Chapter Summaries

Shangjing University mathematics students, including Chen Muhua, arrived at the Mathematics Research Institute to hear Xiao Yi lecture on the abc Conjecture, Far Abelian Geometry, and Automorphic Forms. In the United Kingdom, James Maynard tuned in to learn whether Xiao Yi had solved the problems they discussed, only to discover the lecture would be in Chinese.

• Chen Muhua photographed Xiao Yi's poster and reflected on how far Xiao Yi had surpassed him.
• Lu Ping encouraged Chen Muhua to stop overthinking, and the students went into the park to find the lecture hall.
• Professor Cao Ping met the students, urged them to listen carefully, and left them to find seats in the crowded hall.
• Yu Can helped James Maynard access the livestream, and Maynard explained that Xiao Yi had emailed him about combining Far Abelian Geometry with Automorphic Forms before claiming he had solved all the problems.
• Maynard realized Xiao Yi's lecture covered the abc Conjecture, Far Abelian Geometry, and Automorphic Forms, then arranged to use an AI subtitle program after learning it would be delivered in Chinese.

Xiao Yi gave a lecture at the Academy of Mathematics and Systems Science, explaining why a contradiction in the Hodge Theater invalidated Shinichi Mochizuki’s IUTT proof and left the ABC Conjecture unsolved. He then prepared to move on to his new material connecting Far Abelian Geometry with Automorphic Forms.

• Xi Lan asked Xiao Yi about his lecture preparations, and Xiao Yi said he had memorized his presentation and established a connection between Automorphic Forms and Far Abelian Geometry.
• Xi Lan introduced Xiao Yi to the audience at the Academy of Mathematics and Systems Science, where scholars and students welcomed him with applause.
• Xiao Yi explained the Hodge Theater in IUTT and concluded that its contradiction invalidated the theory, so the ABC Conjecture remained unsolved.
• Ma Zhiyuan and Xi Lan praised Xiao Yi’s clear explanation, and Ma wondered whether Xiao Yi understood Automorphic Forms.
• Xiao Yi set down his cup and prepared to begin the Automorphic Forms portion of the lecture.

Xiao Yi connected Far Abelian Geometry to number theory and analyzed Grothendieck's mysterious functor. He ended his presentation by revealing its form, astonishing the mathematicians who understood his proof.

• Xi Lan told Ma Zhiyuan that Xiao Yi understood number theory better than they had expected.
• Hu Guangde defended Xiao Yi's abilities while arguing with his rival, Liao Huan, about what Xiao Yi would present.
• Xiao Yi explained how Far Abelian Geometry could be applied to number theory and introduced Grothendieck's mysterious functor.
• Xiao Yi analyzed the functor using Far Abelian Geometry and wrote an equivalence on the blackboard.
• Xiao Yi announced that he had found the functor's true form, and the knowledgeable mathematicians in the audience recognized his achievement.

Xiao Yi used a newly established functor to connect Far Abelian Geometry with Automorphic Forms, then demonstrated the connection by raising the Elliott-Halberstam conjecture’s value of θ to 0.75. He left the mathematicians stunned by claiming he had a method to prove the conjecture, but the blackboard had no room for the full derivation.

• Maynard and Yu Can recognized that Xiao Yi had defined a new functor and opened a path toward connecting Far Abelian Geometry with Automorphic Forms.
• Xiao Yi used the Weil conjectures and the new functor to develop his argument while Maynard took notes and the audience struggled to follow.
• Xiao Yi concluded that the Kuznetsov Trace Formula linked étale fundamental groups with Automorphic Forms.
• Academician Ma Zhiyuan asked what the connection could be used for, and Xiao Yi showed that it raised θ to 0.75 in the Elliott-Halberstam conjecture.
• Xiao Yi said he had an elegant way to prove the Elliott-Halberstam conjecture, but the blackboard’s remaining space was too small to write it down.

Xiao Yi ended his lecture by joking that his proof would not fit in the available space, then promised to upload his proof of the Elliott-Halberstam Conjecture in two days. Maynard lamented the delay, while Richman learned that Xiao Yi’s result had already made his thesis outdated.

• Xiao Yi told the audience he would upload the lecture and his Elliott-Halberstam Conjecture proof to arXiv in about two days, and Ma Zhiyuan led the hall in applause.
• Chen Muhua and Lu Ping struggled to grasp the significance of Xiao Yi’s claimed proof, while Hu Guangde bragged to Liao Huan about Xiao Yi being a USTC undergraduate.
• Maynard complained in his Oxford office that Xiao Yi had stopped before revealing the proof and blamed Fermat’s famous margin joke for inspiring the delay.
• Maynard warned his students not to imitate Fermat when asking for signatures, then told Richman that Xiao Yi’s result had pushed the theta value to 0.75 and made Richman’s thesis outdated.

After Xiao Yi’s lecture drew worldwide attention, leading mathematicians backed his claims while awaiting his paper. When the two papers appeared on arXiv, Bombieri read them and confirmed that Xiao Yi had succeeded.

• Richman completed his defense the day before and received Maynard’s signature, avoiding trouble with his dissertation’s originality.
• Reports of Xiao Yi’s lecture spread internationally, and Shing-Tung Yau, Maynard, and Bombieri publicly supported the credibility of his conclusions.
• Chinese mathematicians translated Xiao Yi’s lecture into English, and scholars largely agreed that his result was likely correct.
• Bombieri, Deligne, and other mathematicians waited for Xiao Yi’s paper to appear on arXiv.
• Xiao Yi’s two papers appeared on arXiv, and Bombieri read them before concluding, “He did it.”

Mathematicians at Princeton and Oxford verified Xiao Yi’s two papers and celebrated his proof of the Elliott-Halberstam Conjecture, which reduced the bounded prime gap to 6. The new theory drew broad acclaim, while Terence Tao and James Maynard discussed its limits and possible applications to other prime problems.

• Bombieri told Deligne that Xiao Yi’s two papers were correct and praised their mathematical reasoning.
• Princeton Institute for Advanced Study scholars agreed that the papers were correct after reading them.
• James Maynard followed Xiao Yi’s proof and celebrated the solution of the Elliott-Halberstam Conjecture.
• Terence Tao called Maynard and discussed how Xiao Yi’s new theory advanced the Polymath project and reduced the prime gap to 6.
• Maynard and Tao considered other applications, including the Twin Prime Conjecture, and concluded that the theory could not reduce the gap below 3.
• Mathematicians widely praised the papers, and Bojielun said he would vote for the theory in Science’s annual breakthroughs selection.

Wen Chenghua used Xiao Yi’s étale algebraic-variety automorphic theory to solve a difficult problem in p-adic Hodge theory, completing his dissertation under Andrei Okounkov. He praised Xiao Yi’s work on Quora, while the story left Wen ready to graduate and the theory drawing acclaim from mathematicians.

• Wen Chenghua excitedly told Andrei Okounkov that he had solved the problem of p-adic Galois representations over non-Archimedean local fields.
• Okounkov asked Wen to explain his solution, and Wen used Xiao Yi’s étale algebraic-variety automorphic theory at the blackboard.
• Wen’s solution resolved the problem and completed his dissertation with results worthy of a top journal.
• Wen read praise for Xiao Yi’s theory on Quora, including an answer from Terence Tao, and posted his own thanks to Xiao Yi.

Xiao Yi returned to his dorm at USTC and gave Ye Cheng, Luo Qiao, and Song Ziyang valuable math books he had brought from the Max Planck Institute for Mathematics. The chapter left off when Xiao Yi received a call from the United States as Ye Cheng tried to record him for the Joint Examination Group.

• Ye Cheng played PUBG with Luo Qiao and Song Ziyang, and they accused an enemy of hacking after he defeated them.
• Xiao Yi returned to Room 313, where Ye Cheng welcomed him and asked for a gift.
• Xiao Yi gave Ye Cheng a photocopy of Peter Scholze's Principles of Algebraic Geometry filled with Schultz's notes, and gave Luo Qiao and Song Ziyang books as well.
• Ye Cheng posted photos of his book in the Joint Examination Group, got himself and his roommates muted, and then told Chen Muhua he would record a video of Xiao Yi.
• Xiao Yi's phone rang, and he saw that the call was from the United States.

Xiao Yi received a call from Peter Sarnak, the editor-in-chief of Annals of Mathematics, inviting him to submit two papers to the journal. While Xiao Yi discussed the submission, his friends shared a recording of the call in their group chat, where it caused a stir.

• Xiao Yi answered a call from Peter Sarnak, who introduced himself as the editor-in-chief of Annals of Mathematics.
• Sarnak invited Xiao Yi to submit his two papers to Annals of Mathematics, and Xiao Yi agreed, saying he had already planned to send them there.
• Ye Cheng recorded the call and sent it to the group chat, where Chen Muhua, Jiang Long, and others reacted with shock and envy.
• Jiang Long muted Ye Cheng for an hour after Ye Cheng complained about the group’s reactions.
• Sarnak heard the commotion and asked whether someone was arguing, and Xiao Yi replied that someone might be waging a struggle.

Xiao Yi's two papers underwent an intensive review by a prestigious committee while mathematicians around the world began using his theories. After joking with his roommates about the joy of mathematics, he attended his first university class and was greeted enthusiastically by his classmates.

• The Annals of Mathematics assembled Enrico Bombieri, Jacob Stix, Faltings, Terence Tao, and other mathematicians to review Xiao Yi's two papers.
• Song Ziyang told Xiao Yi that many mathematicians expected his theories to advance their research, and Xiao Yi said he had received at least a hundred emails.
• Song Ziyang, Ye Cheng, and Luo Qiao browsed Xiao Yi's emails and found messages from leading universities and Fields Medal winner Andrei Okounkov.
• Luo Qiao asked Xiao Yi whether being a mathematical genius made him happy, and Xiao Yi said he loved studying mathematics and solving difficult problems.
• Xiao Yi went to his first class, Mathematical Analysis, where his classmates greeted him as Scholar Xiao.

Xiao Yi attended Liu Bin’s Mathematical Analysis class while thinking about how to advance the Twin Prime Conjecture. Liu Bin posed a challenge about prime factors, and the problem sparked an idea in Xiao Yi that might help him cross the remaining gap.

• Liu Bin noticed Xiao Yi in class and greeted the student he had helped recruit to USTC.
• Xiao Yi listened to Liu Bin’s lesson while reflecting on how the Elliott-Halberstam theorem had brought the Twin Prime Conjecture within 6, though he could not see a way across the remaining gap.
• Liu Bin called Ye Cheng and Zhang Zhilan to prove a sequence-limit result, then praised their different approaches.
• Liu Bin wrote a challenge problem involving the distinct prime factors of a product of primes.
• Xiao Yi considered using a sieve method and glimpsed a possible bridge toward solving the Twin Prime Conjecture.

Xiao Yi solved Liu Bin’s difficult problem using two different methods, then stayed at the blackboard to attempt a third approach with a sieve method. He began considering the Selberg sieve, but had not yet found a clear path.

• Liu Bin excluded Xiao Yi from the grade guarantee but invited him to solve the problem if he used at least two methods.
• Xiao Yi used an equivalent limit expression, the Prime Number Theorem, and Stolz’s theorem to complete his first proof.
• Xiao Yi completed a second, more direct proof using techniques from Mathematical Analysis, earning applause from the class.
• Xiao Yi asked to try a third method and began writing a sieve-based approach, considering the Selberg sieve.

Xiao Yi used the Selberg Sieve to finish a solution to the classroom problem, then realized how sieve methods might be pushed further to address the parity problem and the Twin Prime Conjecture. He left class to write down his new ideas.

• Xiao Yi introduced sieve-method notation and applied the Selberg Sieve to classify the terms in the problem.
• Ye Cheng recognized the method as the Selberg Sieve and explained how it helped solve the problem while avoiding the parity problem.
• Xiao Yi paused, then wrote advanced sieve formulas on the blackboard as he developed an idea for classifying the Twin Prime Conjecture.
• Xiao Yi finished the third solution and told Liu Bin that he thought he could push sieve methods further.
• Liu Bin let Xiao Yi leave class, and Xiao Yi gathered his things and went to record his ideas.

Xiao Yi left classes to develop a new approach to sieve theory and the parity problem. A viral video of him solving a prime-related limit problem in three ways drew attention from Chen Beihua and Liu Dongchen, who recognized that Xiao Yi’s third method might be a classified study of the parity problem.

• Xiao Yi studied the parity problem in sieve theory and began writing a new theory based on his work with automorphic étale algebraic varieties.
• Xiao Yi’s classmates noticed he was spending long days on research and covering his desk and walls with notes.
• A Bilibili video of Xiao Yi solving a limit problem involving primes in three ways became popular.
• Chen Beihua solved the problem in two ways, then watched Xiao Yi’s video and was impressed by Xiao Yi’s three solutions.
• Liu Dongchen recognized that Xiao Yi’s third solution appeared to classify the parity problem in sieve methods and said he would ask someone about it.

Liu Dongchen shared Xiao Yi’s sieve-theory formulas with a number theory group, where Professor Tao Quan said they might help address the parity problem and speculated that Xiao Yi was pursuing the Twin Prime Conjecture. The video spread overseas, and at Rutgers University, a student brought the formulas to Henrik Iwaniec after his lecture on sieve theory.

• Liu Dongchen posted a video of Xiao Yi’s formulas in the Hope of Chinese Number Theory group, where Tao Quan recognized them as a classification related to the parity problem.
• Tao Quan noted that the formulas covered cases with both even and odd numbers of prime factors, but warned of proof and technical difficulties.
• The group speculated that Xiao Yi was working on the Twin Prime Conjecture, and Liu Dongchen told Chen Beihua that Xiao Yi was tackling it.
• The video spread through mathematical circles and reached Henrik Iwaniec’s number theory class at Rutgers University.
• A student asked Iwaniec about the formulas and began writing them on the blackboard.

After Xiao Yi’s new approach to sieve methods drew attention across the mathematical community, he completed a proof of the Twin Prime Conjecture in his dormitory. The system confirmed his success, leaving the conjecture established as the Twin Prime Theorem, though the proof had not yet been recognized by other mathematicians.

• A student showed Ivanets the opening of Xiao Yi’s classification-based approach to sieve methods, and Ivanets praised it as an unusually clever first step.
• The student said Xiao Yi’s video only covered that much, and Ivanets expressed hope that Xiao Yi would succeed in proving the Twin Prime Conjecture.
• News of Xiao Yi’s research spread through the mathematical community, and Zhang Yitang called his idea advanced and innovative.
• Faltings told Schultz that he had once redirected Xiao Yi toward the Elliott-Halberstam Conjecture after Xiao Yi answered his questions about the Twin Prime Conjecture.
• On the last day of November, Xiao Yi finished writing a proof that infinitely many twin prime pairs existed.
• The system confirmed Xiao Yi’s achievement, increased his level to lv4, and announced new effects for Ruthless Continuous Study.

After proving the Twin Prime Conjecture, Xiao Yi upgraded his Ruthless Continuous Study buff and began preparing a paper on his proof and improved sieve method. Zhang Yitang discovered the paper on arXiv and realized Xiao Yi had completed the proof.

• Xiao Yi learned that the Twin Prime Conjecture counted as a medium-value result, but his buff upgraded to level four and its daily study requirement dropped to twelve hours.
• Ye Cheng and the other roommates told Xiao Yi that a video of him working on sieve methods had drawn attention to his Twin Prime Conjecture research, and they informed him that his two earlier papers had passed review.
• Xiao Yi signed the copyright transfer agreements for his accepted papers, then began writing his paper, titled "Parity Check Classification Sieve and the Twin Prime Conjecture."
• Several days later, Zhang Yitang searched arXiv for work on sieve methods and found Xiao Yi's paper claiming a proof of the Twin Prime Conjecture.
• Zhang Yitang downloaded the paper and read its abstract, which described a stronger Friedlander-Iwaniec theorem and classifications of parity problems used to establish the result.

Zhang Yitang carefully checked Xiao Yi's Parity Check Classification Sieve and found no flaws, discovering that it could substantially advance his Landau-Siegel zero research. As the mathematical community began discussing the Twin Prime Conjecture proof, Henrik Iwaniec told his students that experts viewed the paper positively and that Princeton planned a special presentation.

• Zhang Yitang worked through Xiao Yi's paper and verified the Parity Check Classification Sieve, then used it to make a breakthrough on a difficult problem in his Landau-Siegel zero research.
• Joey Ayton rushed into Zhang Yitang's office to tell him that Xiao Yi's paper claiming a proof of the Twin Prime Conjecture had appeared on arXiv.
• Zhang Yitang told Joey Ayton that he would review the paper rigorously if asked, despite being a potential beneficiary of its sieve method.
• Scholars and students across the mathematical world began discussing Xiao Yi's Twin Prime Conjecture paper after it was discovered on arXiv.
• Henrik Iwaniec told his Rutgers University students that he and other experts viewed the proof positively but that its correctness could not yet be determined.
• Henrik Iwaniec announced that Princeton planned a special Twin Prime Conjecture presentation for Xiao Yi at its annual mathematics conference next month, exciting his students.

Xiao Yi received Peter Sarnak’s invitation to Princeton’s 2020 Annual Mathematics Conference, where he would give a special lecture explaining his Twin Prime Conjecture proof. He accepted, and the mathematical community eagerly awaited the lecture as January 2021 arrived.

• Peter Sarnak congratulated Xiao Yi on his approved papers and Twin Prime Conjecture proof, then invited him to Princeton’s conference and a special lecture on his Parity Check Classification Sieve.
• Xiao Yi recalled meeting Schultz and Faltings at the Max Planck Institute and looked forward to thanking Faltings for steering him toward the Elliott-Halberstam Conjecture.
• Ye Cheng and Song Ziyang read Sarnak’s email and told Luo Qiao that Princeton had devoted a full day to Xiao Yi’s lecture, prompting Luo Qiao to lament another dimensionality-reduction strike.
• Xiao Yi replied that he would attend, then joined his roommates in playing PUBG.
• News of the lecture stirred excitement among mathematicians, enthusiasts, and journalists as they awaited the proof’s final outcome and January 2021 arrived.