216: Chapter 216 A Positive Number That Will Never Disappear

Jiang Lin first moved the two narrow boxes out of the delivery boundary, checked the box locks and anti-tamper seals one by one, and then placed the three sets of storage media into the offline cabinet.

He packed the paper task card into a separate file bag, and those two lines of text still remained at the very top.

[Primary Task of the Fifteenth Round: Deliver communication satellites to the designated Earth-Moon relay orbit]

[LF-03 Response: Verify after the link is established]

A new countdown had already begun ticking at the edge of his vision.

Twenty-nine days, twenty-three hours, fifty-nine minutes, and over fifty seconds.

The launch vehicle in the LC-04 sealed area would remain parked there according to standard procedures, and T-01 would also keep waiting for that set of wheel-end components.

Jiang Lin added the component model into the procurement list, closed the offline cabinet, and then went to open the metal cabinet outside the boundary line.

His campus card, phone, and hard-shell folder were all in their original places.

He changed out of his work clothes, which were so worn out that their original color was no longer visible, folded them neatly and put them into a sealed box, and then put on the real-world clothes he had left here last night.

When his phone reconnected to the network, the real-world time had just passed six-three.

The last message sent by Gu Mingche last night was still lingering in the dialogue box.

[At eight-twenty, by the side door of the lecture hall, I will pick up the question catalog for you.]

Jiang Lin replied with an okay and unwrapped the rice ball.

Sitting beside the two narrow boxes that had just accompanied him through forty-three years, he finished this real-world breakfast with a cup of soy milk.

The sky outside the warehouse had already brightened.

A park delivery truck passed by the public driveway, its backing-up alert sounding a few times through the rolling shutter door before moving on to the next row of warehouses.

Jiang Lin tidied away the food packaging, locked the Wasteland task card back into the offline cabinet, kept the real-world procurement list on his phone, and packed the report materials for the morning into the hard-shell folder.

At eight-twenty, Gu Mingche was already standing outside the side door of the lecture hall.

Holding that hard-shell folder in his arms and with a computer bag slung over his shoulder, he was lowering his head to look at the newly added entries in the remote question pool.

Jiang Lin took the folder and entered the side door with him.

The first four rows of the lecture hall were already fully seated, people kept coming into the back rows, and staff members were collecting the temporarily added seating cards one by one.

Reservations for on-site seats had stopped the day before, everyone who could enter had a confirmation email, and adding more chairs would only block the evacuation route.

On the main screen, the name of Jiang Lin appeared twice in total.

[Organizer: Jiang Lin Cross-Disciplinary Science Research Center (Preparatory), Tsinghua University]

[Speaker: Jiang Lin]

One staff member at the live broadcast console was still a bit uneasy and asked through the headset whether it was a duplicate entry.

The administrative assistant was guarding beside the console, answering with practiced familiarity.

"Both places are correct, please keep them."

After saying this, she continued to verify the remote lines, and a proof of a job announcement that had not been sent back to the office yet was pinned under the edge of the desk.

Ever since the center began preparations, reports, recruitment, visit applications, and space handovers had not lined up in a neat queue in chronological order; several matters often occupied her desk simultaneously.

When Jiang Lin walked to the outside of the first row, he saw a heavily worn gray folder placed on a reserved seat.

Two labels were pasted on the cover of the folder.

[Clean-room Reconstruction／Round 1]

[Clean-room Reconstruction／Round 2]

Both labels were stamped with a red FAILED.

The owner of the folder was that reconstruction team member who had rushed over from Budapest.

The first two isolation reconstructions did not follow the primes, parameters, and local certificates provided by Jiang Lin, and in the end, they failed to obtain a tower satisfying all conditions.

For the third time, they changed to another set of parameters and independently obtained a fixed positive exponent.

When he came to Beijing, he stored the success records in his computer, but his paper luggage consisted almost entirely of why the first two attempts failed.

"I would like to use one of the counterexamples during the Q&A session."

He confirmed with Jiang Lin in English.

"It is not to question the theorem, but to ask about the applicability range of low-degree obstructions."

"You may."

Jiang Lin opened the folder and saw that termination conditions were written behind each failed branch.

"State the conditions fully, do not just leave behind a failure conclusion."

The researcher nodded and moved the yellow sticky note clipped to the first page to the position prepared for questioning.

Gu Mingche had already opened the remote queue beside him. Hearing this, he casually marked the on-site question from the other party as reserved.

Before nine in the morning, Professor Lin Zhaoye came over from the other side and checked the post-report discussion arrangement with Jiang Lin.

The introduction by Professor Lin Zhaoye was very short, glossing over official publication, independent verification, and unsolved extremal functions with a single sentence each.

"Today we will no longer vote on whether this paper is right or wrong."

Professor Lin Zhaoye placed the last page of the hosting materials face down on the table.

"Those who want to ask questions should try to raise them to a level where we can take action in the next step."

The live broadcast status in the upper right corner of the main screen turned red, and the wall clock of the lecture hall pointed right at nine in the morning.

Jiang Lin walked to the blackboard on the left and first wrote down a very short definition.

Taking n distinct points on a plane, what is the maximum number of pairs of points whose distance is exactly 1? This maximum value is denoted as U(n).

Beneath the definition, he drew nine neatly arranged points and connected several line segments of the same length.

"The problem has remained from 1946 to today, and the statement of the problem has not become complicated."

Jiang Lin put down his ruler.

"What is complicated is that people have never known to what extent this graph can be packed at most."

Three lines of equations subsequently appeared on the main screen.

[Known upper bound: U(n) = O(n^(4/3))]

[Previous lower bound: U(n) ≥ n^(1 + Ω(1 / log log n))]

[Erdős conjecture: U(n) ≤ n^(1 + o(1))]

The gap between the first two lines was very large, while the last line suggested that the tiny fraction exceeding the first power would gradually shrink away as n increased.

Jiang Lin wrote down the main theorem of the paper on the right side.

[There exist absolute constants C > 0 and 38 such that infinitely many n satisfy U(n) ≥ n^(1 + δ) / C.]

Quite a few people in the lecture hall had already read this line of theorem, yet some still picked up their phones and aimed them at the exponent on the blackboard.

"Six times ten to the negative thirty-eighth power is very small, so small that it is difficult to make a finite-scale graph exhibit visible changes to the naked eye."

"Yet it remains fixed there, and the continued increase of n will not shrink it back into o(1)."

Jiang Lin added a horizontal line beneath δ with chalk.

"What is being revised today is the third line."

"As for which exponent U(n) ultimately approaches, the blank space remains."

Without jumping directly from the conclusion into the over four thousand dependencies, he first flipped out the failed drafts of the cubic number fields left over from that class in Chapter 192.

That draft was originally only twenty-seven pages long, its corners softened by thumbing, and the cover still retained the three questions written at the time.

Can the points remain?

Can there be enough unit difference vectors?

After projection, will repeated points and repeated edges swallow up all the gains?

"Fixing the cubic field is not because the parameters have not been tuned well yet."

Jiang Lin wrote several sets of effective gains of different scales on the blackboard, letting them approach zero one by one as n increased.

"In the construction covered by this obstruction lemma, with the degree fixed, the gain that can be achieved will shrink back."

"Therefore, what needs to be changed is not the decimal point, but the degree of the number field."

The left blackboard kept the failed routes, while an upward-extending CM-field tower began to appear on the right blackboard.

The old lattice construction continuously put more primes into a fixed number field, but the direction adopted by Jiang Lin was precisely the opposite.

He fixed the primes satisfying the local splitting condition, letting the number field grow along the infinite tower while controlling the root discriminant within a consistent range.

The infinite tower was not guessed through finite-layer calculations.

Jiang Lin wrote three sets of numbers in the center of the blackboard.

[d(G) = 5]

[r(G) ≤ 6]

[d(G)^2 / 4 = 6.25]

Strict inequalities were left among the number of generators, the number of relations, and the Golod-Shafarevich threshold, ensuring that every subsequent layer had a proof of continued existence.

Thus, the degree of the number field could grow continuously, and the completely split fixed primes would also provide increasingly many available arithmetic directions at each layer.

These directions could not yet be directly treated as points on a plane.

Jiang Lin turned to the second movable blackboard, arranged all conjugate embeddings of the CM field into Minkowski space, and drew a polydisc window outside.

After the lattice points fell into the finite window, the number of points was jointly controlled by the window volume and the root discriminant.

The difference vector satisfying the conjugate norm condition then gives a unit modulus in the chosen complex embedding, and can send a large number of lattice points inside the window to another lattice point still within the window.

The final plane projection cannot be chosen arbitrarily.

Which points will coincide, which unit differences will be counted repeatedly, and which exceptional fibers must be cleared first—the last three of the six proof packages are paying the bill for this step.

Jiang Lin left a reconciliation table of duplicate points on the main screen, and continued to derive the loss terms on the blackboard.

After the loss entered the constant and exponential budget, the remaining δ was still greater than zero.

It was very small, yet it was not swallowed up by any of the accounts in the point counts, difference vectors, multiplicities, and projections.

The ninety-minute presentation did not bring all four thousand three hundred and seventeen dependencies to the stage item by item.

Jiang Lin only stopped at each change of route to explain what the old method specifically lost and where the new conditions made up for it.

The public dependency graph was always kept on the right side of the main screen, and anyone present who clicked into any node could see the corresponding lemma, parameter interval, verification record, and person in charge.

R-003 had not disappeared from the event directory either.

That record still stated that the third public certificate entry had repeatedly pointed to the second one, and the fix contained only a single index link.

The reconstruction team from Budapest did not use those three author certificates from beginning to end, and another set of primes and parameters they adopted in the third round had also been hung under the independent material entry.

After the presentation entered the Q&A session, the first person to stand up was the PhD student who had submitted the failed manuscript of the cubic number field in Chapter 192.

He was holding the original manuscript from back then, with many recently written annotations added to the first page.

"What the low-degree obstruction lemma can rule out is precisely the fixed-degree route in this Minkowski window construction, or all fixed number field routes?"

"The former."

Jiang Lin turned back to the hypothesis page of the obstruction lemma and circled the three conditions again.

"What it constrains are routes of fixed degree, discriminant, and multiplicity satisfying these consistency conditions, and taking points through such windows."

"If someone finds another mechanism in a fixed number field that is not governed by these conditions, this lemma is not qualified to declare failure for that new route."

The PhD student lowered his head, crossed out all occurrences of the two words in his question, and replaced them with window construction.

After sitting down, he did not close the old manuscript, and was still comparing it with the three conditions Jiang Lin had just circled.

The second onsite question came from the reconstructor from Budapest.

He brought the first two rounds of failed materials to the projection table, selecting precisely an attempt where the local splitting condition could no longer be maintained at the higher levels of the tower.

"After failing here, we once suspected whether 101 was a non-replaceable special point in the construction."

He placed the successful parameters of the third round beside it.

"Later, switching to another set of primes still closed, so I want to confirm whether the paper requires such a witness, or must retain the witness you provided?"

"It requires a witness that satisfies all compatibility conditions, without specifying 101."

Jiang Lin placed the materials from both rounds side by side on the screen.

"The first round shows that not just any randomly chosen set of primes can work, and the third round shows that the theorem is not tied to the set of parameters I published."

"If these two failure records are prepared to be made public, write down clearly at which layer the local conditions broke off respectively, and do not compile them into an appendix of the main proof."

That researcher paused and asked: "Sole authorship?"

"Of course, the failed routes were walked out by you again."

Jiang Lin pointed to the center application entry on the report page.

"If you need computing power and public verification interfaces, you can submit them as a short-term visiting project, and the ownership of the materials remains under your names."

Hearing this beside the console, the administrative assistant pulled a sticky note from under the job posting proofs and noted down the other party's name and contact information.

She still had three unanswered phone calls at hand, so this matter could only be scheduled after noon.

In the remote question pool, the question voted to the very top had only one line.

"With δ being so small, what on earth is the use on a finite scale?"

Gu Mingche sent the question to the main screen as it was.

Jiang Lin glanced over it and picked up the transparency sheet on which he had just drawn the lattice points.

"If you really want to arrange ten thousand points tonight to produce as many unit distances as possible, do not build the number field tower according to this paper first."

Someone in the back row chuckled, and then lowered their head again to look at the two lower bounds on the main screen.

"This result does not promise to outperform existing lattice constructions on currently computable scales, and the constant and effective scale are far from reaching an engineering-usable level."

"What it solves is the asymptotic boundary: the previous extra index could continue to decrease as n increases, but δ here will not."

"How to make δ larger and how to compress the effective scale of the construction are all new questions left by this paper."

The question that appeared immediately after came from the on-site media section.

The media section wanted to confirm a statement most easily distorted by headlines.

"Can it now be said that the planar unit distance problem has been solved?"

"No."

Jiang Lin returned to the first blackboard and redrew a vertical line between the upper bound and the new lower bound.

"What can be crossed out at present is the conjecture U(n) ≤ n^(1+o(1)), and U(n) cannot be written as an equality."

"The n^(4/3) above is still there, and the fixed positive exponent below is not necessarily the best answer."

The people in the media section deleted the headline they had half-written and re-entered it.

Subsequent questions became increasingly difficult to fit into the main screen in a single sentence.

Some people asked whether the complete splitting condition could be further weakened, some requested replacing the projection reconciliation with another set of probability models, and others shifted from the repeated distance problem to more general finite graphs.

Those that Jiang Lin could close on the spot returned to specific lemmas, while those exceeding the boundaries of the paper were left on the question page as they were, waiting for real proofs to answer them.

When the originally scheduled one-hour Q&A ended, there were still more than a hundred unprocessed items in the remote queue.

Professor Lin Zhaoye wrapped up the questions at the scheduled time, had technical liaison save the complete queue, and announced the screening rules for subsequent written replies.

Before 12:30, the venue had to be handed back to the afternoon activities.

As the staff began collecting the equipment in the back rows, the PhD student had already spread out the failed manuscripts of the cubic number field on the front row's table, and the researcher from Budapest had also moved two gray folders over.

The two sets of materials were originally separated by countries, languages, and completely different parameters, but now they were both stuck at the edge of fixed degree and local splitting conditions.

Just as they matched the first set of definitions, a staff member came over to remind them that the desktop must also be cleared.

Gu Mingche glanced at his watch and then looked at Jiang Lin.

"Can the new discussion room upstairs be used today?"

"The key is in the preparation office, holding at most eighteen people."

"There are already more than eighteen people here who want to follow."

"Then let those who have truly brought questions and materials go up first, and leave the rest online."

Jiang Lin had technical liaison connect the blackboard, public materials, and remote interface to the discussion room upstairs.

The administrative assistant picked up that stack of unreplied job posting proofs and led the group toward the second floor.

As soon as the door of the discussion room was opened, there were still network cables and two empty cardboard boxes left over from configuring the computing resources in the morning.

The administrative assistant kicked the cardboard boxes to the wall, first clearing out the space in front of the whiteboard for them.

The job postings were still tucked in the crook of her arm before she had time to put them down when the phone rang again.

Jiang Lin placed the two sets of failed materials separately at both ends of the long table, so as to avoid them getting mixed up into the same project attachment before registration.

Gu Mingche connected the remote meeting, and several researchers who couldn't enter the discussion room just now soon appeared on the screen again.

That PhD student had already picked up the whiteboard marker and started rewriting from the first line of the fixed cubic field.

Jiang Lin pulled out a chair nearby and had just sat down when the Budapest researcher pushed the page of the second-round failure in front of him.

"The loss here, I want to calculate it again."

"Then start from this layer."

Jiang Lin turned the page right-side up, and Gu Mingche also brought over a blank sheet of scratch paper.