207: Chapter 207 Speculation about the morning of the day he died

November 14th, 8:00 AM.

Zijing Apartment Building 17, Room 402.

Gu Mingche dragged a 42-page verification report to the left side of the screen, pulled out the initial construction master plan from Jiang Lin's increasingly thick "Public Verification Instructions", pressed it beside the printed paper final draft, and pointed with the tip of his pen to compare them item by item.

That old problem list marked with five hundred dollars was clipped behind the R-003 errata report, the staples piercing through three layers of paper, its edges having grown somewhat frayed from him thumbing through it over the past few days.

"Did they really not use a single certificate?" Gu Mingche looked at the parameter set on the report.

"The author attachments, parameter seeds, and decomposition tables are all in the isolation list."

Jiang Lin scrolled the mouse, bringing up the material declaration at the end of the report: "The reconstruction team could only read the paper definitions, main theorem, and the seventeen public conditions that needed to be met. As for which primes I chose, they couldn't see a single line of it."

Over the past six days, seven researchers from Bonn, Princeton, and Budapest had rewritten more than thirty thousand lines of code in a clean-room environment without seeing the original code.

They chose another set of ramified primes and completely split primes, constructed a pro-2 class field tower from scratch, and then compressed all conjugate embeddings into the same computation window.

In the first round, the tower they selected stopped at the sixth layer, the growth of the class group rank was eaten up by the discriminant cost, and ultimately they could only obtain n multiplied by a string of logarithms, meaning the construction failed.

In the second round, after changing the base field, the tower barely survived to the eleventh layer, but the planar projection hit a batch of duplicate points; after deleting the duplicates, the fixed index shrank back to zero.

Until 3:17 AM today, when the third round of testing ended.

The reconstruction team used a set of labeling primes and projection parameters completely different from Jiang Lin's original text, and successfully obtained another family of planar point sets.

Although the index of this new parameter set was worse than the existence lower bound given in the paper, it genuinely and solidly stopped to the right of zero.

On the first page of the report, a solitary line of bold text summarized all the results:

Clean-room reconstruction successful.

Gu Mingche read this line of English for the third time, his finger sliding down along the material declaration to confirm that those three local decomposition certificates, which had once put them through three hours and forty-three minutes of torment, were indeed lying obediently in the disabled column.

"Recalculating your files at most only proves that you weren't lying in the files." He flipped the report to the third-round parameter table. "Now that they've changed a bunch of numbers and can still produce the same thing, does that finally count as truly having learned it?"

"At the very least, it proves that this route does not depend on that specific set of samples."

"Can you temporarily turn off the word 'at least' for ten minutes?" Gu Mingche pointed at the signature column on the last page. "All seven people signed it, and the German professor who was originally the most skeptical of the pro-2 relation budget even wrote his name neatly right at the very front."

Jiang Lin didn't reply, his gaze lingering on the two reservations in the appendix.

These two opinions were not directed at the mathematical logic, but rather requested bringing forward the boundary conditions already stated in the main manuscript.

It was suggested to simultaneously place "except for a finite number of parameters" from the end of the proof into the first sentence of the planar projection lemma.

It was suggested to move "holds for infinitely many n" from the proof of the main theorem forward to the abstract, lest external media and non-professional readers mistakenly misstate the infinite construction family as being directly constructible at every integer scale.

Both modifications only adjusted the positions where the information appeared; the fixed positive constants, quantifiers of the main theorem, and scope of application had not been changed by a single word since the original manuscript.

Jiang Lin included them in the v1.1 layout modification table.

At this time, the top of the public status page still retained the yellow "Under Review" marker.

Gu Mingche stared at that patch of yellow for a moment, picked up his phone, and sent a message to the Instructor.

"What are you doing?"

"Asking for instructions on how to handle a teaching accident."

"What accident?"

"Last month, the Instructor printed it as a conjecture in the handout and even assigned a problem asking us to explain why all known constructions support n to the power of one plus little-o."

Scarcely ten seconds after the message was sent out, the Instructor's call came straight through to Jiang Lin's phone.

"Have you finished reading the report?" The Instructor's voice was somewhat deep.

"Finished reading it. Independent reconstruction holds, and the two reservations need to go into v1.1."

"I am asking about the main chain."

"The current five sets of external verifications plus the clean-room reconstruction have not found any main chain blocking items."

On the other end of the phone, the Instructor took a long breath, and the sound of rummaging for a chalk box immediately came from the desk drawer.

"Class is at nine-fifty. Could I trouble you and Gu Mingche to come over?"

Hanging up the phone, the Instructor sent a revised electronic handout to both of their email inboxes.

The semi-algebraic sets originally ranked in the third section were moved back as a whole, and the title on the front page was cleanly and neatly changed from "Planar Unit Distance Problem" to "New Lower Bounds for the Planar Unit Distance Problem".

The old conjecture expression was still on the first line, but an eye-catching red deletion marker had been added behind it.

After the call ended, Gu Mingche stared at the screen where the old and new versions of the handout were enlarged side by side, and finally found something even more worth screenshotting than that five-hundred-dollar list.

"When you went to audit for three hours back then, the Instructor laid out three accounts for you." He tapped the screen, his voice suppressing a smile. "Who would have thought that only twenty-eight days later, he'd be forced to rewrite this entire class."

"The three accounts are still there."

"Isolation reconstruction is already done, how can they still be there?"

"The point counts, difference vectors, and multiplicity counts have already been paid off." Jiang Lin's tone was calm. "But the explicit indices, optimal constants, and upper bound gaps have generated new accounts."

Gu Mingche decisively closed the old version of the handout, refusing to accompany someone who had just overthrown a seventy-six-year-old conjecture in developing new academic debts before breakfast.

At 9:02, Professor Lin Zhaoye submitted the third round general report in the public review group.

This report heavily summarized all the turbulence of the past few days.

One hundred and forty-seven valid challenges, two pre-existing boundary forward-shifts, one index errata, five sets of section-by-section verifications, and one set of isolation reconstruction.

Q-001, Q-002, and R-003 were rigorously listed in the three columns of tower relation bounds, explicit constants, and public indexes, respectively.

Q-001 and Q-002 recorded key challenges that had already been closed by independent routes, while R-003 recorded index errors that did not affect the mathematical objects.

Every main chain dependency in the current version had obtained signatures from at least two external researchers, and the clean-room group had used different parameters to produce a second family of instances.

Professor Lin Zhaoye typed out the review group's joint opinion in the conclusion column.

Review Conclusion: Within the scope of the public declaration, existing evidence supports the validity of the main theorem. It is recommended to end the candidate counterexample status and transition to publicly verified status for continued journal peer review.

The editing privileges for the yellow marker were still in Jiang Lin's hands.

He silently verified the seven signatures, version hashes, and two reservations, pulling up all records of the status page since its release once more:

November 6: Controlled verification completed.

November 7: Publicly released. In the afternoon, P-094 triggered a temporary public reproduction block.

November 7: After three hours and forty-three minutes, R-003 was archived, and the block was lifted.

November 8–13: Four sets of cross-attacks completed.

November 14: Isolation reconstruction completed.

At 9:11, the yellow banner at the top of the page finally disappeared, replaced behind the main theorem by a serene dark green marker.

Public Verification Passed | Journal Peer Review in Progress

Jiang Lin submitted v1.1 to the preprint platform, moving the two existing boundary descriptions from the original manuscript forward to the abstract and theorem pages.

All challenge records, independent verification materials, R-003 index errata, and attack directories were retained as they were, without any whitewashing.

Five minutes after the status page update, Terence Tao sent an email.

He had already read the isolation reconstruction report, personally and independently checked the index accounts between the new parameter family and the main theorem, and explicitly acknowledged that the existing proof and external reproductions were sufficient to support the fixed positive constant lower bound.

At the end of the email was an update link to his personal academic page.

The title of that short essay from six days ago was still "Jiang Lin Claims to Have Provided a Counterexample to the Unit Distance Conjecture", but Gu Mingche keenly noticed that two English words were missing from the end of the first paragraph of the main text.

He dug out the old webpage cache and placed it side-by-side with the newly updated page.

The old version on the left read: "if correct."

In the new version on the right, that sentence landed directly on the conclusion right before the period, no longer carrying any conditional adverbials.

"9:23." Gu Mingche captured the system time of the two screenshots. "These two words survived for six days and eight hours."

From "if correct" to "already established."

The deletion of those two words was undoubtedly a formal endorsement given by a top mathematician.

Jiang Lin ignored this exclamation, simply adding the subsequent questions listed in Terence Tao's email to the public directory.

"He has listed increasing δ as an open problem."

"That belongs to the trouble of the next paper." Gu Mingche closed his laptop, grabbed his schoolbag, and walked toward the door. "Today's class will first let the old conjecture die completely."

At 9:47, in the classroom originally scheduled for sixty-two people, more than ninety people had already crammed in.

In the hallway, there were still students hugging their computers and leaning against the wall, standing on tiptoe to peer inside.

The Instructor guarded the doorway tightly, insisting on letting only teachers and students of the school enter, and directly handed over the two media outlets that had rushed over upon hearing the news carrying cameras to the school's publicity department for handling.

"Last time we borrowed the six-hundred-person auditorium, people in the back rows couldn't even see the formulas on the blackboard clearly. This time, whoever wants to listen can squeeze in themselves."

The Instructor moved out a folding backup chair from under the podium and shoved it to Gu Mingche.

"You sit on the side of the podium; don't take up the seats of the students enrolled in the class."

Gu Mingche hugged the chair and passed through the gaps in the front row, suddenly finding it somewhat amusing.

As the attachment maintainer of this academic-sensation-causing paper, his treatment was surprisingly stable.

Before becoming famous, he checked letter subscripts in the audience; after becoming famous, he brought his own stool to sit on the side stage.

At 10:00 sharp, that newly revised handout appeared on the projection screen on schedule.

The first person invited onto the podium by the Instructor was precisely that doctoral student who had handed Jiang Lin three failed counterexamples of number fields four weeks ago.

He participated in the fifth group's external verification and was also responsible for the planar projection part of the isolated reconstruction; the first image currently opened on his computer was still his own failed construction from back then.

"Last month, I used this example to argue against letting the number field degree continue to grow. Because after the third expansion, other embedding directions would expand much faster than the planar window, making the cost unbearably large."

The doctoral student held a laser pointer and switched the image to the third round of results released by the clean-room group at 3:00 AM.

"Jiang Lin's proof forces the various embeddings into the same controlled window, and then repays the discriminant and class number costs through the tower's growth. I initially thought this account would be overdrawn again at some layer. But after we changed his base field, marked primes, and projection parameters, the first two rounds indeed failed, yet the third round yielded another family of fixed positive constant lower bounds."

The PPT turned to the last page, bearing his handwritten signature.

"All the main chain objections I submitted have been closed. There are two remaining reservations, which only concern the prominence of the existing boundaries on the abstract and theorem pages and do not affect the mathematical essence."

A student in the back row raised a hand to ask: "So you think the conjecture has been overthrown?"

"As a verifier, I believe the main theorem of the paper holds." The doctoral student closed his own failed construction diagram, his gaze sweeping across the audience, "But as a researcher, I hope someone continues to attack it, because I still want to know just how far this path can go."

The sound of keyboards in the classroom instantly merged into one, and several students by the door raised their phones, aiming at the signature column on the big screen and pressing the shutter button.

The Instructor picked up a white chalk, walked to the left side of the blackboard, suspended the wrist, and brought the chalk down.

Rewrote 1946, the grid point construction, and $n$ to the power of one plus small-o of one on the blackboard again.

The layout of the blackboard writing was identical to the beginning of that class four weeks ago.

The whole class quieted down.

Everyone knew what was going to happen next.

Everyone quietly waited for what was about to happen.

The Instructor's wrist sank, and the chalk tip pressed heavily onto the conjecture formula.

Screech—

A thick, straight horizontal line was drawn from the first power of $n$ to the end of $o(1)$.

Snap!

The chalk head snapped and fell onto the podium, making a crisp sound.

Without pausing, the Instructor switched to the flatter broken end and continued to write a new line of words forcefully below.

Theorem (Jiang, 2022): There exists a fixed constant $\delta > 0$ such that infinitely many $n$ satisfy $U(n) \ge n^{1+\delta}$.

"From today, from this very moment, the old conjecture is void." The Instructor turned around, patting the chalk dust off the hands, with barely suppressible excitement in the voice, "Starting today, this line officially enters the known results of this course, becoming a cornerstone in the edifice of human mathematics."

The moment the words fell, the classroom fell completely silent for two seconds.

Chalk dust was still slowly settling in the slanted skylight, the fingertips of the pen-clutching students in the front row paused on the paper, those holding phones to photograph the blackboard forgot to press the shutter, and even several students from other departments squeezed and tiptoeing near the door subconsciously held their breath.

Everyone's eyes were fixed on the blackboard.

That thick horizontal line spanning across the conjecture formula, and the brand-new theorem below it that still carried the force of the chalk.

A seventy-six-year-old verdict had become history in the span of this single stroke.

It was unknown who started the first round of applause, and immediately afterward, the scattered points of applause merged into a piece.

It wasn't deafening, but it carried a solid, burning passion beneath it.

Someone clapped while locking eyes with the classmate next to them, their eyes startlingly bright.

Several graduate students in the back row who had stayed up for days guarding the public status page smiled and shook their heads, their fingertips gently tapping the covers of their notebooks filled with derivations.

People crowded in the corridor who couldn't get in heard the applause and also gently began to clap, the muffled sound passing through the door panel into the classroom.

When the applause subsided slightly, the Instructor raised a hand to press downward, smiling and adding: "Since the old conjecture is gone, naturally the exercises assigned in the third section are all void. However, students who have already finished writing can hand them in, and I will still grade them as extra assignments."

As soon as the words fell, the classroom first burst into a low chuckle, mixed with a few sighs of sudden realization.

Many people directly laid the pens in their hands onto their notebooks, exchanging wry and helpless glances with one another.

Who would have thought that the exercises chewed through by staying up until midnight last week would turn from course highlights into historical documents in just a few short days, and even the opportunity to properly submit homework was directly voided.

Amidst the laughter, a boy in the front-left row didn't stop; he lowered his head to unzip his schoolbag, and a rustling sound of pages being searched through was exceptionally clear amid the classroom discussions.

When he straightened up, he held three neatly stapled sheets of homework paper in his hand, the edges of the pages already slightly curled from being thumbed through, and the surface of the paper densely filled with formulas.

Blue ink traced the main derivations, red ink marked the sources of lemmas, and several lines of small print were added in the blank spaces between lines for corrections and notes, even with two steps of upper-bound estimations calculated at the very corners—he had obviously chewed through all the problems with utmost earnestness.

He leaned forward from his seat, handed the homework to the edge of the podium, and scratched his head a little sheepishly: "Instructor, I just finished writing it yesterday."

The Instructor took the destined-to-be-invalidated set of answers, wrote a line of reserved questions on the first page, and handed the half-piece of chalk to the doctoral student, asking him to explain to the whole class starting from the clean-room construction why the original quadratic number field route would get stuck near the zero exponent.

On the podium, the doctoral student was just talking about the second-layer embedding constraints of the tower construction, with soft whispers of discussion floating in the classroom, when Jiang Lin's phone in his pocket suddenly vibrated six times in succession under silent mode, the vibrations short and dense.

He took it out to take a look.

The first email came from the maintainer of the global electronic archive for unit distance problems.

In this academic entry that had been used for decades, the words "verification in progress" had been officially replaced by "disproved".

Jiang Lin's original paper, the v1.1 verification report, the full attack catalog, and the R-003 index errata were listed together in the evidence column, becoming the official support for this conclusion.

As for that open problem which had hung there for many years marked with a five-hundred-dollar prize, its status had been updated to "waiting for management confirmation of payout".

The second was a system notification from the preprint platform: on the eighth day of the paper going online, the full-text download count officially broke 300,000 times.

The third came from the editorial department of a top combinatorics journal in the field.

They hoped Jiang Lin would submit the paper to them, and they would add a green channel for this paper to strive for publication as soon as possible.

The fourth was a screenshot sent by Liang Zhixia; the backend visit curve of the Low Entropy Workshop official website jumped steeply.

A large number of visitors flooded into the robot product page and Jiang Lin's paper description page simultaneously, and the customer service representatives who were originally only responsible for replying about delivery cycles were now frantically explaining to callers over and over again.

The prize money for the mathematical conjecture was not linked to production capacity, nor would the production scheduling order be advanced because of it.

[Liang Zhixia]: "I plan to add a pinned FAQ on the official website—mathematical conjectures and industrial production scheduling orders do not affect each other."

[Jiang Lin]: "Okay."

The remaining two vibrations were notifications from the department and the seminar; he glanced at them and turned off the screen.

Gu Mingche next to him nudged him with an elbow, handing over half a piece of paper torn from a notebook.

On the paper, a detailed account was crookedly listed in pencil: seven steamed buns, four cups of soy milk, two bowls of instant noodles, and two printing fees. Each item was marked with its RMB price at the end, and the last line was specially converted into US dollars using today's real-time exchange rate in small print, with the total adding up to less than one-twentieth of that five-hundred-dollar prize list.

"The public verification has passed, so my debt should transition from candidate status to the payment process, right?" Gu Mingche lowered his voice, a teasing smile at the corners of his mouth.

Jiang Lin glanced at the slip of paper, then spotted a blind spot: "Why are the printing fees counted twice?"

"That index error and omission in R-003 forced me to reprint a version of the construction diagram."

"Your signature is on the errata sheet, which counts as a joint mistake."

"That's why I only charged the material costs of paper and toner, without calculating emotional distress damages."

Jiang Lin didn't reply further, tucking the crumpled slip of paper into the back of the five-hundred-dollar problem list.

At 11:32, after the class-ending bell had rung twice, the crowd gathered in front of the podium gradually loosened up.

Someone chased after the doctoral student to ask about the details of the projection parameters, and a few students hesitated and took two steps towards Jiang Lin's side. Seeing that he was lowering his eyes to flip through the subsequent problem entries in his notebook, with his fingertips still pressing on an unfinished line of parameter estimations on the page and an expression of focused concentration, they felt too embarrassed to step forward and interrupt, standing just a few steps away to discuss in low voices.

The Instructor put the three-page homework answers into the lecture folder, turning back to see that the property's janitor had already pushed the cleaning cart and was quietly waiting by the back door.

"Instructor, there are still classes this afternoon for this board. Should I wipe it?" The janitor pointed to the right half of the blackboard, which was densely filled with formulas.

The Instructor took out a phone, stepped back two steps, photographed the complete blackboard writing, and then turned back to instruct the doctoral student to copy the subsequently derived tower construction into the electronic lecture notes for preservation.

"The right side can be wiped." The Instructor put away the phone and pointed to the left, "Leave the left side; for this afternoon's combinatorial geometry seminar, we will continue lecturing from here."

The janitor nodded and pressed the board eraser onto the right side.

Chalk dust fell layer by layer into the slot below, and the complex derivation process on the right gradually turned into a blurry white shadow.

Meanwhile, in the untouched upper-left section, the handwriting remained clear.

That thick, straight chalk line spanning across the old conjecture formula, like a quiet boundary marker, cleaved open the old era of seventy-six years.

And beneath it was the theorem that had just been written into the course lecture notes, belonging to Jiang Lin.