Chapter Summaries

Xiao Yi joined the Institute of High Energy Physics to help analyze quark-gluon plasma data and noticed unusual patterns in experimental errors. His calculations suggested that W and Z boson production rates and the W boson mass did not match Standard Model predictions, so Guo Ding went to fetch Tang Yifang.

• Tang Yifang asked Guo Ding to process QCD phase-transition data and recruit Xiao Yi to the Institute of High Energy Physics.
• Xiao Yi spotted patterned errors in superconductivity data and asked Guo Ding to find the experimental equipment specifications before agreeing to join the project.
• Guo Ding secured Academician Pang's permission to take Xiao Yi to the Institute of High Energy Physics, where Tang Yifang welcomed him.
• Guo Ding showed Xiao Yi collision data from BEPC, and Xiao Yi noticed several measurements that deviated from expected distributions.
• Xiao Yi found that W and Z boson production rates seemed unusually high and that the results fit only if the W boson mass increased by about 0.1%, prompting Guo Ding to call Tang Yifang.

Xiao Yi explained his novel statistical methods to Tang Yifang, who arranged a presentation and a research team’s recalculation of the W boson mass. As Xiao Yi prepared to refine his work, he wondered whether the apparent mass excess might relate to the Yang-Mills mass gap, but decided to verify the result with more data first.

• Guo Ding brought Tang Yifang to hear Xiao Yi explain his calculations suggesting the W boson mass was more than five standard deviations above the Standard Model prediction.
• Tang Yifang questioned Xiao Yi’s unfamiliar error-handling and statistical methods, and Xiao Yi explained his improved estimation algorithms and error analysis.
• Tang Yifang scheduled Xiao Yi to present his methods in ten days and said researchers would recalculate the W boson mass; a CERN team led by Giorgio Parisi might also provide access to LHC data.
• After Tang Yifang left, Xiao Yi considered a possible connection between the W boson mass and the Yang-Mills mass gap problem.
• Xiao Yi decided to focus first on confirming the W boson mass result with substantially more experimental data.

Xiao Yi presented a new method for analyzing and processing experimental errors, using it to show that 2,000 BEPC data points suggested the W boson was heavier than the Standard Model predicted. Giorgio Parisi said the result needed more data and offered to help Xiao Yi obtain LHC data if he published a convincing paper.

• News spread through the Institute of High Energy Physics that Xiao Yi would give a presentation arranged by Tang Yifang, and Giorgio Parisi arrived with his CERN team for academic exchange.
• Before the presentation, Parisi asked Xiao Yi about the result that differed from the Standard Model, but Xiao Yi saved the details for his talk.
• Xiao Yi demonstrated his error-analysis method on W boson decay data and calculated a mass of 80.471±0.027 GeV for one data point.
• Xiao Yi showed that processing 2,000 data points produced a W boson mass of 80.438±0.011 GeV, prompting shocked reactions from the audience.
• Parisi said the result was promising but needed more data, and offered to persuade CERN to provide LHC data if Xiao Yi published a convincing paper.

Xiao Yi’s new error-analysis method led PRL to publish his claim that the W boson mass exceeded the Standard Model prediction, prompting CERN and the China Institute of High Energy Physics to begin a joint investigation. The collaboration began processing a million LHC data samples, with results expected in six months as physicists worldwide studied Xiao’s method.

• Researchers at the Institute of High Energy Physics and the visiting CERN team studied Xiao Yi’s Error Handling Method while awaiting his paper.
• PRL editor-in-chief Robert Griester and associate editors accepted Xiao Yi’s paper for the next issue despite concerns that it used only 2,000 data sets.
• After reading the paper, Frank Wilczek judged Xiao Yi’s analysis method sound and discussed using LHC data to test the W boson mass result.
• Giorgio Parisi connected Xiao Yi and Tang Yifang with Wilczek, and CERN agreed to provide LHC collision data and collaborate with the China Institute of High Energy Physics.
• CERN and the China Institute of High Energy Physics began jointly processing data, targeting one million samples and estimating that verification would take six months.
• Physicists worldwide debated the possible discovery and rushed to study Xiao’s Error Analysis Theory.

Xiao Yi began his doctoral studies under Xun Yong, joined the mathematics lab, and continued struggling with the Yang-Mills mass gap problem. In October, Pang Cheng visited with news that their project had been submitted to Nature, and Xi Lan brought Xiao Yi an invitation to the National Science and Technology Awards Conference.

• Xun Yong welcomed new master's students Gong Weihua and Zou Qiao, discussed their research plans, and introduced them to the mathematics lab.
• Xiao Yi arrived as Xun Yong's doctoral student, surprised the new students and lab members, and exchanged WeChat contacts with them.
• Xiao Yi tried several approaches to the mass gap problem but stalled at difficult points in each one.
• Pang Cheng visited Xiao Yi and announced that their research project had been completed and submitted to Nature.
• Xi Lan arrived with an envelope, and Xiao Yi discovered it contained an invitation to the National Science and Technology Awards Conference.

Xiao Yi attended the National Science and Technology Awards Conference and unexpectedly received the First Prize of State Natural Science Award for proving the Goldbach Conjecture. After the ceremony, online discussion focused on why his Twin Prime Conjecture had not won an award.

• Xi Lan and Pang Cheng argued over whether Xiao Yi should attend with the Institute of Physics or the Academy of Mathematics and Systems Science.
• At the conference on November 2nd, Xiao Yi listened as Academician Gu Songfen and Academician Wang Dazhong received the State Supreme Science and Technology Award.
• The host announced that Xiao Yi had won the First Prize of State Natural Science Award for his proof of the Goldbach Conjecture, and Xiao Yi went onstage to accept it.
• Pang Cheng congratulated Xiao Yi, and Xiao Yi promised to treat researchers from the Academy of Mathematics and Systems Science and the Institute of Physics to dinner.
• The ceremony presented two more Natural Science Award first prizes and then concluded, while online discussion turned to why Xiao Yi's Twin Prime Conjecture had received no award.

After receiving a major science award, Xiao Yi developed a mathematical explanation for the W boson’s excess mass and presented CERN researchers with a possible new field. He left them with the proposal that the X field could explain the anomaly, though confirming it directly would be difficult.

• Xiao Yi ignored the online discussion of his Goldbach Conjecture award and focused on the Yang-Mills mass gap problem, using topological quantum field theory to develop a new approach.
• CERN and the China Institute of High Energy Physics reported that analysis of one million data sets confirmed the W boson’s mass exceeded the Standard Model prediction.
• At a CERN meeting, Fabiola Gianotti approved making W boson mass verification a priority, and Tang Yifang agreed that the China Institute of High Energy Physics would continue collaborating.
• Tang Yifang relayed Xiao Yi’s warning that the LHC could not reveal the cause of the W boson anomaly, and CERN invited Xiao Yi to explain by video.
• Xiao Yi derived a new vector field from Yang-Mills theory and said it could account for the W boson’s measured mass, naming it the X field or Xiao field.

At CERN, Xiao Yi revealed that the X field emerged from his research on the Yang-Mills mass gap, and CERN decided to search for evidence of it despite his doubts about the project. Back in his dorm, Xiao Yi connected his remaining proof problem to the Hodge standard conjecture and began developing a new approach using vertex algebras and homological algebra.

• Xiao Yi told Frank Wilczek that the X field might mediate the unification of the strong and weak interactions, though his inference rested on intuition for now.
• Edward Witten inferred that Xiao Yi was close to proving the Yang-Mills mass gap, and Xiao Yi confirmed that the X field had emerged during that research.
• CERN researchers questioned Tang Yifang about Xiao Yi's progress while speculating that Xiao Yi might present a proof at the International Congress of Mathematicians.
• Fabiola Gianotti put the matter to a vote, and CERN chose to continue the W boson project as a search for the X field, with Wilczek urging a further LHC upgrade.
• Xiao Yi returned to his draft and realized that proving the Hodge standard conjecture could resolve his theorem about the Yang-Mills mass gap.
• Xiao Yi connected the Hodge standard conjecture with vertex algebras and homological algebra, then began writing out a new approach based on a generalized theory of norms.

News of Xiao Yi’s proposed X field and near-completion of the Yang-Mills mass gap proof spread through the physics and mathematics communities. As the International Congress of Mathematicians approached, Xiao Yi submitted a talk title announcing a strictly positive mass gap, and his study system awarded him a new effect.

• CERN released Xiao Yi’s derivation of the X field, prompting widespread attention to its possible links between the Higgs field and weak and strong interactions.
• Edward Witten and Peter Higgs praised the significance of the X field, while Witten said Xiao Yi was close to proving the Yang-Mills mass gap problem.
• At Princeton, Sarnak, Deligne, Bombieri, and Witten discussed Xiao Yi’s research and whether solving the mass gap problem could inspire new mathematical theories.
• CERN researchers reexamined collision data, prepared for another LHC run, and sought European Union funding to investigate the X field.
• The International Congress of Mathematicians organizing committee discovered that Xiao Yi had submitted the talk title “There Exists a Strictly Positive Mass Gap in Yang-Mills Theory,” leaving them convinced he had finished the proof.
• Xiao Yi’s study system recorded a high-value paper, raised his level to lv5, and granted him the extra effect Material Mastery.

Xiao Yi completed his proof of the Yang-Mills existence and mass gap problem and submitted a presentation on his new Hodge-Vertex Algebra Analytical Method to the International Congress of Mathematicians. The announcement stirred excitement among mathematicians and physicists worldwide as the conference approached in July.

• Xiao Yi finished proving the Yang-Mills mass gap problem and gained the Material Mastery effect from his upgraded study buff.
• Xiao Yi submitted his presentation topic and abstract just before the International Congress of Mathematicians deadline, then confirmed to Alexander Berezovsky that he had found a proof.
• Berezovsky arranged for Xiao Yi's presentation to be held on the first day of the conference and published its topic and abstract.
• Faltings and Schultz learned of Xiao Yi's result and examined the abstract describing his Hodge-Vertex Algebra Analytical Method.
• Mathematicians and physicists around the world reacted to the announcement, and the International Congress of Mathematicians drew applications from physicists.
• July 2022 arrived, and the International Congress of Mathematicians was about to begin.

Xiao Yi arrived in Saint Petersburg for the International Congress of Mathematicians, where he was set to present his proof of the Yang-Mills existence and mass gap problem. The report began before an audience of more than seven thousand, and Xiao Yi started laying out his proof.

• Chen Muhua, Lu Ping, and Jiang Long shared photos from their flight to Saint Petersburg, while Ye Cheng revealed that Xiao Yi had paid for the USTC students’ trip.
• Xiao Yi arrived in Saint Petersburg, was welcomed by Berezovsky and other mathematical leaders, and told the Joint Examination Group that his report would take place in three days.
• Xiao Yi sent ten thousand yuan in red packets to the group as he traveled to his hotel, while the group celebrated his arrival and upcoming report.
• On July 6, Chen Muhua, Ye Cheng, Lu Ping, and Jiang Long joined thousands of attendees at the EXPOFORUM Convention Center, where Xiao Yi arrived and greeted them.
• Berezovsky introduced Xiao Yi and described the significance of the Yang-Mills existence and mass gap problem before handing the stage to him.
• Xiao Yi opened his report by discussing Saint Petersburg’s White Nights and the scientific pursuit of truth, then displayed the claim that Yang–Mills theory had a strictly positive mass gap and began his proof.

Xiao Yi presented his proof of the Yang-Mills mass gap problem, tracing it through several failed approaches before introducing the Hodge-vertex algebra analytical method. He completed the proof, extended the method into Quantum Hodge theory, and derived a new particle he named the X particle, leaving the audience in a standing ovation.

• Faltings told Schultz that Xiao Yi had gained the calm bearing of a master, and Schultz reflected on what that state of mind meant.
• Xiao Yi described his attempts using Lattice QCD, Schwinger-Dyson equations, renormalization groups, and AdS/CFT before turning to Topological Quantum Field Theory.
• Xiao Yi derived an equivalent theorem for the mass gap problem from Yang-Mills theory on S^4 and introduced the Hodge standard conjecture as a clue to proving it.
• Xiao Yi combined Hodge decomposition, vertex algebra, and moduli spaces into the Hodge-vertex algebra analytical method, astonishing Deligne, Bombieri, Faltings, Schultz, and the other scholars.
• Xiao Yi used the method to prove the Yang-Mills existence and mass gap problem, then extended it into Quantum Hodge theory and derived a new particle he called the X particle.
• Xiao Yi ended his presentation, and the audience rose to applaud him thunderously.

Xiao Yi completed his lecture proving the Yang-Mills existence and mass gap problem, presenting methods and predictions that leading scholars said could have far-reaching effects. As the news spread worldwide, attention turned to the International Congress of Mathematicians’ opening ceremony and whether Xiao Yi would win the Fields Medal.

• Deligne, Bombieri, and Witten applauded Xiao Yi, while Witten described Quantum Hodge theory as a major advance for physics and string theory.
• Ye Cheng and his classmates celebrated Xiao Yi’s achievement and resolved to follow his example.
• Berezovsky ended the lecture, praised Xiao Yi’s contributions, and directed the audience to the banquet before the evening opening ceremony.
• At the banquet, Schultz congratulated Xiao Yi and asked whether he considered himself a master mathematician; Faltings signaled that Xiao Yi’s denial sounded like humility.
• News outlets and online platforms rapidly reported Xiao Yi’s proof, and Chinese netizens began speculating that he might win the Fields Medal that evening.

At the 2022 International Congress of Mathematicians opening ceremony, Xiao Yi received the fourth Fields Medal for his achievements across multiple areas of mathematics, including proving the Twin Prime Conjecture, the Goldbach Conjecture, and the Yang-Mills existence and mass gap problem. He returned to his seat with the medal as news spread that he had become the youngest recipient in history.

• Berezovsky found Xiao Yi at the banquet and brought him to the opening ceremony, where Schultz asked whether he expected to win the Fields Medal.
• Carlos Kenig presented the Chern Prize to Barry Mazur and the Gauss Prize to Eliot Leib before announcing the Fields Medal recipients.
• Kenig announced James Maynard, Hugo Duminil-Copin, and Hyeon-jun Huh as the first three Fields Medal recipients, leaving one award unsettled.
• Kenig praised Xiao Yi's broad mathematical achievements and announced him as the fourth recipient, prompting applause from the audience and congratulations from Schultz and other mathematicians.
• Kenig placed the Fields Medal around Xiao Yi's neck, and Xiao Yi accepted the medal, certificate, and prize money before returning to his seat.
• News spread that Xiao Yi had won the Fields Medal at nineteen, making him the youngest recipient in history.

After Xiao Yi won the Fields Medal and proved the Yang-Mills existence and mass gap problem, scholars and media debated whether he could also receive the Nobel Prize in Physics. Xiao Yi dismissed the speculation and gave CCTV an interview about his studies, the award, and his advice to scholars.

• Online discussions celebrated Xiao Yi as the first mainland mathematician to win the Fields Medal, while news outlets around the world covered his achievement.
• Terence Tao, Faltings, Deligne, and Edward Witten defended Xiao Yi's award, and Witten said his work qualified him for the Nobel Prize in Physics.
• Ye Cheng told Xiao Yi that people were calling him a Nobel Prize candidate, and Liang Qiuhao introduced himself and invited Xiao Yi to a CCTV interview.
• Xiao Yi finished his meal and joined Liang Qiuhao's interview, describing his enjoyment of difficult problems and his diligence as a student.
• Xiao Yi discussed his reaction to winning the Fields Medal, explained the academic value of his Yang-Mills proof, and urged scholars to remain curious about problems.

After Xiao Yi’s Fields Medal interview aired, Luo Mingya reconsidered her stalled research and decided she wanted to graduate. Xiao Yi continued investigating high-temperature superconductivity, preparing to return home and test his emerging theory.

• Ye Cheng shared photos of Xiao Yi’s interview with the Joint Examination Group, where his friends joked about touching his medal.
• Luo Mingya and the USTC students watched Xiao Yi’s interview, which explained that his work on the mass gap problem had grown out of his research into superconductivity.
• After hearing Xiao Yi urge people to remain curious about problems, Luo Mingya told Sun Lina that she wanted to graduate.
• Xiao Yi’s interview became a trending topic, while Ye Cheng found him working through stacks of research notes in his hotel room.
• Xiao Yi connected his analysis of quark-gluon plasma phase transitions with superconducting-insulating transitions in FeSe/SrTiO3 monolayers and planned experiments after the congress.
• Xiao Yi opened a funding application for research on high-temperature superconductivity and FeSe/SrTiO3 monolayer materials, eager to see how Material Mastery would help.

After giving his Fields Medal Lecture and returning to Shangjing, Xiao Yi met Yang Zhenning, who congratulated him and shared stories about physics. Xiao Yi then graduated from his doctoral program and accepted research appointments from both the Mathematics Research Institute and the Institute of Physics, becoming Professor Xiao.

• Xiao Yi gave his Fields Medal Lecture at the International Congress of Mathematicians, then attended its closing ceremony and flew home with Ye Cheng and the others.
• Xi Lan, Pang Cheng, and Xun Yong welcomed Xiao Yi at Shangjing International Airport and brought him to Basic Science Park.
• Pang Cheng introduced Xiao Yi to Yang Zhenning, who congratulated him on proving the Yang-Mills mass gap problem and spoke with him about life and physics.
• Xi Lan and Pang Cheng held a joint graduation ceremony for Xiao Yi, who received a Doctor of Science degree certificate from the two research institutes.
• Xi Lan and Pang Cheng invited Xiao Yi to become a researcher at their respective institutes, and Xiao Yi accepted both appointments.

After graduating, Xiao Yi returned to Fei City, where USTC’s president and senior faculty offered him a professorship and funding to establish a laboratory. Xiao Yi revealed his plans to study high-temperature superconductivity and asked the university and province to support a condensed matter-materials lab.

• Xiao Yi reviewed the costs of building his own laboratory and decided to wait for his project funding before proceeding.
• Xiao Yi returned to Fei City and reunited with his parents before USTC’s president Bao Hexin and several academicians and professors visited him.
• Bao Hexin offered Xiao Yi a university laboratory, a research team, guaranteed funding, and other incentives to return to USTC.
• Xiao Yi said he wanted a condensed matter-materials laboratory to investigate high-temperature superconductivity and develop a practical superconducting material.
• Bao Hexin agreed to discuss Xiao Yi’s request with provincial leaders, and USTC’s deans presented him with professorship appointments in mathematics and physics.

Xiao Yi accepted a professorship at USTC while preparing to investigate high-temperature superconductivity. USTC leaders approved a 500-million-yuan laboratory for him on Fei City Science Island, named Science Island Laboratory.

• USTC students celebrated Xiao Yi becoming a professor, while Jiang Long mistakenly muted him in the Joint Examination Group before unmuting him.
• Xiao Yi told Pang Cheng that he would remain a researcher at the Institute of Physics while working in Fei City, then considered how more experiments and material samples could help his superconductivity research.
• Bao Hexin defended the funding for Xiao Yi’s laboratory at a USTC meeting and revealed that Xiao Yi hoped to develop a high-temperature superconductor with industrial applications.
• Academician Li Jiangan argued that Xiao Yi’s research could benefit both quantum computing and nuclear fusion, persuading Pan Wei, Academician Chen, and the other opponents to support the funding.
• Anhui Province and national sources allocated 500 million yuan for Xiao Yi’s laboratory, which was named Science Island Laboratory after consultations with Xiao Yi.

Xiao Yi helped design the Science Island Laboratory while developing a model that linked local electron density waves and spin density waves to explain high-temperature superconductivity. He named the model XSC theory and planned to test and refine it using experiments in Luo Mingya's laboratory.

• Xiao Yi contributed design ideas to the Science Island Laboratory and helped accelerate its plans, while orders for equipment were placed and construction was scheduled to take about a year.
• Xiao Yi reviewed BCS theory and Cooper-pair formation as he investigated how local electron density fluctuations could strengthen electron interactions.
• Xiao Yi modeled iron-based superconductors using Absolute Electronic Computation and DMRG, then found that iron-vacancy-induced density fluctuations and spin density waves could produce an effective attraction near the Fermi Surface.
• Xiao Yi combined electron density waves and spin density waves in a unified model, named it XSC theory, and found that it could explain nearly all the high-temperature superconductivity phenomena he had listed.
• Xiao Yi concluded that experiments were needed to refine the coupling constant and decided to borrow Luo Mingya's laboratory.