168: Chapter 168 The End of the Ghost Machine

September 12th, the first afternoon after classes officially began.

6:07 PM.

On the large screen in the project team's meeting room hung a single proposition written back in 1990.

[People will never be able to prove: Σ(5)=4098, S(5)=47176870.]

Below it was a neatly signed name.

Allen Brady.

For anyone working on the Busy Beaver problem, this was an unavoidable name.

In 1983, he completed the proof for the 4-state Busy Beaver.

Meanwhile, the famous champion machine for the 5-state Busy Beaver was discovered in 1989 by Marxen and Buntrock.

It was a machine akin to a miracle, running on a blank tape for exactly 47,176,870 steps, and at the instant it halted, leaving behind 4,098 ones.

The champion had long been found, admired by people for over thirty years.

But the problem was that no one could offer an ironclad proof in mathematics and logic: within this vast search space—containing over sixteen trillion transition tables before equivalent reduction, and still requiring the processing of hundreds of millions of representative machines even after tree normalization—no one could rule out another machine that was hidden deeper and ran for even longer.

Five states.

Two symbols.

A single table with only ten transition entries.

This was the entire structure of a 5-state Turing machine.

Its rules were so simple.

Anyone could copy its rules onto a sticky note in just a few minutes.

Yet, it was precisely this near-primordial simplicity that birthed a complexity that even modern supercomputers could not exhaust.

Generations of top researchers had stepped forward one after another, attempting cycle detection, symbol compression, closed tape language, finite automata reduction, and formal verification, yet they could never stamp the final seal of approval on these two numbers.

Thirty-two years ago, after exhausting countless efforts, Brady simply wrote it into his list of predictions.

Never to be proven!

Professor Qiao Wenduo had put this sentence back up today.

The veteran professor, who had spent half a lifetime in the field of formal verification, now rested both hands on the edge of the conference table, silently staring at the big screen.

The reason he displayed this sentence was that on the right side of the screen, the final line of status prompt after the project team's full library re-verification was still displayed.

[Unresolved_Machines: 1]

On September 10th, while freshman Jiang Lin was just finishing returning his military training gear and listening to the noisy department orientation on the sports field, two implementation teams strictly separated physically had quietly completed independent blind tests of the shared trusted kernel.

It was a fierce war without smoke.

In the middle, a discrepancy arose that was enough to break the entire team out in a cold sweat.

At the time, when handling the logic for trimming blank cells at both ends of a tape, the OCaml implementation team—due to an extremely hidden blind spot in thinking—mistook the starting index of the underlying container for logical coordinates, directly swallowing a shift offset that should have been preserved.

The third-party testing group unhesitatingly extracted a minimal reproduction case.

Jiang Lin was urgently recalled; without looking at the code from either side, he provided an exact interpretation based directly on the specification document locked in the safe.

The two teams were forced to discard their previous progress, recompile the executables, and rerun all test cases from the very first hidden test case.

Those were a suffocating few hours.

When the final results came back, everyone held their breath.

Rust passed.

OCaml passed.

The hash values of the two adjudication summaries were compared bit by bit in the validator, and a green MATCH lit up, declaring consistency.

On September 11th, safety education and student registration updates proceeded methodically according to the academic calendar.

Meanwhile, the full-database re-verification was launched with a roar on the same day.

88,664,064 seed database machines, along with the non-halting witnesses generated day and night by the project team over the past few months, were fed in batches like ore dumped into a giant furnace into two mutually isolated verifiers.

Decisive cycle mechanism initiated.

Translated cycle mechanism initiated.

Backward unreachable decision tree expanded.

Millions upon tens of millions of machines disappeared from the pending queue after rigorous examination, turning into lines of confirmed green records in the database.

Format errors in old certificates, machine hash mismatches, and even coordinate convention pollutions left over from years ago were ruthlessly rejected one by one by the verifier and forced to be regenerated.

Every machine marked as non-halting had to provide irrefutable evidence.

A proof path that could be retraced from the machine's low-level semantic level allowed absolutely no probabilistic guesses.

By the afternoon of September 12th, re-verification of all existing certificates was complete.

In the vast sea of stars formed by over eighty-eight million machines, only a single one remained.

In the status bar of the final machine, there was no REJECT, only UNKNOWN.

To this day, the project team could not produce any verifiable witness.

Professor Qiao Wenduo turned off Brady's heavy prediction, picked up the presenter pen, and enlarged the transition table of the last machine until it occupied the center of the entire screen.

[Skelet #17]

Only five rows and two columns.

Ten transition entries.

The table was small enough to fit on a single sticky note.

"This machine comes from the forty-three holdouts published by Georgi 'Skelet' Georgiev in 2003, hence named Skelet #17."

Professor Qiao Wenduo stood to the left of the screen, his voice sounding particularly deep in the spacious meeting room.

"We tried all conventional weapons: exact cycles didn't work, translated cycles didn't work, and we forcibly expanded the backward unreachable set to the ninth level, but after that, the number of states exploded exponentially into a terrifying surge, causing direct memory overflow."

Deep exhaustion leaked from Professor Qiao Wenduo's tone.

"The longest existing trajectory has already exceeded twenty billion steps without halting. What's even worse is that it hasn't entered any repetitive structure that our current rules can capture."

Zhou Shu took over the operation from the side.

Taking a deep breath, he pulled up a massive space-time diagram.

The black-and-white tape trajectory instantly filled the entire wall screen, resembling a complex genetic map of some ancient creature, or perhaps the ruins of an alien city viewed from high above.

"This is its twenty-billion-step life journey," Zhou Shu said, pointing at the dense textures on the screen. "The tape head sweeps back and forth over ever-widening intervals like a tireless weaver. It writes, erases, and turns back. The outer boundary of the pattern expands continuously, and the tape grows longer and longer."

As Zhou Shu spoke, he zoomed in on one of the regions.

"Yet internally, no two identical cross-sections can be found. Continuing the simulation is pointless."

Zhou Shu turned to look at everyone, his tone firm.

"Twenty billion steps, two hundred billion steps, or even two trillion steps—there is no fundamental difference here. As long as it can still generate new tape structures at the boundaries and as long as its behavior shows no closed periodicity, we will never reach a cycle."

At the other end of the conference table, Ye Ning rubbed her bloodshot eyes and pulled up three thick analysis reports that the project team had worked overnight for days to produce.

"The team attempted three analytical routes. Some viewed it as a generalized binary counter, trying to bound it using carry rules; others suspected it was simulating a dynamic iteration similar to the Collatz conjecture, a so-called 3n+1 variant."

Ye Ning continued flipping through pages.

"We tried three cutting-edge compression models to explain its trajectory. Locally, these models fit perfectly. But as soon as we push to the extreme cases of boundary merging—whenever the tape head touches that specific boundary between 0 and 1—the model instantly breaks down, and all predictions fail."

She flipped the last report to its final page, showing it to everyone.

[Conclusion: Current abstractions are insufficient to rule out future halting.]

In the upper right corner of the screen, that number remained 1.

Like a Damocles' sword hanging over the entire team.

As long as that 1 remained there, Brady's prediction held true.

4,098 could only be called a lower bound, and 47,176,870 was merely a candidate value.

The forty-year history of the 5-state Busy Beaver problem and the efforts of several generations were all crushed under the weight of this tiny ten-cell transition table.

Professor Qiao Wenduo turned his head and directed his gaze to Jiang Lin, who sat at the far end of the conference table.

This nominal freshman was wearing a plain solid-color T-shirt, his face as calm as an unruffled lake.

Yet over the past few days, it was he who had repeatedly demonstrated sharpness and decisiveness far beyond ordinary people.

"The shared trusted kernel has successfully completed its mission," Professor Qiao Wenduo said to Jiang Lin, his tone filled with expectation and trust. "Now, what we lack is a witness that can be checked by it and withstand the scrutiny of the mathematical community."

Jiang Lin's gaze swept over the transition table, rapidly reconstructing the algebraic structure behind these ten transition rules in his mind.

After a dozen seconds or so, he raised his head and said, "Turn off the single-step animation."

Zhou Shu froze for a moment, then pressed the spacebar to pause.

The dizzying black-and-white trajectory on the screen displayed a tape state so vast it felt hopeless.

"Only keep the configuration each time the tape head returns to the leftmost delimiter," Jiang Lin stood up and walked toward the whiteboard at the front. "Hide all those intricate back-and-forth sweeps in between, while keeping the state properties."

Understanding instantly, Ye Ning immediately pulled up the trajectory filter in the system, her hands flying across the keyboard to reset the sampling conditions.

The dense execution logs rapidly shrank until only a single column of tape snapshots with increasingly long time intervals remained on the screen.

The first snapshot.

The second snapshot.

The third snapshot.

When the filtering finished, everyone's attention was drawn to it.

In every snapshot, the tape was no longer a jumble of chaotic black-and-white pixels, but instead displayed a bizarre regularity.

Blocks of consecutive ones of varying lengths were precisely separated by single zeros.

As the machine's simulation rounds continued to run, some consecutive segments of ones grew, some shrank, and others were ruthlessly pushed to the right.

"Looking at these lengths alone, they still seem like chaotic values," Jiang Lin walked to the whiteboard and picked up a black marker. "But if you look closely at where the changes occur, you'll find a clear sequence hidden within."

With that, he put pen to whiteboard and wrote down a basic tape symbol representation.

[0 1ᵃ⁰ 0 1ᵃ¹ 0 1ᵃ² 0 ...1ᵃᵏ 0]

"It's not creating random patterns on the tape."

Jiang Lin turned and labeled an integer above each block of consecutive ones.

"These consecutive segments form an integer table written in unary. Every time the tape head completes a long-range round-trip scan, it appears to modify thousands of cells, but at the macro level, it only modifies the data of a single entry. On the next complete scan, it will modify an adjacent entry. When a carry occurs at one end, the direction of modification reverses."

Zhou Shu stood straight up abruptly, his eyes wide as his gaze moved rapidly across the filtered list of integer snapshots, his brain working at top speed.

A dozen seconds later, as if struck by electricity, he rushed over to the big screen and pointed to the changing locations across eight consecutive snapshots, his voice trembling slightly with excitement: "Zero, one, zero, two, zero, one, zero, three."

Around the conference table, several researchers who were typing just a moment ago instantly froze with their hands hanging in midair.

"Gray Code," Zhou Shu blurted out, turning to look at Jiang Lin.

"Right," Jiang Lin nodded.

In computer science, Gray code is a special binary numeral system.

Its most prominent feature is that two successive values differ in only one binary bit. This characteristic is commonly used in hardware design to eliminate glitches.

Yet this extremely primitive Turing machine with only five states had used its clumsy-looking, back-and-forth tape head and strings of ones of varying lengths to seamlessly conceal an entire Gray code update sequence within the structural evolution of the tape across twenty billion frantic steps!

Over the past dozen-plus years, all researchers' analyses had been bogged down in the mud because they were all trying to track what happened to every single tape cell in every single nanosecond.

That was a microscopic hell of exploding details.

Whereas Jiang Lin had raised the scale of observation by an entire dimension.

[part:gemini-3.5-flash-lite]

"After a round of long-range scanning, how many 1s were specifically written in this interval is no longer important."

Jiang Lin's speaking speed was not fast, but every word hammered into everyone's mind like a nail.

"We need to strip away irrelevant variables. The three types of information that need to be retained are: the active segment index i existing as a symbolic parameter, the current scanning stage, and the parity class of the boundary markers. The eight macro-states only describe the combination of the control stage, scanning direction, and parity class, while the active segment index and the length of each segment still exist as symbolic parameters in the certificate."

As a veteran of formal verification, Ye Ning had completely figured out what he wanted to do at this moment.

"But the length of this list will grow infinitely!" Ye Ning pointed to the edge of the screen, a flash of excitement glittering in her eyes, "If we only record the position and parity of the active segment, can it cover all newly generated consecutive segments in the future?"

"Yes." Jiang Lin answered without hesitation, "Because the generation of each new segment must pass through the same fixed carry template."

With that, on the right side of the whiteboard, he rapidly wrote out the outlines of three categories of local rewriting rules.

1. Ordinary flip.

2. Carry to the right.

3. Reverse scan after reaching the boundary.

"The list can extend infinitely to the right, and it won't stop even if the universe is destroyed. But the rules for generating this endless list are only these three."

Jiang Lin turned around and looked straight at everyone.

"Therefore, our non-halting witness does not need to hold a snapshot of the entire tape, which the memory cannot hold. We only need to prove mathematically that all legitimate list states, after undergoing such a macro-transformation, still belong to the same set we defined. As long as this set is closed, the machine can never escape."

"What about the halting condition?" Professor Qiao Wenduo suddenly stood up from the edge of the table and walked quickly to the front of the screen.

The composure of years of academic study was broken at this moment.

Jiang Lin walked over, his finger pointing directly at the conspicuous undefined entry at the end of the transition table.

"For this machine to crash into the halting state, the conditions are extremely harsh. It must simultaneously satisfy three preconditions: first, it must reach the right-end delimiter of the current encoding area during the recycling stage; second, it must read the blank character in a specified state; third, it must let the boundary marker fall into the even class."

"Only when these three conditions are met simultaneously will that undefined transition be triggered."

Jiang Lin turned around and deduced layer by layer along the macro-transformation logic just written out.

The pen traced strict logical chains one after another on the whiteboard.

"According to the three categories of macro-transformations just extracted, every time the active segment advances to the right, the boundary parity bits will flip with the corresponding carry template. This conclusion comes from the machine's own local rewriting relation, rather than the general properties of Gray codes."

"Therefore, we can conclude: when the machine reaches the right-end delimiter of the current encoding area to form a macro-state, its parity marker always falls into the odd class, which is the non-halting class. And if the parity marker happens to match the even class required for halting, I'm sorry, at this time, between the active segment and the boundary, there is still at least an insurmountable delimiter separating them, and it simply cannot touch the boundary."

"That undefined halting entry is indeed right there, wide open."

Jiang Lin looked at everyone, his tone so calm that it made people involuntarily get goosebumps.

"Yet it does not belong to any macro-state reachable from the initial configuration."

Professor Qiao Wenduo thought for a moment, picked up another red whiteboard marker on the table, and walked to Jiang Lin's side.

Next to the macro-states listed by Jiang Lin, he drew a circle and added a degenerate case.

He keenly perceived a risk point in this grand architecture.

"Your theory is exquisite. But what if this middle section of consecutive 1s shrinks to zero? The two 0 delimiters merge directly, the active position will jump, directly crossing a layer, and the direction marker will also flip simultaneously during the cross-layer merger."

Jiang Lin smiled slightly, as if he had anticipated this problem long ago.

He took the red pen and added a row of rigorous algebraic derivations below the merge rules.

"After the cross-layer occurrence, although an extra flip occurred, because the position jump compensated for the phase loss, the calculated parity class of the new boundary still remains opposite to the halting class. It still falls into our invariant set."

Zhou Shu raised his hand immediately and pointed out the second problem at the system implementation level.

"It holds logically, but how does the verifier recognize it? If two delimiters merge into consecutive 00s, when the read-write head stops at any of the cells, both left and right encodings may hold. The verifier will produce ambiguity when matching rules; how to avoid recognizing two different macro-states as the same one?"

"Very simple. Force the orientation of the read-write head staying on the delimiter to be written into the state definition." Jiang Lin drew a square with an arrow on the whiteboard, "We do not do symmetric merging; we eliminate ambiguity."

The six macro-states originally on the whiteboard became eight because of this adjustment.

Without any nonsense, Ye Ning directly retrieved the longest previously run trajectory from the server, and extracted all extreme samples where boundary merging had occurred from it.

She packaged this data and fed it one by one into the new category of macro-state logic just defined by Jiang Lin for batch testing.

Thirty-seven severe boundary mergers.

All landed safely, without a single omission falling into the two boundary paths just written out by Jiang Lin.

Of course, these thirty-seven trajectories could prove that the new abstraction was consistent with known data, but still could not replace the closure proof for all symbolic parameters.

The typing sounds of several keyboards that were originally open at the same time in the conference room gradually stopped.

Everyone stopped their current work and stared dumbfounded at the derivation on the whiteboard.

Professor Qiao Wenduo looked back at the whiteboard.

"The historical samples have passed. Now, how do we hand over the infinite parameter cases to the verifier?"

The third question he asked at this moment was no longer doubting whether this invariant held, but had directly leaped to the level of engineering implementation.

Jiang Lin drew a large box on the periphery of the eight categories of macro-control states, enclosing them as a whole.

"In the system, establish a fourth type of non-halting witness independently—the macro-state invariant witness."

"What about the underlying public trusted core?" The person in charge of the third party immediately asked, because this concerned the credibility of the entire verification architecture.

"The public core remains frozen and must never be modified."

After finishing, Jiang Lin wrote down three cold and rigid checking conditions outside the box.

[Condition 1: Verify whether the initial configuration safely enters the macro-state set.]

[Condition 2: Exhaust all local rewriting rules and verify whether they can keep the set closed.]

[Condition 3: Verify whether the macro-state set is completely mutually exclusive with the halting entry.]

"The specific engineering scheme is as follows: the new witness file is only responsible for submitting the definition of macro-states, the division of symbolic blocks, and the templates of local rewriting. The public core continues to be responsible for verifying the one-step transition semantics and finite base cases of the Turing machine; the new macro-state verifier is responsible for checking the induction conditions, set closure, and halting exclusion of the parameterized rewriting templates. The two layers are connected only through the frozen machine semantic interface."

Looking at Zhou Shu and the person in charge of the third party, Jiang Lin spoke eloquently.

"The Rust team and the OCaml team independently implement verifier extension modules targeting these eight categories of macro-control states. Third party, your responsibility is to use every possible means to generate mutation samples that can break through these eight state boundaries."

"Can the two groups of developers see your whiteboard derivation?" Professor Qiao Wenduo asked.

"Only give the abstract specification document and the serialized certificate format. The isolation must be complete." Jiang Lin said calmly.

Professor Qiao Wenduo raised his wristwatch and glanced at the time.

Seven twenty-six in the evening.

From Jiang Lin ordering the shutdown of single-step animation to him writing out complete verification boundaries by hand on the whiteboard sufficient to suppress Skelet #17, it took one hour and nineteen minutes.

Professor Qiao Wenduo turned around, his gaze like torches, sweeping across Zhou Shu and the person in charge of the third party.

"Did you all hear clearly? Open a new rule number, the two sets of implementation teams continue to maintain absolute isolation, and cut off all private communications. Seal the specification document before nine o'clock tonight, and start blind testing early tomorrow morning."

Zhou Shu took a deep breath and closed the stack of analysis reports on the table that had once consumed countless efforts of theirs and were now invalid.

He turned his head and glanced again at the red number 1 in the upper right corner of the large screen.

"This time it should finally be its turn to return to zero." Zhou Shu muttered to himself, his palms covered in sweat.

The next three days were days bordering on madness.

The eight categories of macro-control states drawn by Jiang Lin on the whiteboard were disassembled by strict mathematical language, and finally turned into seventeen pages of densely packed specification documents, forty-three local rewriting templates covering all edge cases, and six categories of vicious hidden samples specially used by the third party to attack the boundaries.

The confrontation of the blind test was extremely fierce.

The first round of blind testing started.

Just four hours later, the verifier of the Rust implementation team suddenly sounded an alarm, crashing while verifying set closure.

The Rust implementation first rejected the certificate.

The reason was that the finite leading run segment entering the first category of macro-states from the all-white tape was not written into the serialization format.

The macro-state invariant itself was not affected, but the certificate lacked the initial entry witness.

After receiving the report, Jiang Lin had no emotional fluctuations, and simply added back the leading run segment and its hash, then resealed the certificate.

The second round of blind testing.

The OCaml implementation team launched a challenge.

Their logic engine rejected a merge path spanning double delimiters, considering that a logical break existed in that path.

The third-party testing group immediately intervened; they didn't look at the code, but instead directly called the underlying machine replay function to compare frame by frame.

Three hours later, it was confirmed that the problem was not a specification error, but rather that the OCaml team had inadvertently missed a direction marker when serializing the certificate, resulting in data misalignment during deserialization.

Corrected the code.

Recompiled.

Resealed the hash.

Started the blind test again.

The time came to September 15th, at 11:41 PM.

The project team meeting room was once again filled with people.

No one spoke, and even the sound of breathing was suppressed to the lowest.

The air was permeated with the strong smell of black coffee and the irritability brought by continuous staying up late.

On the left side of the large screen was still Brady's pessimistic judgment from 1990.

On the right side were the real-time running windows of the two physically isolated verifiers.

On the black-background green-text terminal screen, the verification logs constantly refreshed.

Suddenly, the Rust window on the left stopped scrolling.

The progress bar reached 100%.

[Gray _ Invariant _ Witness / ACCEPT]

[Skelet_17 / NONHALT]

Zhou Shu's hand holding the mouse stopped on the desktop.

In the iron law of formal verification, the passing of one implementation could only prove that one program recognized this witness.

Without the corroboration of another independent code, it was still mathematically incomplete.

The OCaml window on the right was still unfolding the last six most complex boundary merging template tests one by one at a maddeningly slow speed.

The 40th check passed.

The 41st check passed.

The 42nd... verifying macro-state mutual exclusivity... passed.

The 43rd...

The whole place was dead silent.

Ye Ning bit her lower lip unconsciously.

The cursor abruptly moved down one line.

[Gray _ Invariant _ Witness / ACCEPT]

[Skelet_17 / NONHALT]

Immediately after, as if prearranged, the systems on both sides simultaneously began to generate the final result verification summary.

A comparison process popped up at the bottom of the screen.

Two seconds later.

[Cross _ Impl _ Result _ Digest / MATCH]

After receiving this impeccably double-confirmed witness, the underlying logic of the database master control program triggered.

It mercilessly classified the last entry in the directory marked as UNKNOWN into the NONHALT category.

In the upper right corner of the large screen, that nightmare-like number that had hung for an unknown number of days gently jumped once.

[Unresolved _ Machines : 0]

Zhou Shu leaned back abruptly, leaning heavily against the back of the ergonomic chair.

The huge force made the chair wheels emit a screech on the hardwood floor, sliding backward more than a meter.

"It really returned to zero!"

Zhou Shu's voice became hoarse due to the eruption after extreme suppression.

Ye Ning had no time to cheer.

She immediately threw herself at the keyboard, dragging the finally generated certificate package back into the network-disconnected offline environment.

Her fingers trembling, she continuously calculated two sets of digests, SHA-256 and BLAKE3.

The screen returned completely identical hash strings.

Only then did she let out a long breath, raised her head, looked at Professor Qiao Wenduo standing in front, and nodded heavily.

The third-party person in charge said nothing and directly reached out to unplug the external network cable interface connected to the test server.

Skillfully, he began to seal the binary files of the two compiled entity programs, the hidden test sets, and dozens of gigabytes of complete execution logs.

This is ironclad proof to deal with any academic skepticism in the future.

Professor Qiao Wenduo stood in place, like a statue.

His gaze lingered on that 0 for a long time, then slowly moved to Brady's prediction on the left, and finally fell back onto the main control page of the database.

Over forty years.

Ever since he was a young scholar, these two candidate values of the 5-state Busy Beaver had always been like two ghostly lighthouses, printed in the marginal notes of countless papers, blog forums, and classic textbooks on computational theory.

Every time they were mentioned, the same glaring word helplessly followed behind.

Unproven!

And now, tonight.

The last arrogant ghost machine was put on the shackles of non-halting.

Every single one of the 88,664,064 seed database machines obtained a traceable and independently verifiable non-halting judgment path.

The machines previously confirmed to halt by simulation were retained in another index of the complete enumeration directory.

Professor Qiao Wenduo took a deep breath and stepped over to the main seat at the very front of the conference table.

"Report the final data, enumeration directory."

"Dual-path independently generated, the quantities are completely consistent, the underlying hashes are consistent, with no omissions." The third-party person in charge answered loudly.

"Known champion verification."

Ye Ning projected the final states of the two independent simulators to the center of the main screen.

[Maximum_Halting_Steps: 47,176,870]

[Maximum_Ones_On_Tape: 4,098]

"Non-halting classification status."

Zhou Shu hit the Enter key with both hands and opened the final global index table.

"All non-halting entries can be one hundred percent traced back to the corresponding certificate file, verifier version number, and machine original hash, with none isolated. The UNKNOWN classification is confirmed to be zero."

Professor Qiao Wenduo slowly took off the reading glasses on the bridge of his nose, pulled out an eyeglass cloth from his pocket, and carefully wiped the lenses.

When he put his glasses back on, those three lines of decisive results on the screen still stood quietly there, proclaiming eternal mathematical truth.

He picked up the thick project master record book on the table and turned to the last blank page.

Pulling out his fountain pen, he wrote stroke by stroke in the final conclusion column—

[S(5) = 47,176,870]

[Σ(5) = 4,098]

Date: September 15, 2022.

The moment the pen tip left the paper, Professor Qiao Wenduo raised his head and looked around at every team member in the conference room.

"Everyone." His voice trembled slightly, "Starting from this very second, these two numbers can be deleted from the candidate list and officially incorporated into the theorem column of human mathematics and computational theory."

Around the conference table, thunderous applause suddenly erupted.

The first to clap was the third-party person in charge, who was usually extremely harsh and picked faults everywhere; his palms were clapping red.

Followed by Ye Ning and Zhou Shu.

Immediately afterwards, in several large video communication windows beside the conference room, the external collaborators who had continuously participated in the reproduction for months and were distributed around the world also stood up and applauded one another.

Someone took off their headphones and walked back and forth excitedly in the narrow room.

Someone skillfully used a screenshot tool to capture the three lines of results on the screen into their own work records as a private medal witnessing history.

Other people couldn't wait to type in the public channel, starting to ask for the final encrypted certificate package.

Jiang Lin sat quietly in the corner position close to the whiteboard.

The applause surrounded him like a tide, but he just quietly looked at the formulas on the whiteboard, as if all this commotion had nothing to do with him.

The applause lasted for a full dozen seconds.

Professor Qiao Wenduo pressed his hands down, and when the sound subsided slightly, he immediately regained the composure of the project leader and began allocating wrap-up tasks overnight.

"The on-duty personnel will complete archiving in groups according to the contingency plan, and the rest will take turns resting. The certificate package, the core source code of the two sets of verifiers, the standardized machine directory, and the complete build instructions are divided into four copies for cold backup archiving. Remember, tonight's data will only be sent to the few external top-tier reproduction nodes scheduled on the list. The final title of the paper is yet to be determined, but before the external nodes return verification passes, not a single word of our conclusion is allowed to leak to foreign media and the academic community."

Zhou Shu asked while packaging files: "What about the school side?"

"I'll report it in person."

Professor Qiao Wenduo picked up his phone, strode to the end of the corridor outside the conference room, and dialed a number.

At midnight, the phone rang only twice before being answered.

Two minutes later, the person in charge of the Department of Computer Science on the other end of the phone had obviously received and opened the briefing email sent by Professor Qiao Wenduo.

"Old Qiao, that 0 attached at the end of your email, does it mean that the entire database of machines we discussed earlier has been completely ruled out?"

"All of them, without a single dead angle." Professor Qiao Wenduo answered decisively.

"Then who chewed down that last hard bone, Skelet's ghost, what new algorithm did you guys use?"

Across the glass wall of the conference room, Professor Qiao Wenduo took a deep look at the young figure standing in front of the whiteboard.

On the whiteboard, the eight categories of macro-control state definitions and algebraic derivations written by Jiang Lin freehand were still clearly visible.

"Jiang Lin." Professor Qiao Wenduo said into the phone.

The other end of the phone fell into a dead silence lasting several seconds.

"It's that kid again?" The voice on the other end sounded incredible, "The shared trusted core architecture you reported a few days ago was also determined by him?"

"Yes, the base architecture, plus the final mathematical witness, it's all him."

"If I remember the school calendar correctly, today should only be his fourth day of formal classes, right?"

Professor Qiao Wenduo's lips raised a proud smile that he didn't even notice: "Yes, a freshman, fourth day."

In the conference room, the first batch of external reproduction emails targeting top global scholars had been sent through encrypted channels.

The recipient list was very short, with only a few names.

But the weight of these few names was enough to shake the entire theoretical computer science community.

Among them were both veteran researchers who had maintained the BB(5) machine database like ascetic monks for decades, and theoretical giants renowned in the fields of formal verification and small Turing machines.

The email content was very short.

There were no long-winded press releases, no exaggerated introductions of achievements, and not even any pleasantries.

There was only a line of temporary read-only addresses pointing to the code repository, four sets of hash checksums for anti-tampering, and a request that was simple to the point of arrogance.

[Please independently build the underlying environment and try to overthrow this conclusion.]

At 0:17.

System monitoring logs showed that the first external node located in Europe began downloading the newly added macro state certificates, two sets of verifier source code, and the final index increment.

At 0:43.

The second North American node sent back the handshake protocol, confirming the completion of the preliminary verification of the machine's massive directory.

At 1:06 AM.

An email from an external reproduction node entered Professor Qiao Wenduo's inbox.

The sender had been continuously tracking the operating rules of Skelet #17 since 2003.

He did not make any grand comments on the conclusion of the entire BB(5) problem, but directly skipped the first eighty-eight million machines, focusing his gaze straight to the point on the Gray code invariant witness written by Jiang Lin.

In the body of the email, behind the page numbers of the specification document, he sharply marked out three mathematical boundary problems that he thought might have fatal vulnerabilities.

Extremal collapse during empty segment merging.

Backward lookup ambiguity under dual delimiter states.

Phase loss during boundary parity flipping.

These three problems went straight to the core.

They were all dangerous passes that the previous third-party blind testing group spent countless energies to attack.

Professor Qiao Wenduo smiled slightly, without any hesitation, and directly packaged the certificate numbers of the corresponding forty-three rewriting rules, as well as the detailed execution records and log back-inference assertions of the underlying layers of the two different implementations in Rust and OCaml when handling these boundary cases, as attachments, and clicked reply.

Twenty-six minutes later.

The inbox over there flickered again.

This time, the body of the email sent by the other party was even shorter, with only two lines of text.

[All three boundaries have formed a logical closure. Flawless.]

[Please tell me, is the author of the specification document, that Jiang Lin, that Jiang Lin?]

Zhou Shu finished reading the email on the screen, turned his head, and turned the laptop screen toward Jiang Lin.

"Jiang Lin, how do you think I should reply to this old senior's question?" Zhou Shu joked, his fatigue from days seemingly swept away.

Jiang Lin was packing his backpack; he turned his head slightly and took a casual glance at the name on the screen.

"According to the rules of the academic community, just reply by normally providing a project contribution description." He said calmly, without a trace of showing off in his tone.

At 3:22 AM.

The sky outside the window was still pitch black, but dawn was not far away.

Ding.

A crisp system prompt tone exploded in the conference room.

The first external node holding the complete database mirror completed independent construction and full-scale index review.

In the review remarks sent along with the batch run results, the first sentence was written in bold English font.

"If your whole-database hash index continues to remain consistent without conflict in the next few days, this will be the first newly determined Busy Beaver value in computational theory for over forty years. This is an incredible miracle."

Because the other party used a completely different system architecture and compiler version, the hash of the final binary file of the Rust compiled entity was naturally different from the domestic one.

But when they sent the finally generated logical adjudication summary into the verification and comparison tool—

Completely consistent.

[External_Reproduction_01 / PASS]

The moment they saw this green PASS pop up, the air in the conference room boiled.

This time, no one deliberately maintained that serious and quiet academic atmosphere anymore.

Someone slammed a heavy palm onto the hard wooden desktop, their palm turning red.

Someone tilted their head back and drank the coffee in their cup prepared for staying up the rest of the night in one gulp.

Ye Ning even directly copied and pasted that green PASS sign fiercely into the project team's highest-level master record archiving file.

Professor Qiao Wenduo walked back in front of the big screen.

1. Brady's prediction with a despairing color still hung quietly there.

Professor Qiao Wenduo pulled up the red annotation tool in the system, and with a trembling yet powerful hand, slashed a bright red horizontal line across the six huge characters of "can never be proven".

Afterwards, beside the crossed-out prophecy, he wrote with an electronic pen—

[The first global external independent reproduction confirmation has been completed.]

[September 16, 2022, early morning.]

The crowd was cheering and hugging each other.

At the edge of this historical moment, Jiang Lin quietly picked up his schoolbag from the table, slung it over one shoulder, and prepared to push the door open to leave and return to Zijing Apartment to catch up on a few hours of sleep.

Passing by the whiteboard in front filled with derivation formulas, his footsteps paused for a moment.

Picking up the eraser placed in the trough, he reached out and gently wiped away the old state marked UNKNOWN written at the very bottom of the whiteboard two days ago.

The 5-state Busy Beaver.

This noun, like a curse, had lingered and stayed like a fog on the boundary of entire human computation theory for over forty years.

Countless genius minds had met their Waterloo in front of it.

And tonight, right in this inconspicuous conference room, the huge question mark behind it finally vanished into thin air amidst rigorous logic and absolute isolated blind testing.

And the person standing in front of the whiteboard wiping it away had just finished the fourth day of college classes.