203: Chapter 203 The Second Six

When Gu Mingche returned to the office carrying three cold steamed buns, the Instructor was completely covering up the page numbers in the first attack package.

The printed manuscript left only the theorem statement, a list of permitted external results, and a blank dependency graph.

The phone call was already connected; Professor Zhou on the other end studied class field towers and non-commutative Iwasawa theory, and the complete document he had received half an hour ago had now been placed by him into a read-only directory.

"I'll make the rules clear first."

Professor Zhou flipped to the first page of the attack package.

"The six written by the author can only be treated as a conclusion to be verified, not as the starting point for my calculations."

"Where are you planning to start?"

The Instructor asked.

"From the definition of the maximal pro-2 extension, the Shafarevich relation bound, and the splitting set you selected."

The pages turned once on the other end of the phone.

"Jiang Lin can tell me which folder the original literature is placed in, but he cannot tell me which number to substitute in the next step."

Jiang Lin sent over the offline literature index, version hash, and the locations of the original texts of the six external theorems.

After receiving the materials, Professor Zhou closed the instant Q&A window with the author.

Gu Mingche placed the steamed buns under the whiteboard slot.

"Then what am I responsible for?"

"Definition tables, symbol tables, and citation paths."

The Instructor handed him a new registration page.

"If any notation requires Jiang Lin to explain, record it as a reproduction failure first; don't guess for the reader."

"My job has finally upgraded from checking page numbers to stopping mathematicians from asking the author."

"Compensation remains unchanged."

Gu Mingche signed his name in the responsibility column and renumbered the first issue.

[ Q-001: Independently recalculate d ( G ) and r ( G ) from the original relation bound, and verify all finite prime places, real infinite places, and the 101 complete splitting condition. ]

This problem was aimed right at the narrowest part of the entire proof.

In the Wasteland, Jiang Lin obtained five independent generators using six allowable ramified primes, and then compressed the relation rank to within six, while the Golod-Shafarevich threshold was twenty-five divided by four.

The margin was only a quarter.

If any local condition paid one more relation, the infinity of the tower could not be deduced from the existing proof.

In his first step, Professor Zhou dismantled the three rows of numbers Jiang Lin had arranged.

The six primes in T were rewritten as local Kummer classes, their mutual product relations were listed separately, and the real infinite place and 101 each occupied a column.

The original Frattini matrix was temporarily sealed, and Professor Zhou recalculated the elementary quotient using another set of bases.

At 3:42 PM, he sent back the first fifth-order matrix.

The row and column order of the matrix differed from Jiang Lin's, yet after elimination, five pivots still remained.

[ d ( G ) = 5 ]

The Instructor wrote the result into Q-001, marking only the generator part as passed.

Gu Mingche picked up a meat-sauce steamed bun; the thoroughly chilled grease had stuck back onto the filling.

"The first five is out, how far is it from the second six?"

"Relation bounds cannot be calculated according to a progress bar."

Jiang Lin erased the original six on the whiteboard.

"Then why did you keep the five?"

"It has already been calculated from another set of bases."

Gu Mingche took a bite of the bun and decided to leave this distinction for the mathematical community to digest.

The recalculation of the relation rank dragged on until evening.

Professor Zhou fed the local relations potentially contributed by each allowable ramification point into the upper bound item by item, and restored the real splitting conditions to the narrow range used in the original theorem.

The column where 101 was located took the longest time.

After the original manuscript added the rational prime 101 into the splitting set, more and more prime places above it would appear in subsequent finite layers; if every high-level prime place were treated as an independent input again, the number of relations would naturally spin out of control along with the tower layers.

However, the definition of the tower constrains the closure of the bottom-level decomposition group, and the complete splitting of the high levels is the result of this closure being quotiented out, without requiring the same batch of relations to be repeatedly added at each layer.

Professor Zhou still rewrote the complete conjugate orbits once more.

He checked the relation module generated by the orbits along the group action, and verified the image of 101's Frobenius in the maximal elementary quotient, confirming that it fell within the Frattini subgroup.

This splitting condition increased the relation budget, but did not trim away the five generators already obtained.

At 10:13 PM, the phone rang into the office again.

"That sentence in the author's document, '101 contributes one relation,' was written too briefly; I expanded the orbit module into eleven pages."

Professor Zhou reported the hash of the new file.

"After expansion, it is still a single independent relation, not repeatedly charging for the same bottom-level closure condition according to the number of high-level prime places."

The Instructor asked: "What about the real infinite places?"

"They have entered the Shafarevich upper bound; nothing was missed, nor was anything counted twice."

"So what is r?"

"I currently obtained that r ( G ) does not exceed six."

The Instructor rewrote the numbers that had been erased on the whiteboard back up.

[ d ( G ) = 5 ]

[ r ( G ) ≤ 6 ]

[ d ( G ) ² / 4 = 6.25 ]

Professor Zhou's recalculation order, bases, and local condition expansion methods were all different from Jiang Lin's, yet in the end, he still arrived at the same strict inequality.

He refused to sign a full pass behind Q-001.

"I have only completed one route."

"Who will do the second one?"

"I have already transferred the theorem statement and original input to another person who works on restricted splitting class field towers; they can only see your parameters and cannot see my eleven-page expansion for the time being."

"How long will it take?"

"Whenever he finishes calculating, he finishes calculating."

Before hanging up the phone, Professor Zhou added another sentence.

"Jiang Lin's manuscript at least wasn't propped up by a calculator showing 6.25; the preceding relation bounds truly carried the entire weight."

Of the three steamed buns in the office, two had been eaten.

The third one was put into a paper bag by Gu Mingche, intended to be taken back as evidence that he still possessed a normal life today.

At 11:47 PM, Professor Lin Zhaoye sent the first round of results on the geometric end from Nanjing.

He split apart the possible point coincidences, direction coincidences, and forward-backward edge repetitions that might occur after projecting the Minkowski lattice onto the plane, and re-proved that each loss could only enter a fixed constant C.

[ G-001: Injectivity of planar projection passed. ]

[ G-002: Loss from converting directed edges to undirected edges is at most a fixed factor of 2. ]

[ G-003: Window boundary loss does not change the main exponent. ]

The geometric end sealed the front-end inputs of the tower lemma into a hypothesis to be verified, and every step of the planar part expanded independently from this hypothesis.

That hypothesis was still waiting for a second expert to give an independent signature.

On the morning of October 31, Gu Mingche went to teach his own class, while Jiang Lin stayed in the office to organize the citation licenses for the public version.

Since he could not alter the main proof already submitted for review, he established read-only mappings for all received issues by module.

At 12:06 PM, Gu Mingche pushed the door open and returned, holding a bowl of hot rice and two dishes of food.

"The third bun has been retired."

"Where did you throw it?"

"The dormitory trash can."

"Is there a photo?"

"Such failure witnesses do not enter public attachments."

Gu Mingche pushed one portion of the food to Jiang Lin, and as soon as the computer was plugged into power, the first batch of files from the second recalculation route came through.

The other party abandoned the relation module expansion used by Professor Zhou, starting directly from the local decomposition groups of the finite layers, and then checking via inverse limits whether the added relations were compatible among the tower layers.

The first round only reached the seventh page, and a five was already written in the conclusion column.

At 4:00 PM, the second five obtained a complete certificate.

The relation rank side was much slower.

The materials requested by the other party for supplementation were only two items: the original residue symbol table of the six ramified primes, and the independent calculation record of 101 being zero in the elementary quotient.

Jiang Lin handed the locations of the two files to Gu Mingche.

Gu Mingche checked the hashes, and then they were sent out by the Instructor, without a single line of author explanation attached throughout the entire process.

At 9:00 PM that evening, the second route compressed the upper bound of the relations to seven.

On the right side of the whiteboard, a number sufficient to violate the strict inequality appeared for the first time.

Gu Mingche stared at that row of sevens.

"Did this discover one extra?"

"Tentative upper bound."

The Instructor expanded the other party's calculations.

"He collected the same real infinite place condition once in the totally real constraint and once in the local relation bound."

This sentence could only show that the two sets of notations might overlap, and could not delete that item on behalf of the other party.

The Instructor sent the location and the page numbers of the original theorem back, and the status remained yellow.

Forty-seven minutes later, the other party replied.

He confirmed that he had repeatedly counted the definition condition of the totally real tower and the corresponding term in the Shafarevich relation bound; after deleting the duplicate items, the upper bound returned from seven to six.

This was not repairing the proof for Jiang Lin.

The error remained in the second verifier's own recalculation draft, while Jiang Lin's original manuscript had only collected it once from start to finish at the same position.

The other party put the two calculations before and after revision together into the verification record, and then continued to check the splitting condition of 101.

November 1, at 10:26 PM.

Professor Zhou called for the third time.

Only Jiang Lin, Gu Mingche, and the Instructor remained in the office; the three rows of numbers on the whiteboard had been kept for two days, and the upper right corner of the six still lacked a second signature position.

"The second route has ended."

Professor Zhou switched the phone call to a multi-person call.

The other verifier confirmed their scope in person.

"I reconstructed from finite-level decomposition groups and inverse limits, and did not adopt Professor Zhou's relation module expansion."

"The six allowable ramified primes give five independent generators, and the upper bound of the relation rank after combining all local conditions and real infinite places is six."

"The relations added by the complete splitting of 101 have been accounted for, and the generator rank of the Frattini quotient remains five."

The Instructor asked: "Did you use the r ≤ 6 conclusion provided by the author?"

"No."

"Are there any lemmas that need to be supplemented by the author?"

"No."

"What is your signature scope?"

"The tower infinity lemma and its root discriminant inputs."

Gu Mingche wrote the signatures into Q-001, with Professor Zhou's name falling on the first line and the second verifier's name falling on the second line.

[ Q-001: Two independent routes completed. ]

[ d ( G ) = 5: Confirmed. ]

[ r ( G ) ≤ 6: Confirmed. ]

[ 6 < 25 / 4: Confirmed. ]

The Instructor changed the status of Q-001 to green, and the 173 nodes below tower infinity in the main dependency graph were consequently opened up for the next round of cross-verification.

Professor Zhou asked Jiang Lin over the phone: "How long did you spend on this part during the internal review?"

"Indeed quite a lot of time."

"No wonder the attack package put everything I might ask upfront."

"Earlier versions left behind failure records."

"Those failures belong to the path you took to reach this proof, not to the loopholes of this proof."

Professor Zhou put his authorization for the eleven-page expansion into the public verification attachment.

"Now the next group can be allowed to move forward from here."

After the call ended, Gu Mingche looked at the two rows of signatures on the whiteboard.

The first six in the Wasteland was hidden behind more than 400 pages of proofs, over 4,000 dependencies, and sixteen years of work.

The second six in reality was calculated from different entrances by two researchers, and even the repeated counts made in the middle remained in the verification records.

He picked up the whiteboard marker and added back the less-than sign between six and 6.25.

The pen tip pressed across that quarter-fraction gap, leaving a very short horizontal line of black ink on the whiteboard.