper today."

The Instructor asked Jiang Lin to prepare a fifteen-minute speech, talking only about what the problem was, what the paper claimed, which parts had been verified, and which parts remained open.

Jiang Lin hung up the phone, closed the complex proof that was over six hundred pages long, and created a new presentation with only four pages in total.

Page one: A regularly arranged grid lattice of points.

Page two: Connecting lines drawn with a distance of exactly 1 between points.

Page three placed n^(1+o(1)) and n^(1+δ) side by side.

Page four: Only the preprint public link, the attack catalog entry, and a line of bold text below.

[The conclusion is still undergoing inspection by the whole world.]

Ten in the morning.

The long line at the entrance of the Grand Auditorium had extended all the way down below the steps.

When Jiang Lin entered from the side door, Zhao Chengyu, Lin Yizhou, and Xu Zhiyuan had already saved a seat for Gu Mingche.

"I heard you held the first grassroots academic briefing session in the cafeteria this morning?" Zhao Chengyu asked Gu Mingche with a smile.

"That's right, the organizer was a broken soy milk maker."

"Equipment failure effectively promotes academic exchange. This news material can be casually reported to the publicity department of the Low Entropy Workshop," Zhao Chengyu joked.

The Grand Auditorium was bustling with people, packed with students from the Mathematics, Physics, Computer Science, and even Electronics departments. Teachers and researchers who rushed over upon hearing the news were also standing in the back rows and aisles.

Near the central aisle, the university television station set up three professional cameras.

The university authorities intercepted external media outside the inner arena, but the online viewer count of the official live-streaming room fiercely crossed the two hundred thousand mark before the opening.

Jiang Lin's four-page presentation started from the grid point model originally proposed by Erdős in 1946, deduced all the way to the geometric meaning of the fixed normal number, and then hung the two key problems Q-001 and Q-002 after the third page.

He explicitly stated to the whole audience that the current version had only completed controlled cross-verification, and external reviews could still restart from any logical step.

Students in the front row raised their phones to capture the algorithm boundaries on the screen, professors in the back rows aimed their lenses at the podium, and people in the live stream's bullet comments continuously pressed with questions.

Since multiple groups of top experts had checked it, why did the university authorities still not use the four words "already solved"?

The Instructor took the microphone and projected the current main theorem's logical dependency graph and internal verification scope completely onto the big screen.

"Mathematical proofs never vote based on the number of reviewers; they only recognize whether each step of derivation can be independently reconstructed. We only sign for the parts we have strictly checked, and we cannot waive later people's right to question on their behalf."

The Instructor's voice spread throughout the Grand Auditorium.

"Anyone sitting here today, or anyone in front of the live stream, as long as they can find a blocking loophole in the tower lemma or subsequent interfaces, the public reproduction status can be hung with a red card at any time."

The impetuous bullet comments in the live stream were quickly pushed away by subsequent rigorous discussions, while the crisp sound of Enter keys from dozens of laptops simultaneously downloading the paper's attachments instantly rang out at the scene.

Professor Lin Zhaoye began taking the microphone to explain the planar projection interface, while the remotely connected European tower direction expert was responsible for answering the technical details of the pro-2 construction.

Jiang Lin sat quietly on the side of the podium, only speaking out to report the page numbers when someone asked about the specific axioms and definition locations.

At eleven twenty-six, a high-priority email suddenly popped up in the upper right corner of the master control computer on the podium.

The sender had a long-term collaboration lasting several years with Erdős himself, and was also one of the most authoritative researchers on the unit distance problem today.

After carefully reviewing the main theorem, construction master plan, and existing verification records, he CC'd this email to Jiang Lin, Professor Lin Zhaoye, and Terence Tao simultaneously.

The first paragraph of the email body confirmed:

If the fixed normal number given in the paper can withstand broader public verification by the global academic community, it will formally negate Erdős's most well-known core conjecture on this problem.

And the evaluation in the next paragraph went further than all previous "if correct" evaluations.

This old scholar believed that the profound connection between high-degree number fields, class field towers, and planar geometric unit distances was completely unexpected, and the existing construction was fully deserving of being regarded as a major milestone achievement that could completely solve the classical problem.

The attachment to the email was a scanned page.

An old, yellowed list of academic problems from over thirty years ago.

After obtaining consent, the Instructor directly projected this scan, heavy with a sense of history, onto the big screen.

Right in the center of the paper was printed the old number for the unit distance problem, alongside the conjecture of n^(1+o(1)) that had ruled the academic community for seventy-six years, and at the end of the formula was marked an easily overlooked symbol and number.

[$500]

In 1946, Erdős used a few concise rows of grid point deductions to give a lower bound that had been maintained for seventy-six years, and also tossed this difficult problem to future mathematicians.

Years later, he offered a cash prize of five hundred dollars for this conjecture.

And the δ that Jiang Lin placed on the third page of the presentation was now like a sharp sword, nailing into the other side of this old list of problems.

From the back row of the Grand Auditorium came first a slight sound of a folding desk board closing.

Immediately following, someone clapped the first pair of hands.

Applause quickly spread from a few scattered seating areas, and then surged wildly like a tide along the stepped seating.

The remote audio pickup level bar on the stage was pushed into the red zone representing the peak in an instant.

Gu Mingche had seen top researchers in reality use all kinds of methods to recalculate five, six, and that string of thirty-seven zeros for the lower bound, and he knew even better that the five hundred dollar prize on the big screen might not even be enough to cover the paper costs for printing papers and materials over these past few days.

But that piece of paper, starting from the hands of Erdős and walking all the way to Jiang Lin today, had walked for a full seventy-six years.

The applause lasted for a very, very long time. The host raised his hands twice in succession, but failed to pull the atmosphere at the scene back into the rhythm of academic explanation.

Jiang Lin rose from his seat, smiled and greeted the scholars participating in the verification remotely and the on-site review group, and then handed the microphone back to the Instructor.

At twelve noon, the university's official website released a public statement, with the title continuing the formal phrasing used on Jiang Lin's presentation.

[Tsinghua University's Jiang Lin Gives Super-Linear Lower Bound for Planar Unit Distance Problem]

The statement clearly listed the core claims of the paper, the specific scope of internal cross-verification, the public preprint version, and subsequent global review arrangements.

Then, the media began a new round of carnival.

[Eighteen-Year-Old Freshman Challenges Seventy-Six-Year Mathematical Conjecture]

[From Rescue Robots to Unit Distance: Jiang Lin Enters Public View Again]

[Why a Positive Number of Unknown Magnitude Can Shock the Entire Mathematical World]

[From Qingya Tunnel to the Grand Auditorium: Decoding the Dual Lives of a Genius Freshman]

Netizens who were still fiercely debating on social platforms in the morning about when the Low Entropy Workshop rescue robots would open for individual purchases swarmed into the ArXiv preprint page en masse following the same name.

Someone dug up his live photos at the World Robot Conference, and someone even edited the shocking footage of unit 4 propping up as a communication relay station during the Qingya Tunnel rescue into the same video with those four clean presentation pages in the Grand Auditorium today.

Among them, the comment with the most likes had only two extremely brief sentences.

[I originally thought that him running a company already delayed his studies a lot.]

[Now it seems it really only delayed him just enough to overthrow a seventy-six-year mathematical conjecture.]

At one in the afternoon, the preprint download count crossed the one-hundred-thousand mark.

In the public attack catalog, ninety-three new academic doubts with clear page numbers were added.

Twenty-one of them had already received real-time replies from Jiang Lin's team, and six had entered independent deduction and recalculation, with no blocking items capable of shaking the main chain structure appearing temporarily.