204: Chapter 204 As long as it is greater than zero

"Screech—creak—"

In the Department of Mathematics office, the old laser printer in the corner wheezed and spat out the latest version of the theorem pages.

Gu Mingche caught the paper and immediately began counting one by one from the decimal point onward, following the meticulous formatting of the second line of the formula.

38]

"One, two, three..." He muttered under his breath, his fingertips sliding across the still-warm paper surface, "Thirty-seven."

Thirty-seven zeros, not a single one missing.

This string of numbers, almost too small to be observed in the physical world, had undergone a high-precision rational interval certification of a full 2,048 digits alongside Jiang Lin over unknown years.

Meanwhile, the tower infinity, which served as the cornerstone of the entire edifice, also welcomed the final confirmation of two independent real-world verification routes late last night.

Gu Mingche spread the theorem pages out on the desk, his gaze sorting down step-by-step along the dense string of dependency markers to the right of δ₀.

The base field, ramification set, marked primes, class number upper bound, window magnification factor, and denominator exponent had all been fixed before the tower layers began to grow.

From then on, as the algebraic structure continuously changed, only the field degree f_j remained.

According to Jiang Lin's proof architecture, he wrote the unit direction and the number of window points as quantities growing exponentially with f_j, and then defined δ₀ using the difference between the two growth rates.

As long as this difference strictly fell to the right of zero, the o(1) error term in the unit distance conjecture, which had plagued the mathematical community for over half a century, could no longer prevent the establishment of the counterexample.

Gu Mingche clicked open the interval certification log package on the screen, the light spots of the screen reflecting in the depths of his eyes.

"The tower has already been rebuilt by someone else," he rubbed the dry corners of his eyes, "This thirty-eighth power also has to be recalculated by someone else."

"It's already being calculated," Jiang Lin's voice came from beside him, "In Q-002."

As they spoke, the mouse in Jiang Lin's hand clicked lightly, dragging a blankly formatted independent verification page to the center of the monitor.

[Q-002: Bypassing the author's numerical program, independently certify the strict positive lower bound of log B / log A.]

Gu Mingche glanced at the system creation timestamp in the upper right corner of the page: "When did you create this last night?"

"Three and a half minutes after Q-001 turned green."

Before the words fell, the heavy wooden door of the office was pushed open.

The Instructor walked in carrying a heavy insulated bag, from which wafted the aroma of hot soy milk and freshly cooked pastries.

The third steamed bun from yesterday, which had to be retired from the consumption sequence because it sat out too long, had clearly left a deep lesson; today's soy milk, sesame flatbread, and eggs were all tightly locked inside the insulation layer.

"Who was Q-002 handed over to?" The Instructor put the insulated bag down on the desk and asked straight to the point.

"An algorithmic team specializing in rigorous numerical verification, and a researcher leaning towards the direction of analytic number theory."

Jiang Lin listed two strings of hash reception columns in the side panel of the screen.

"Neither side can get my original calculation program; they can only read a few fixed algebraic parameters in the main text of the theorem, as well as the basic rational inequalities permitted for invocation."

"Very good, that's how it must be done."

The Instructor reached out and grabbed a golden-brown sesame flatbread from the bag, handing it to Gu Mingche.

"You continue to check the definition tables and symbol mappings today, but be careful not to touch any numerical results."

Gu Mingche took the sesame flatbread and took a bite, mumbling: "I thought checking zeros suited me better."

"Zeros cannot be counted out." The Instructor glanced at him, tone solemn, "Decimal points obtained by counting are worthless in formal proofs."

The progress of the first verification team was far more difficult than imagined.

They directly discarded the interval arithmetic library Jiang Lin used in the preliminary draft of the paper, choosing a route bordering on self-torture.

Re-selecting the rational upper and lower bounds with logarithmic functions, expanding all estimates involving transcendental functions into huge integer numerators and denominators, and tightly locking the direction of rounding errors on the side most unfavorable to the conclusion.

This method eliminated all hidden dangers of floating-point overflow and rounding errors, but the cost was that the calculation was extremely slow.

At twelve noon, the progress bar on the status page had barely advanced to the root discriminant constant.

At four in the afternoon, the unified upper bound of the class number term barely passed validation.

By nine in the evening, the finally derived numerator had turned into a huge integer hundreds of digits long, which could not be directly and fully displayed in a regular text document.

The second group took a completely different technical route.

They first used an arbitrary-precision floating-point interval library to quickly locate the positive lower bound, and subsequently stripped out the nine most sensitive key terms in the entire calculation chain that were most likely to change the positive and negative signs, replacing them with strict formal algebraic certificates.

The two groups reached a consensus agreement before starting.

Floating-point calculations could only be used to determine search locations; any positive sign that ultimately appeared had to be supported by strict integer inequalities.

At 2:12 AM on November 3rd, the green pass signal popped up first on the terminal screen of the second group.

The final lower endpoint they calculated was—

[δ₀＞6.00041×10^－38]

This numerical value exceeded the six point zero initially claimed in the paper by a very narrow margin, but this extra six point zero zero zero four one was enough to make the strict greater-than sign on the theorem page that determined success or failure stand firmly in place.

The Instructor underlined it in the backend verification records, marking it as [Single Route Passed].

The first group did not deliver the complete certificate until 10:48 AM that same day. The order of rational approximation they used was lower, resulting in a wider error interval, and the finally calculated lower endpoint stopped at six point zero zero zero zero seven times ten to the negative thirty-eighth power.

The intermediate process integers, rounding truncation strategies, and error allocation schemes of the two certificates were completely different, but the two sets of independent computing power exercises ultimately landed precisely on the same decimal place count.

[Q-002: Two independent routes completed.]

[38: Confirmed.]

Gu Mingche turned the new version of the theorem page over.

"After this string of numbers is preserved," he turned his head to ask Jiang Lin, "Is there any direct use for it in mathematics or in reality?"

"To continue overthrowing conjectures." Jiang Lin replied.

"Aside from overthrowing conjectures?"

"Not for the moment."

Gu Mingche looked at the numbers on the paper and chuckled softly: "Then it's quite single-minded."

Jiang Lin did not pick up the conversation, merely casually packaging the two newly generated numerical certificates into the explicit constant attachments.

Everyone was clear that numerical verification was merely the easiest and most intuitive section for outsiders to understand in the entire lengthy proof.

The skeleton reconstruction that truly drained energy was only just beginning.

The two researchers in the tower direction had previously signed off on Q-001, but the class number and unit modular element counting group only signed off on the algebraic modules they were responsible for, and Professor Lin Zhaoye, far away in Nanjing, had only confirmed the interface of the geometric projection.

Everyone could guarantee that the bridge piers they inspected were solid enough, but whether the entire bridge still connected into a single pathway from the first line of derivation to the last line required a complete reconstruction without relying on any prompts from the author at all.

The Instructor slapped a table listing the heads of the six modules onto the desk.

"Subcontracted review ends here," the Instructor clapped his hands, "Next round, cross-module review."

"The tower group will check the window average," Jiang Lin added, "The geometry group will trace backward from the class number upper bound, and the explicit constant group is only allowed to read the parameter interface in the main text."

The Instructor looked at Jiang Lin: "Where does the author sit?"

Jiang Lin pointed to the back row: "The back row."

"Not even reminding them if someone explains it wrong?"

"They can check the table of contents at any time," Jiang Lin said calmly, "but they cannot request any intermediate process conclusions from me."

"Then leave your page directory book behind," the Instructor waved his hand, "Sit further away and speak less."

On the afternoon of November 4th, a small classroom in the Science Building capable of holding thirty people was temporarily requisitioned.

To prevent any unverified information interference, the networked recording and broadcasting system in the classroom was completely cut off.

The two workstation terminals by the window only received remote image signals, and the computer connected to the projector had its wireless network card directly unplugged.

Jiang Lin's thick paper directory was left desolately on the desk in the back row.

Professor Zhou, the second tower-direction researcher, and the explicit constant verification group accessed via a dedicated remote link.

Professor Lin Zhaoye made a special trip from Nanjing to the site.

The Instructor personally presided over the entire session; Gu Mingche sat on a spare chair to the side of the podium, with three comparison documents open at hand, ready to maintain the definition table at any time.

At the very top of the blackboard, the standard form of the main theorem was neatly written in white chalk.

[There exists an absolute constant 38, and an arbitrarily large planar finite point set P, such that U(P)≥|P|^(1＋δ₀)／C.]

Below the main theorem, it was clearly divided into three columns by vertical lines.

[Can the tower be infinitely long]

[Can the unit direction grow exponentially]

[Can high-dimensional windows fall back into the plane]

The person speaking about the tower started from the orders five and six already confirmed by Q-001.

According to regulations, he could not use any explanatory notes written during his previous round of derivation.

Professor Lin Zhaoye, in charge of the geometry end, proactively stepped up to the podium and took over this part of the explanation.

He reconstructed the input according to the original relation bounds, and intentionally put the splitting condition of 101 at the very end, not allowing six less than six point two five to enter the next column until confirming that the generator rank would not decrease because of it.

Professor Zhou in the remote video feed interrupted three times in succession.

Every time, he demanded Professor Lin Zhaoye to explicitly point out whether what was currently being invoked was an existing external theorem, a self-created lemma by the author, or an independent certificate derived from Q-001.

When Professor Lin Zhaoye turned past the thirty-seventh page, he put the chalk back into the slot, clapped his hands, and said: "I can walk through this tower part entirely from public inputs."

In the remote screen, Professor Zhou raised an eyebrow: "Do you accept my judgment?"

"Not accepted." Professor Lin Zhaoye turned around and heavily circled the number six he had just rewritten on the blackboard with a piece of chalk, "I only accept this blackboard."

The counting of unit module elements in the second column was taken over by the second researcher in the tower direction.

Under the same norm, a large number of complex ideal combinations were first accurately grouped according to ideal classes, and then ratios were extracted from each class to obtain the generators of principal ideals.

The complex conjugate transformation accurately mapped these elements onto the unit circle under all complex embeddings.

In this step, there were three most likely mathematical coincidences to occur.

Roots of unity would cause different generators to correspond to the same direction, general units might enter the kernel, and plane projections could also compress different objects in higher dimensions together.

During the initial internal verification, Jiang Lin packed the losses of the first two items into a uniform constant, while the third item was eliminated relying on the injectivity of field embeddings.

At this point, the second researcher re-disassembled these three items one by one on the blackboard.

The number of roots of unity was successfully incorporated into the fixed factor.

The kernel of general units was given unified control by the algebraic structure of the CM field.

Moreover, the images of different algebraic numbers under the specified embedding were also proved to be non-coincident.

The positive and negative losses of direction lost a factor of two when directed edges were converted into undirected edges, but the exponential term remained firmly in place.

Professor Lin Zhaoye sat in the front row and listened quietly, then reverse-engineered the same set of deduplication logic from the plane end.

The calculation processes of the two eventually converged perfectly in the expression of constant C.

When the explanation progressed to window averaging, Gu Mingche pointed at the screen and suddenly spoke up: "Wait a minute, which D is this here?"

The left side of the screen displayed the common denominator D, while the right side displayed the absolute value of the discriminant |D_K|.

Under the fuzzy light and shadow of the projector, the two symbols looked extremely similar, and several students sitting in the back row had already copied them onto the same line in their notebooks.

Gu Mingche quickly opened the definition table and replaced all the discriminant terms in the text back with expressions carrying complete subscripts.

The symbols originally used in the text were logically separable from each other, but the presentation had removed the subscripts to save typesetting space, nearly creating an additional logical chaos on site.

The Instructor wrote down N-017 in the presentation records.

[The symbols used in the report are over-abbreviated and will not be included in the paper revisions.]

"Even the author's manuscript wasn't touched today," Professor Lin Zhaoye turned his head to tease, casually completing the symbol on the right blackboard as D_K, "to think you could actually find work to do."

Gu Mingche released the clicker: "The scope of responsibilities has stability."

"The treatment is also very stable," Professor Lin Zhaoye continued.

At five o'clock in the afternoon, only the last connecting arrow remained between the three columns on the blackboard.

The infinite tower provided CM fields with continuously increasing degrees and root discriminants that were always controlled.

The completely split 101 provided exponentially growing unit module directions.

Window averaging then turned some of those directions into unit edges within the planar point set.

The explicit constant array re-projected the parameter page onto the large screen.

The base field, ramification set, marked prime numbers, window radius magnification, and denominator exponent were all strictly fixed before selecting the tower layer.

As the tower layers extended upward, the number of unit directions and the number of window points each grew exponentially according to the field degree, and the difference between the two growth rates always used the same set of global parameters.

In the remote video feed, the researcher responsible for numerical verification erased the tedious decimal expansion and only wrote down two concise integer inequalities on the right side of the blackboard.

The first ensured that the denominator was always positive.

The second ensured that the numerator was strictly greater than zero.

The positive margin sandwiched by the two formulas was then divided by the fixed log A, finally yielding the absolute constant δ₀ completely independent of the tower layers.

"The numerical program only expanded it to the thirty-eighth decimal place in decimal notation," the verifier turned to face the camera and said, "the positive sign was already established in the algebraic structure before the expansion."

The second researcher in the tower direction remained rigorous; he requested changing the field degree to another increasing subsequence and re-checking whether all parameters still remained fixed.

The person on the podium did as requested.

The point set scale changed, the tower layer index changed, but the items on the parameter page did not move a single fraction.

The δ₀ on the right side of the blackboard was still nailed above six times ten to the negative thirty-eighth power.

"I concede on this one."

The second researcher let out a long sigh of relief and signed his name after his verification scope.

When dinner was delivered to the classroom door, the first complete reconstruction was officially declared finished.

All six modules left comments signed by the new presenters, twenty-three objections were recorded in the catalog, nineteen of which were closed on the spot by independent certificates, and the remaining four needed to return to the original literature to check their scope of application.

Jiang Lin stood up from the last row, walked to the door to take two boxed meals, and silently sat back in his original seat.

Gu Mingche walked over from the side of the podium to collect his meal, looked at him and smiled: "Your only job today was to carry two boxes of meals?"

"I also provided the page number booklet," Jiang Lin patted the thick stack of paper on the desk.

"The page number booklet was placed on the desk the entire time, and you didn't even touch it."

"Therefore, it shows that the whole process had no malfunctions." Jiang Lin opened the lid of his boxed meal.

The remaining four objections were all closed the next afternoon.

One questioned whether the upper bound of the class number holds uniformly for the current CM field family, one checked the quantifiers when window translation changed from average counting to deterministic existence, and the other two fell respectively on directed edge deduplication and point set scale rounding.

Every single objection had its status changed to green personally by the proposer, and Jiang Lin only submitted the access permissions for the certificate file locations throughout the process.

The subsequent cross-reconstruction swapped the modules once again; the person originally responsible for the tower checked the plane projections, and the numerical group traced back the inputs along the class number estimations. No single conclusion could have a second seal stamped for oneself by the original person in charge.

At 8:00 PM on November 6th, the Instructor collected the last signed page.

The signed pages stated the facts that each verifier was willing to undertake.

[The pro-2 tower infinitude lemma has been reconstructed by two independent routes.]

[All interfaces for class numbers, unit module elements, window counting, and plane projections have completed cross-reconstruction.]

[The explicit lower bound 38 has been confirmed by two sets of independent rigorous interval certificates.]

[To date, the review group has found no gaps sufficient to block the main theorem; it is recommended to make the preprint public and accept broader review.]

The Instructor neatly organized the papers and handed the signed pages to Jiang Lin.

Jiang Lin put the verification notes into the public appendix, listing all reviewers in the acknowledgments and independent verification records.

Gu Mingche's name appeared in the column for definitions, citations, and version consistency verification.

Gu Mingche leaned over and carefully checked the pinyin spelling of his name: "Is the treatment still unchanged?"

"This document can at best be considered your work-related injury certificate," the Instructor closed the last folder and joked.

The preprint could finally be publicly uploaded.

Gu Mingche helped Jiang Lin do the final directory jump before publication.

The paper title had been changed from [Counterexample to Unit Distances] in the internal folder to its formal name.

[A Superlinear Lower Bound for Unit Distances in the Plane]

The first sentence of the abstract clearly retained that tiny explicit lower bound.

[We prove that there exists an absolute constant 38...]

Gu Mingche jumped from the abstract to Q-001, and then jumped from Q-002 back to the theorem page.

"People on the outside will be able to see at first glance that you spent a full two days recalculating six and six point two five."

"They will launch their attacks from the exact same place," Jiang Lin said.

"You put the answer right at the very front, aren't you afraid they'll copy your train of thought?"

"Verification records cannot replace the proof itself," Jiang Lin organized Q-001, Q-002, and all open objections onto the home page of the appendix, "if anyone doesn't believe it, they can recalculate it themselves."

At eleven o'clock at night, the preprint platform system returned the submission success number.

Jiang Lin simultaneously posted the public version on his personal academic homepage and sent the link to the review chain experts who had previously only received attack packages.

At the very top of the web page, six entrances were minimally listed: the main text of the paper, proof appendix, verification records, attack catalog, hash values, and version history.

Gu Mingche manually refreshed the page twice, and the download count jumped from three to eleven.

Eight of those probably came from the review group members who had just received the email notification.

The twelfth download came from an unknown IP outside the review chain.

Three minutes later, an English email rang its notification tone and landed in the public inbox.

The sender came from the Department of Mathematics at a well-known European university, and the body text was as brief as a single sentence.

[Please confirm my understanding of the abstract: ε here is a fixed positive constant, so is what you constructed a counterexample to the Erdős conjecture?]

Jiang Lin tapped the keyboard and replied with only a single word.

[Yes.]

Just as the email status changed from sent to read, the download count on the personal homepage instantly skyrocketed from twenty-seven to one hundred and six.

Gu Mingche tore open the two pairs of disposable wooden chopsticks he brought back, handed one pair to Jiang Lin, and casually refreshed the page once more.

The number on the screen had already become two hundred and eighty-four.