Chapter Summaries

Xiao Yi drew inspiration from a Ferris wheel and conceived the Generalized Modular Curve, a higher-dimensional analogue of a modular curve. He began deriving its properties and linked its zeta function to extended L-functions, bringing him closer to resolving the remaining obstacle in his work on the Riemann Hypothesis.

• After riding a roller coaster, Xiao Yi realized that a new geometric concept resembling modular curves might help him continue his research.
• While riding the Ferris wheel with Luo Mingya, Xiao Yi imagined a higher-dimensional modular curve and named it the Generalized Modular Curve.
• Xiao Yi wrote down a preliminary definition of the Generalized Modular Curve using Siegel modular forms and concluded that it parameterized certain higher-dimensional Abelian varieties.
• Xiao Yi formulated a theorem relating the extended L-function of an Abelian variety to the zeta function of its Generalized Modular Curve.
• Xiao Yi began working toward the final proof, believing that the Artin Conjecture could no longer block his progress.

Xiao Yi completed his proof of the Artin Conjecture and then derived a proof of the Riemann Hypothesis. Afterward, he proposed a new conjecture linking automorphic representations, L-functions, and geometric objects, naming it Xiao's Conjecture.

• Xiao Yi completed his Artin Conjecture proof by showing that elliptic curve Galois representations came from modular forms.
• Xiao Yi derived a new automorphic representation and spent the following month connecting his results to the Riemann Hypothesis.
• Mathematicians in the Prime Pioneer Program made limited progress, while their teams published results on their website and media mocked Xiao Yi for not sharing his work.
• In the early hours of July 15, Xiao Yi finished his proof that all nontrivial zeros of the zeta function lay on the critical line.
• While reviewing his proofs, Xiao Yi proposed that an automorphic representation's L-function equaled a natural L-function of a corresponding geometric object, and he named the idea Xiao's Conjecture.

After resting for a week, Xiao Yi returned to the laboratory and spent ten days preparing and uploading five papers presenting his proof of the Riemann Hypothesis and its supporting work. With that project complete, he chose Xiao's Conjecture as his next research problem.

• Xiao Yi took a week off, relaxing with Luo Mingya and Sun Lina before telling Luo Mingya he would return to the laboratory.
• Xiao Yi resumed his ten-hour study routine, planned four papers for his proof, and completed them over the next ten days.
• Xiao Yi uploaded the four new papers and his earlier Elliptic Inverse Analytic Continuation paper to a dedicated website.
• Xiao Yi considered future research and his family's aging, then decided to investigate whether Xiao's Conjecture was true.

Sanjay sought Manuel Balgava’s help with his work on the Gamma function, and Manuel recommended Xiao Yi’s Elliptic Inverse Analytic Continuation. While searching Xiao Yi’s website, Sanjay and Chen Muhua discovered papers claiming proofs of the Artin Conjecture and the Riemann Hypothesis, leaving Manuel and the Princeton mathematics students stunned.

• Manuel identified unexpected properties in Sanjay’s Gamma function problem and suggested using Xiao Yi’s Elliptic Inverse Analytic Continuation.
• Sanjay and Chen Muhua visited Xiao Yi’s personal website and found five papers, including “Proof of the Artin Conjecture” and “Proof of the Riemann Hypothesis.”
• Sanjay and the Princeton mathematics students reacted with shock to the claimed proof, while their attempts to read the paper proved too difficult.
• Manuel saw the Riemann Hypothesis paper on Xiao Yi’s website, accepted that it was real, and asked where the students had found it.
• Manuel shared Xiao Yi’s website and news of the proof with the Prime Pioneer Program discussion group, then began reading the papers on high-dimensional modular curves and the Artin Conjecture.

Xiao Yi announced a five-paper proof of the Riemann Hypothesis, stunning mathematicians and the public. Terence Tao began studying Xiao Yi’s work, while mathematicians worldwide immersed themselves in the papers and the news spread across social media.

• Manuel shared news in the Prime Pioneer Program discussion group that Xiao Yi had posted five papers proving the Riemann Hypothesis.
• Terence Tao downloaded the papers and began reading Xiao Yi’s work on high-dimensional modular curves and extended L-functions.
• Terence Tao praised the paper in the discussion group and continued reading it with growing admiration.
• Mathematicians in the Prime Pioneer Program spent the following days studying Xiao Yi’s papers, disrupting their usual work.
• The news spread internationally, and social media users reacted with disbelief, debate, and praise for Xiao Yi as the God of Mathematics.

After Xiao Yi published his five-paper proof of the Riemann Hypothesis, mathematicians and the public awaited confirmation. Leading mathematicians praised his work, while Manuel Balgava explained why Xiao Yi had outpaced the Prime Pioneer Program.

• Netizens across China and abroad celebrated Xiao Yi’s claimed proof of the Riemann Hypothesis and called him the God of Mathematics.
• Academician Xi Lan said the proof’s success would be a major mathematical breakthrough, while noting that the community still needed to verify it.
• After nearly ten days, Terence Tao praised the three papers he had read as works of mathematical art and said he believed Xiao Yi had succeeded.
• Schultz praised Xiao Yi’s papers after skimming all four, and Manuel Balgava acknowledged that the Prime Pioneer Program had fallen behind.
• Manuel Balgava told a reporter that mathematics followed an Anti-Barrel Principle, with Xiao Yi’s exceptional ability outweighing the program’s many mathematicians.

Mathematicians praised Xiao Yi’s proof of the Riemann Hypothesis, while the Prime Pioneer Program shifted to reviewing his papers and the president worried about the political fallout. Xiao Yi learned he was a candidate for both academies, then asked Luo Mingya what she thought about their wedding, leaving her blushing and speechless.

• Deligne and other mathematicians praised Xiao Yi’s work and described how his abilities had continued to improve.
• The Prime Pioneer Program announced it would stop trying to prove the Riemann Hypothesis and focus on reviewing Xiao Yi’s papers.
• The president angrily blamed the mathematicians for praising Xiao Yi, worried about Vance’s falling poll numbers, and learned the CIA could not obtain Xiao Yi’s genetic material.
• Xiao Yi rejected media interviews and told Luo Mingya he planned to research miniaturized nuclear fusion in the future.
• Liu Cong told Xiao Yi he was on the candidate lists for both academies and asked when he would marry Luo Mingya.
• Xiao Yi asked Luo Mingya for her opinion about their wedding, and she blushed without answering.

Xiao Yi announced a November 1 symposium in Fei City to explain his proof of the Riemann Hypothesis, drawing mathematicians and government support from around the world. At the symposium’s opening, Shing-Tung Yau praised Xiao Yi as a genius and said he believed the proof had succeeded.

• Xiao Yi appeared on the candidate lists for both the Chinese Academy of Sciences and the Chinese Academy of Engineering, prompting widespread discussion.
• Mathematicians called for Xiao Yi to explain his Riemann Hypothesis proof, and he announced a symposium for November 1 in Fei City.
• Peter Scholze and Gerd Faltings decided to travel to China to attend the symposium, while mathematicians elsewhere reflected on how Xiao Yi’s proof had changed their own ambitions.
• The Riemann Hypothesis symposium opened at Anhui Province International Conference Center, where international and Chinese mathematicians and reporters gathered.
• Shing-Tung Yau told a CCTV reporter that Xiao Yi was a genius and that he believed the proof had succeeded, then went inside to attend the symposium.

Xiao Yi began his symposium on the Riemann Hypothesis by explaining how Elliptic Inverse Analytic Continuation connected the zeta function to Galois representations, then introduced the Generalized Modular Curve as a key new theory in his proof. He left the audience absorbed in his account of how amusement-park structures had inspired the higher-dimensional approach.

• Terence Tao, Shing-Tung Yau, and Feierman reminisced about attending Xiao Yi’s previous symposium and discussed the timing of his presentation before the upcoming election.
• Xiao Yi began the symposium, stated the Riemann Hypothesis, and started explaining Elliptic Inverse Analytic Continuation.
• Xiao Yi showed how CM elliptic curves, Hecke features, and Galois representations connected the Riemann zeta function to the proof.
• After an hour, Xiao Yi finished covering the method’s basics and introduced the Generalized Modular Curve.
• Xiao Yi described how he had searched for a geometric object related to extended L-functions and how a roller coaster and Ferris wheel at an amusement park inspired his higher-dimensional approach.

Xiao Yi explained how he developed Generalized Modular Curves and used them to prove the Artin Conjecture. After a lunch break, he began the final part of the symposium, which pointed toward his proof of the Riemann Hypothesis.

• Xiao Yi described how geometric ideas, Shimura varieties, and Siegel modular forms led him to define Generalized Modular Curves and connect them to extended L-functions.
• Xiao Yi paused the symposium for lunch, and the mathematicians discussed his proof strategy while they ate.
• Xiao Yi resumed the symposium at 14:00 and presented the final steps of his proof of the Artin Conjecture.
• Xiao Yi concluded that the Artin Conjecture had been proven, prompting astonished reactions and discussion among the mathematicians.
• Xiao Yi began the final part of his report, which pointed toward the proof of the Riemann Hypothesis.

Xiao Yi announced that he had proven the Riemann Hypothesis, then defended his argument during a question-and-answer session. After Scholze accepted Xiao Yi’s explanation of a concern about non-CM elliptic curves, Wiles raised a deeper question about whether all elliptic curves could be embedded into Generalized Modular Curves.

• Xiao Yi reduced the Riemann Hypothesis to the automorphy of Hecke features for CM elliptic curves and announced that his proof was complete.
• The audience applauded prematurely, and Xiao Yi opened the floor to questions.
• Peter Scholze questioned how Xiao Yi’s argument covered ordinary elliptic curves, and Xiao Yi explained that his proof used generalized L-functions and broader conditions.
• Scholze said he understood Xiao Yi’s answer, and a few other mathematicians also appeared to grasp the argument.
• Andrew Wiles asked whether Xiao Yi had implicitly assumed that all elliptic curves could be embedded into a Generalized Modular Curve.

At Wiles's challenge, Xiao Yi acknowledged that his paper had implicitly assumed every elliptic curve could be embedded into a Generalized Modular Curve. He began an on-the-spot proof of that claim, leaving the audience stunned as he started outlining his argument.

• Deligne and Bombieri discussed Wiles's question and the flaw that had once derailed Wiles's proof of Fermat's Last Theorem.
• Xiao Yi told Wiles he had implicitly treated the embedding claim as obvious, based on his mathematical intuition.
• Xiao Yi announced that he would prove that every elliptic curve could be embedded into a Generalized Modular Curve.
• After the hall fell silent, Xiao Yi began his proof by discussing elliptic-curve L-functions and defining a proposed Generalized Modular Curve.

Xiao Yi completed a proof of the Riemann Hypothesis, winning the mathematical community’s acceptance and prompting the audience to celebrate its transformation into the Riemann Theorem. Before ending his presentation, he introduced a new conjecture linking automorphic representations, L-functions, and geometric objects, leaving future research to pursue it.

• Xiao Yi finished his proof by showing how the modularity of an elliptic curve could be embedded into the modularity of a generalized modular curve.
• Andrew Wiles praised Xiao Yi’s answer, and the audience applauded as leading mathematicians acknowledged the proof.
• Xiao Yi declared that the Riemann Hypothesis had become a monument in mathematical history, and the audience celebrated its proof on November 1, 2028.
• Xiao Yi presented a conjectured equality linking the L-function of an automorphic representation to a natural L-function of a geometric object.
• Xiao Yi concluded his presentation and left the hall as the standing audience continued applauding.

After Xiao Yi proved the Riemann Hypothesis and answered challenges at the conference, the livestream sparked worldwide discussion and a nationwide celebration in China. In a post-conference interview, Xiao Yi explained that he had anticipated questions from Wiles and Schultz, though he derived the answers on the spot.

• Viewers around the world discussed Xiao Yi's proof and his answers to Wiles and Schultz, while mathematicians debated how difficult it would be to build on his paper.
• International reactions spread across social media, and people in China celebrated Xiao Yi's achievement as television stations reported on the conference.
• A Fei City TV reporter interviewed Xiao Yi after the conference and asked how he had found the Generalized Modular Curve's auxiliary tools.
• Xiao Yi said the Shimura variety and Siegel modular form had been easy to think of from the Generalized Modular Curve's properties and the proof's next steps.
• Xiao Yi explained that he had already considered likely questions, including Wiles's and Schultz's, but had only recalled his solution approaches before deriving the answers during the conference.

Xiao Yi discussed the five remaining questions about his proof of the Riemann Hypothesis and said he had already resolved them, then spoke about possibly researching biomedicine to extend human lifespan. His interview spread worldwide, and after a celebratory banquet he talked with Luo Mingya, who supported his choices while encouraging him to consider materials research.

• Xiao Yi explained five potential challenges in his proof and told the reporter that he had solved them before publishing his paper.
• Xiao Yi said he might study biomedicine and develop medicine to extend human lifespan so he could witness more truths before he died.
• CCTV aired Xiao Yi's interview, prompting discussion in China and overseas about his proof, his mortality, and the president's hostility toward him.
• Xiao Yi attended a banquet with Terence Tao, Schultz, Deligne, and other mathematicians, who discussed his proof and Xiao's Conjecture.
• Luo Mingya praised Xiao Yi's mathematical ability and said she still hoped he would consider materials research, while supporting whatever he chose.

After Xiao Yi proved the Riemann Hypothesis, mathematicians and public figures praised his achievement as historic. As global attention continued, online arguments over whether he should pursue pure research or practical applications grew, while Xiao Yi himself moved on from the proof.

• Musk warned that Xiao Yi’s scientific achievements made him hope Xiao Yi would not turn to aerospace research, while Terence Tao, Schultz, Deligne, Wiles, and other mathematicians praised the proof and discussed its questions.
• The world marked November 1, 2028, as a historic day, and the chapter noted that Xiao Yi’s proof could influence mathematics for years to come.
• Xiao Yi set aside the Riemann Hypothesis achievement, which he considered less significant than controllable nuclear fusion.
• Online arguments split into four groups: the Extreme Conservative Party, the Extreme Application Party, the Scholar Xiao Free Will Party, and the Science Contribution Party.
• Xiao Yi ignored the arguments and maintained that he was not a celebrity or internet influencer, while his supporters’ respect for him continued to grow.

Xiao Yi took another break while China announced a nationwide electricity price reduction made possible by nuclear fusion, intensifying foreign criticism of the stalled JNEO project. By November, the new academician selection process had ended, and Xiao Yi had been elected to both academies and could be called Academician Xiao.

• The outgoing president had Steven Cowley arrested after discovering he diverted JNEO funds to support the Democratic Party, and Cowley's removal left the organization unable to operate.
• The president-elect promised to restore JNEO after taking office, while China refused foreign requests for nuclear fusion technology for at least five years.
• China announced that the average electricity price would fall to one cent per kilowatt-hour starting October 1, prompting widespread celebration at home and envy abroad.
• Foreign netizens blamed JNEO for failing to develop fusion, and a JNEO member who defended the researchers was driven to shut down his private messages.
• Xiao Yi took another break, then learned in the latter half of November that he had been selected as an academician of both academies.

After becoming an academician of both Chinese academies, Xiao Yi attended a State Council meeting about the 15th Five-Year Plan and the upcoming 16th Five-Year Plan. He explained how science and engineering related to the social sciences, then reassured planners that no current technology besides nuclear fusion was likely to disrupt their plans so dramatically.

• Xiao Yi attended meetings of the Chinese Academy of Sciences and the Chinese Academy of Engineering, where academicians congratulated him on being elected to both.
• At the State Council meeting, Lin Jingxi asked Xiao Yi how he viewed the relationship between science and engineering and the social sciences.
• Xiao Yi said science and engineering created resources while the social sciences decided how to use them, then told Lin Jingxi that his research lay beyond what planners could account for.
• Luo Bangrong explained that Xiao Yi's nuclear fusion technology had disrupted policy plans but also gave the country valuable new options.
• Xiao Yi assured Luo Bangrong that no current technology besides nuclear fusion could have a comparable impact on society.

Xiao Yi attended a national meeting where officials and economists discussed how nuclear fusion could strengthen China’s economy and improve livelihoods. After revealing that Luo Mingya was Luo Bangrong’s granddaughter, Xiao Yi agreed to consider suitable research requests, and the country’s leaders pledged to support his work.

• Luo Bangrong revealed that Luo Mingya was his granddaughter and urged Xiao Yi to set a date to marry her, while Xiao Yi said the decision depended on him and Mingya.
• Lin Jingxi tried to promote his own granddaughter to Xiao Yi, and he and Luo Bangrong argued until staff separated them.
• At the meeting, experts discussed using cheap fusion electricity to expand access, attract investment to poor regions, reduce inequality, and support pollution treatment.
• The senior leader asked Xiao Yi to share his thoughts, and Xiao Yi offered to consider developing technologies that interested him if their requests were feasible.
• The senior leader promised the country would support Xiao Yi’s research, and Xiao Yi called on everyone to strive together to create more miracles for China.

After agreeing to lead China’s effort to develop EUV lithography technology, Xiao Yi helped establish China New Core Technology Company and its research institute. A year later, he proposed to Luo Mingya on Valentine’s Day, and she accepted, leaving them engaged.

• The old leader asked Xiao Yi to develop chip technology, especially EUV lithography machines, and Xiao Yi agreed.
• The old leader told Xiao Yi that Project H-20 would make its maiden flight on National Day and mentioned Xiao Yi’s relationship with Luo Bangrong’s granddaughter.
• Project H-20 made its maiden flight, and Xiao Yi returned to Fei City to begin researching chips.
• China New Core Technology Company and its semiconductor research institute were established, bringing together talent and technical materials from major technology companies.
• On February 14, 2029, Luo Mingya visited Xiao Yi in Fei City, and he asked her to marry him; she accepted.