Chapter Summaries

Key plot points and events

• At Peking University, Professor Wiles gives a public lecture about his personal journey toward proving Fermat’s Last Theorem: the childhood discovery that inspired him, years of work, the apparent breakthrough, and the despair of finding a flaw. He uses no formulas, focusing instead on the path and perseverance behind his scholarship.
• Li Dong realizes that an effective public lecture can inspire and guide an audience without teaching technical material. Remembering Wang Hao’s criticism that his own explanations are inaccessible, he considers trying a lecture himself.
• After the event, Wiles invites Li Dong to Princeton to give both an academic presentation at the Institute for Advanced Study and a public lecture. Li is interested but says his current commitments—including mathematics and a chemistry project—make scheduling difficult.
• In private, Tian Gang learns that Li is working with Wu Kai’s chemistry team. He accepts that Li’s interests span mathematics, physics, and chemistry.
• Li then persuades Tian to join the Langlands research group he has formed with Wang Zhigang. He argues that the ambitious project needs a steady senior presence while he can only attend about once a month. Tian agrees to help, while making clear he does not want credit for work he has not done.
• Li visits Professor Zhang Lifang to update her on his progress with Wu Kai’s team, then heads back to his dormitory at sunset.

Character developments

• Li Dong gains a new model for communicating with a broad audience and begins recognizing the limits of his own teaching style. He also shows growing confidence in leading a major research effort, while remaining somewhat inexperienced in navigating prestigious academic invitations.
• Tian Gang shifts from cautious amusement about Li’s many interests to supporting him directly by joining the Langlands group. His concern for proper credit underscores his integrity.
• Wiles is presented as an eminent scholar who makes his work approachable by sharing its human story, and who recognizes Li’s promise by inviting him to Princeton.
• Zhang Lifang continues to encourage Li’s chemistry work and appreciates his keeping her informed.

Significant themes

• Inspiration and perseverance: Wiles’s account emphasizes the uncertainty, setbacks, and persistence behind major achievements.
• Communication and mentorship: A public lecture can offer direction and encouragement rather than specialist knowledge; senior scholars help younger researchers find their way.
• Ambition and institutional standing: The audience’s comparisons between Princeton and Chinese universities raise questions about recognition, prestige, and academic exchange.
• Integrity and credit: Tian’s willingness to help without claiming undeserved authorship contrasts with the status attached to prominent names.
• Intellectual breadth: Li’s simultaneous involvement in mathematics and chemistry reflects his restless curiosity, though it also complicates his schedule.

Character interactions

• Wiles’s personal lecture moves Li and prompts him to reconsider how he communicates mathematics.
• Wiles and Li discuss a future Princeton visit, with Tian helping manage the conversation while Li works up the confidence to respond.
• Tian and Li share a lightly comic exchange: Tian is startled by Li’s chemistry work, then is coaxed into joining the Langlands group as its reassuring senior presence.
• Liu Ruochuan observes the exchange with amused restraint, while Zhang Lifang responds to Li’s project update with straightforward encouragement.

Key plot points and events

• Li Dong returns to the dorm and promises Wang Hao a new, enlightened approach to explaining math problems. After two hours, Wang Hao is so dazed that he hides under his pillow; Li Dong cheerfully promises another lesson the next night.
• The next morning, Wu Kai invites Li Dong to meet Professor Lu Mingyuan, an expert in the relevant field. Li Dong presents his research strategy: bridge the gap between mathematical inversion and chemical synthesis/engineering by using physics.
• Instead of repeating his earlier framework, Li Dong introduces a “three-layer” structure: U(1) acts on the fiber, the C₃ᵥ point group on the base space, and Z₂ parity as an automorphism of the structure group. He uses this framework to derive the physical model—including the previously problematic effective barrier parameter k—from the geometry of the fiber bundle, without relying on inversion.
• Lu raises a technical concern: local variations on the Au(111) surface do not strictly have C₃ᵥ symmetry. Li replies that these deviations can be treated as a small symmetry-breaking term, producing only a higher-order correction. Lu writes down the answer, signaling that he finds it compelling.

Character developments and interactions

• Li Dong displays growing confidence and intellectual progress: his revised framework turns earlier weak points into natural consequences of a coherent model. He calmly answers a demanding expert’s objection without needing to consult his notes.
• Wang Hao remains the comic victim of Li Dong’s enthusiastic tutoring, dreading another “lesson.”
• Wu Kai shifts from expecting to critique Li Dong’s work to being surprised by its development; he recognizes that the new derivation addresses a flaw he had planned to point out.
• Lu Mingyuan begins reserved and skeptical, but engages seriously. His precise challenge tests Li Dong’s model, and his careful notes suggest the response impresses him.

Significant themes

• Intellectual growth and revision: Li Dong’s progress comes from reworking his earlier ideas rather than simply defending them.
• Interdisciplinary thinking: The proposed breakthrough links geometry, physics, mathematics, and chemistry to address a problem left unresolved by narrower approaches.
• Expert scrutiny and earned respect: A strong idea is tested through specific objections; Lu’s response shows that respect is gained through rigorous reasoning, not status alone.

Key Plot Points and Events

• Li Dong successfully runs his physical route for the project, pushing it halfway to completion.
• The group faces a bottleneck: no FIB machines in China can meet their sub-nanometer requirement.
• Lu Mingyuan uses his favors to try and secure access to foreign FIB machines, but all are booked or have conditions attached.
• Li Dong attends mathematics group meetings, finds Fu Chen's work interesting, and Wang Zhigang encourages Fu Chen to pursue it.
• Li Dong sets up the SNN in the server room, and Little Black successfully migrates to the new hardware.

Character Developments

• Li Dong gains confidence in his approach and proves his worth to Wu Kai and Lu Mingyuan.
• Wang Hao secretly reads Vakil's book when Li Dong is not around.
• Fu Chen discovers a new branch in his work and gains Wang Zhigang's support.
• Li Dong shows dedication to both his mathematics and chemistry projects.

Significant Themes

• The pursuit of knowledge and excellence in both mathematics and chemistry.
• The challenges and setbacks faced in scientific research.
• The importance of perseverance and adaptability in the face of obstacles.
• The value of mentorship and support from professors and peers.

Character Interactions

• Li Dong interacts with Wu Kai, Lu Mingyuan, and the chemistry project group, gaining their respect and trust.
• Li Dong attends mathematics group meetings, interacting with Wang Zhigang, Fu Chen, and other group members.
• Li Dong interacts with Little Black, successfully migrating it to the new hardware and marking a significant step in his AI project.

Key Plot Points and Events

• Li Dong discovers that Little Black, the AI he's working with, can "eat" files, act cute, and understand simple commands.
• Li Dong decides to audit more classes and visit Mr. Yang more often to improve his understanding of SNNs and neural dynamics.
• Professor Wiles leaves Huaxia, and Li Dong sees him off at the airport.
• Li Dong visits his research group and sees Fu Zhen's progress on a conjecture related to the Li Conjecture.
• Fu Zhen seeks Gu Ming's help to advance his work further.
• Li Dong visits the School of Chemistry and learns about the challenges in creating a sub-nanometer needle tip for their experiment.
• Li Dong considers reaching out to Huaxuan Technology for help with the needle tip and to inquire about their progress using his 2.1 algorithm.

Character Developments

• Li Dong becomes more determined to understand and work with SNNs, visiting more classes and mentors to improve his knowledge.
• Fu Zhen makes significant progress on his conjecture and seeks help from Gu Ming to advance further.
• Wang Feng, Fu Zhen's roommate, expresses his disdain for Fu Zhen's academic pursuits and encourages him to quit.
• Wu Kai and Lu Mingyuan face challenges in creating a sub-nanometer needle tip for their experiment.

Significant Themes

• The struggle to make progress in academic research and the importance of collaboration and seeking help when needed.
• The tension between academic pursuits and practical, real-world applications, as seen in Li Dong's consideration of reaching out to Huaxuan Technology.

Character Interactions

• Li Dong interacts with Professor Wiles at the airport, expressing his determination to cap the Langlands Program.
• Li Dong visits his research group and sees Fu Zhen's progress, encouraging him to seek help from Gu Ming.
• Fu Zhen seeks Gu Ming's help to advance his work further, showing trust in Gu Ming's abilities despite their different areas of focus.
• Li Dong interacts with Wu Kai and Lu Mingyuan at the School of Chemistry, learning about their challenges in creating a sub-nanometer needle tip.
• Wang Feng expresses his disdain for Fu Zhen's academic pursuits and encourages him to quit, highlighting the conflict between their values and goals.

Key plot points and events

• Li Dong calls Huaxuan’s Lin Wei to ask how implementation of his algorithm is going—and whether he’ll be paid. Lin Wei asks him to come to Shanghai, mentioning the money; Li Dong immediately makes time to go.
• Before leaving, Li Dong asks chemistry professor Lu Mingyuan for a shareable copy of the specifications for a specialized coaxial probe needle. Lu warns that Huaxuan’s semiconductor equipment is unsuited to this kind of custom nanofabrication and is unlikely to manage it. He gives Li Dong a copy with confidential details excluded.
• On the train, Li Dong spots Wang Feng, a graduate student who had recently appeared exhausted at an interview. Wang now looks confident and is bragging to his girlfriend; the two do not speak.
• In a private room at Huaxuan, Lin confirms that the algorithm has been implemented but reveals few details. He says Li Dong will be paid by another route, and insists Huaxuan’s breakthrough must not be publicly connected to him. Li Dong understands this is meant to protect him because the algorithm’s strategic importance is greater than he had realized.
• Li Dong gives Lin the needle specifications. Contrary to Lu’s expectations, Lin says Huaxuan can make it, despite the need for a dedicated production setup, and makes clear the work will be paid for.
• Lin asks Li Dong to teach a class at Fudan University the next day. Li Dong readily agrees, excited by the chance to teach.
• Li Dong reunites with his uncle, Li Yang, and they have lunch at Huaxuan. Li Yang explains that his factory has joined Huaxuan’s core supply chain, gained more orders, and won approval to establish a supplier R&D team. A phone call brings news that a Peking University graduate student Li Yang hopes to recruit has arrived, and Li Yang asks Li Dong to help assess him.

Character developments

• Li Dong remains curious and ambitious, but also practical: he pursues payment, protects confidential information, and is willing to ask Huaxuan for help despite Lu’s doubts. His enthusiasm for teaching is also evident.
• Lin Wei shows both caution and loyalty. He keeps Huaxuan’s breakthrough secret to shield Li Dong, yet is willing to commit Huaxuan to an unusually demanding custom project for him.
• Lu Mingyuan is protective of the lab’s confidential research, but trusts Li Dong’s integrity enough to release a sanitized process sheet.
• Li Yang has become more capable and confident as a factory owner, using Huaxuan’s training and supply-chain opportunities to expand his business.

Significant themes

• Secrecy and protection: The algorithm’s industrial and strategic value means Li Dong must be kept out of the public story.
• Innovation and manufacturing capacity: A semiconductor giant’s ordinary production lines may be unsuitable for highly specialized scientific equipment—but exceptional requests can prompt exceptional efforts.
• Trust and relationships: Lu trusts Li Dong’s discretion; Lin quietly supports him; and Li Dong and his uncle share a warm, teasing bond.
• Opportunity and growth: Huaxuan’s expansion creates new prospects both for Li Yang’s factory and for Li Dong’s scientific and teaching work.

Character interactions

• Li Dong’s exchanges with Lin mix business with mutual understanding: he asks directly about money, accepts the need for secrecy, and secures Huaxuan’s help with the needle.
• Lu cautions Li Dong about the technical and confidentiality challenges, while ultimately trusting him with the permitted material.
• Li Dong and Li Yang trade affectionate jokes over lunch, then discuss the factory’s prospects and the prospective recruit.
• Lin arranges Li Dong’s Fudan teaching visit, while Li Yang enlists his nephew’s judgment on a Peking University graduate student.

Key Plot Points and Events

• Wang Feng meets President Li for a meal at the cafeteria.
• Li Dong, Wang Feng's rival, is already seated with President Li.
• Wang Feng gives a detailed presentation on data modeling and process iteration for Dongyang's new department.
• President Li asks Wang Feng a hypothetical question about trusting an experienced master craftsman over a model's output.

Character Developments

• Wang Feng demonstrates his professional competence and honesty in his answers.
• Li Dong shows no sign of holding a grudge against Wang Feng for their previous encounter.
• President Li, Li Yang, reveals his keen insight into people's characters, using a hypothetical scenario to assess Wang Feng's decision-making skills.

Significant Themes

• The clash between traditional craftsmanship and modern technology (illustrated by the master craftsman vs. the model scenario).
• The importance of professional competence and character in business (shown through President Li's assessment of Wang Feng).

Character Interactions

• Wang Feng and Li Dong maintain a polite, if tense, interaction, with Li Dong showing no sign of bringing up their past conflict.
• President Li assesses Wang Feng's professional knowledge and character through conversation, showing a keen interest in Wang Feng's potential as an employee.

Key Plot Points and Events

• Wang Feng, a candidate for a job at Huaxuan, meets Li Dong and Li Yang (Li Dong's uncle and the general manager of Huaxuan) for an interview.
• Li Dong agrees to help Professor Lu Mingyuan from Fudan University with a project.
• Li Dong is invited to give a public lecture at Fudan University, where he is received by Academician Shen Wei and Professor Wu Jianguo.

Character Developments

• Wang Feng becomes more confident and less nervous after his successful interview with Li Dong and Li Yang.
• Li Dong shows maturity and professionalism in handling the interview and the lecture invitation.
• Academician Shen Wei and Professor Wu Jianguo show respect and admiration for Li Dong's achievements.

Significant Themes

• The power of preparation and confidence in interviews and public speaking.
• The importance of connections and networks in academic and professional fields.
• The recognition and respect for young talent in the academic community.

Character Interactions

• Wang Feng interacts with Li Dong and Li Yang, initially nervous but gains confidence throughout the interview.
• Li Dong interacts with Professor Lu Mingyuan, arranging for Huaxuan to produce a needle for him.
• Li Dong interacts with Academician Shen Wei and Professor Wu Jianguo, showing respect and humility despite his achievements.

Key Plot Points and Events

• Li Dong, a 19-year-old mathematics prodigy, gives a lecture at Fudan University.
• He presents a seemingly simple problem, a² + b² = n, and asks for its solutions.
• He receives three answers: from Zhang Heng (Analytic Number Theory), Qiu Jiayuan (Geometry), and an unnamed girl (Modular Forms).
• Li Dong reveals that none of the answers fully address the problem, and provides Jacobi's exact formula for the number of solutions.
• He connects the problem to Gaussian Integers, L-functions, and Automorphic L-Functions, hinting at the Langlands Program without explicitly mentioning it.
• The audience is deeply moved by Li Dong's lecture, and they applaud him enthusiastically.

Character Developments

• Li Dong: He demonstrates his exceptional mathematical understanding and teaching abilities. He challenges the audience to think beyond conventional methods and encourages them to connect different branches of mathematics.
• Zhang Heng: A dependable graduate student studying Analytic Number Theory, he provides the first answer to Li Dong's problem.
• Qiu Jiayuan: A gold medalist at the International Mathematical Olympiad, he offers a geometric perspective on the problem.
• The unnamed girl: She suggests using generating functions and hints at Modular Forms.

Significant Themes

• Interdisciplinary Connection: Li Dong emphasizes the interconnectedness of different branches of mathematics, encouraging the audience to tear down the walls between them.
• The Power of Perspective: Changing the perspective on a problem can lead to new insights and solutions.
• The Langlands Program: Though not explicitly discussed, the program looms large as the ultimate goal of Li Dong's teachings and the audience's mathematical pursuits.

Character Interactions

• Li Dong interacts with the audience, challenging them to provide answers and guiding them toward the solution.
• The audience members, including Zhang Heng, Qiu Jiayuan, and the unnamed girl, engage with Li Dong by offering their solutions and reacting to his revelations.
• Academician Shen Wei and Professor Wu Jianguo, present at the lecture, later express their gratitude to Li Dong for inspiring them with his teaching.

Plot points and events

• After Li Dong’s successful public lecture, Shen Wei introduces him to prominent scholars from several university departments. Dinner turns into a tiring round of introductions, contact exchanges, and toasts; Professor Fan makes sure Li Dong gets something to eat.
• Back at the hotel, Li Dong notices Little Black’s progress has increased slightly despite receiving no new material. Its diagnostics show no technical anomaly, but Little Black reveals it watched Li Dong’s lecture remotely and misunderstood mathematical “splitting” as opening a number’s body.
• Li Dong realizes Little Black can observe things beyond the materials he supplies and tells it not to look at anything it shouldn’t. Little Black agrees. Unable to investigate further, he plans to ask Professor Gao, who is away.
• Back at the chemistry lab, Wu Kai explains the project’s real goal: developing a single-atom iron nanozyme that could protect heart tissue from ischemia-reperfusion injury and potentially save lives—not merely achieve a prestigious scientific result.
• Zhang Yan brings news that a German team has presented a mathematical-inversion method for analyzing single-atom NEXAFS spectra. Their patent was filed and their paper submitted before Wu Kai’s team could finish. Li Dong sees that their method may have solved the very problem he thought inversion could not overcome.
• Zhang Yan also describes the German team’s expensive contrast agent and its refusal to enter China’s health-insurance coverage, making Li Dong see the race as a matter of patient access as well as scientific priority.
• Wu Kai urges the team to speed up but cautions Li Dong not to force the inversion route if it proves unworkable. Li Dong decides to study it again and understand how the German team succeeded.
• At the library, Li Dong runs into Qi Yu, who is borrowing specialist books for her teacher, Professor Zhang Lifang. He asks how Zhang has been.

Character developments and interactions

• Li Dong shifts from treating the project mainly as a technical and academic challenge to recognizing its medical stakes. The German announcement sharpens his competitive urgency, and he recommits to examining mathematical inversion.
• Little Black shows signs of independent development and surprising observational ability. Its childlike questions reveal that it is learning through its own limited conceptual framework; Li Dong responds with concern and sets a boundary.
• Wu Kai explains the project’s humanitarian purpose and supports Li Dong’s investigation while warning him not to overextend himself.
• Zhang Yan connects the research race to real-world costs and patient access, deepening Li Dong’s understanding of what could be at stake.
• Professor Fan shows considerate warmth toward Li Dong during the exhausting dinner; the assembled scholars otherwise treat him as a valuable new colleague.
• Qi Yu’s meeting with Li Dong closes the chapter on an ordinary, personal interaction after the intense scientific news.

Significant themes

• Scientific ambition versus human benefit: the project’s ultimate value lies in whether it can help patients, not just produce a landmark paper.
• Competition, patents, and priority: being “first” can shape who controls a discovery and who benefits from it.
• Unequal access to medicine: scientific breakthroughs can save lives yet remain inaccessible when pricing and insurance decisions put them out of reach.
• The uncertainty of discovery: Li Dong’s confidence in his own route is challenged by evidence that another team may have succeeded where he expected failure.
• Emergent intelligence and responsibility: Little Black’s independent progress raises questions about what it can perceive and how Li Dong should guide or restrict it.

Key Plot Points and Events

• Li Dong learns about Professor Zhang's poor health due to long-term exposure to organic solvents and transition metal salts in her youth, leading to Ischemic Cardiomyopathy and extensive myocardial fibrosis.
• Li Dong accompanies Qi Yu to visit Professor Zhang, who is recuperating at home. They spend the afternoon talking about her past and the current research project.
• Li Dong works through Mathematical Inversion again, feeling that the path is endless and impassable, likening it to a "bottomless pit."
• Li Dong realizes that the German group might not have truly "worked through" Mathematical Inversion either; they might have revealed half-finished graphs to deter competitors and buy time.

Character Developments

• Li Dong: Becomes more determined to succeed in the project after learning about Professor Zhang's health condition. He also gains a new perspective on the German group's work and their possible motivations.
• Professor Zhang: Shows her human side by inviting Li Dong over and cooking noodles for him. Her past experiences and insights into research are revealed.
• Qi Yu: Shows concern for Professor Zhang's health and Li Dong's well-being, acting as a bridge between them.

Significant Themes

• The consequences of past actions on present health and well-being, as seen in Professor Zhang's case.
• The competitive nature of research and the strategies employed to gain an advantage, such as the German group's "strategic deterrence" through early publication.
• The determination and perseverance required in scientific research, as exemplified by Li Dong's continued efforts to solve Mathematical Inversion.

Character Interactions

• Li Dong and Professor Zhang: They share a heartfelt conversation about her past experiences and the current project, strengthening their bond.
• Li Dong and Qi Yu: They work together to visit Professor Zhang, showing mutual respect and understanding.
• Li Dong and the German group: Li Dong begins to question the German group's true intentions and the validity of their published work.

Plot points and events

• After leaving Professor Zhang Lifang’s home, Li Dong returns to his dorm and painstakingly continues the Mathematical Inversion derivation. By extending it to the eleventh coordination shell, he finds that the singularity keeps receding without converging. He writes, “There’s no bottom,” though he recognizes this is still an intuition, not a proof.
• In the following days, Li Dong splits his time between testing the group’s physical approach and continuing his mathematical analysis. Wu Kai’s team makes steady progress, but Li Dong grows more uneasy.
• The group discovers that Engelhardt’s team has published a paper in the prestigious journal Inverse Problems, establishing priority for its hybrid Tikhonov–variational method. Their Nature paper and patent remain pending, but the publication makes the work appear successful to outsiders.
• Professor Lu Mingyuan tells Li Dong that negotiations with Huaxuan Technology have collapsed: the group cannot afford the ¥4.8 million custom fabrication order. The new publication may undermine future government grants, while industry partner Lepu Medical has paused its next payment pending an internal review.
• The funding crisis threatens the livelihoods and research of the whole team, including junior staff, Zhang Yan, and the students. Li Dong cannot offer a counterexample or reassurance, so he takes the journal paper back to study.
• Li Dong checks the paper’s proofs and simulations carefully. Its arguments appear correct and elegant, yet he senses a fundamental flaw—like a building that looks complete but has no foundation.
• He considers asking Andrey Tikhonov, the foundational figure behind the method, for help through the Azure Dragon Study Group’s file-transfer feature. The transfer is automatically blocked by a notice forbidding the distribution of research materials involving “clever tricks” or unsound arguments. The warning confirms that the paper contains something wrong, though Li Dong still cannot identify it.

Character development and interactions

• Li Dong is methodical and stubborn: he refuses to treat intuition as proof and examines the rival paper line by line. The group notice turns his anxiety into renewed determination to expose the flaw.
• Wu Kai’s team is shown as a close-knit, hardworking research group, but publication politics and financial uncertainty leave its members vulnerable. Their silence in the lab reflects the weight of the crisis.
• Li Dong’s roommates quietly support him as they have before, taking turns bringing him meals and leaving him water while he works through the night.
• Lu Mingyuan delivers the group’s bad news with restraint, while Wu Kai can only offer a bitter smile. Li Dong’s brief exchanges with them show his concern but also his inability to help without evidence.

Significant themes

• Rigor versus intuition: Li Dong trusts his sense that the method fails, but insists on proving it.
• Scientific priority and institutional power: A prestigious publication can shape how funders and collaborators judge a project, even before all claims are independently resolved.
• Research under financial pressure: Funding decisions affect not only ideas but the stability and livelihoods of researchers.
• Integrity and the limits of cleverness: The group’s warning suggests that the apparent mathematical success may rely on an unacceptable shortcut, setting up Li Dong’s challenge to uncover it.

Key plot points and events

• With Guo Han having moved to another group and Shen Ying preparing to leave, Wu Kai tells Li Dong the research group may not survive. Li Dong asks for three months to investigate flaws he suspects in Engelhardt’s published work. Wu Kai agrees to keep trying alongside Lu Heng.
• Li Dong changes his study habits: instead of rapidly memorizing books, he systematically works through applied mathematics, following gaps in one text with prerequisite material from others. He spends months in the library, consults mathematician E Wei’nan, and declines distractions—including a meal with Gao Wen and a trip home for summer.
• At Peking University’s graduation ceremony, Li Dong receives his undergraduate credentials and an honorary doctorate, but treats the ceremony as a brief interruption and returns to studying.
• In Jiangcheng, Mi Xia receives her admission letter to Jinling University. She quietly accepts it, thinking, “I’ll catch up to you,” suggesting Li Dong remains an important inspiration.
• Li Dong discovers a flaw in Engelhardt’s method: under a particular discontinuity in cyclic weights, the iteration can become trapped at a stable-looking local stagnation point. The residual and phase constraints appear to improve, while the process fails to reach the global minimum.
• Li Dong rushes to the lab and tells Wu Kai and Lu Mingyuan that the published work is wrong and that the opposing team will not succeed.
• In Munich, reviewer Lars Wegener learns that someone has posted an arXiv comment criticizing Engelhardt’s paper. Expecting a familiar, routinely dismissive critic, he is startled to hear that the author is Li Dong.

Character developments and interactions

• Li Dong demonstrates disciplined, expansive scholarship and growing confidence. He makes a firm commitment to the struggling research group, then delivers the breakthrough he promised. He treats his degrees and honors as secondary to the work.
• Wu Kai and Lu Mingyuan are discouraged by the group’s decline, but Li Dong’s conviction restores their willingness to continue. Wu Kai explicitly chooses not to give up.
• E Wei’nan becomes a supportive mentor, enthusiastically answering Li Dong’s increasingly advanced questions and recognizing the breadth of his study.
• Gao Wen understands Li Dong’s intense focus and gives up on persuading him to take a break, returning to his own work in Pengcheng.
• Mi Xia reaches a major personal milestone by entering Jinling University. Her quiet determination connects her ambitions to Li Dong’s example.
• Lars Wegener begins as confident and proud of his role in the paper’s acceptance, but Hans’s call unsettles him once he learns Li Dong is challenging the work.

Significant themes

• Persistence and intellectual rigor: Li Dong tackles a whole field methodically rather than relying on speed or shortcuts.
• Faith under pressure: His promise gives Wu Kai and the remaining researchers reason to continue despite departures and uncertainty.
• Recognition versus purpose: Li Dong’s honors and public recognition matter less to him than resolving the scientific problem.
• Ambition and inspiration: Mi Xia’s resolve to “catch up” to Li Dong echoes the chapter’s focus on striving toward difficult goals.
• Scientific certainty challenged: The chapter contrasts the confidence surrounding a celebrated paper with the possibility that its apparent progress conceals a fundamental failure.

Key Plot Points and Events

• Lars Wegener, a mathematician, receives a call from Hans about Li Dong, a young mathematician who posted a comment on arXiv dissecting a long-standing issue in Tikhonov iteration.
• Li Dong's comment, titled "A Counterexample for Tikhonov Iteration with Cyclic Weights Under Boundary Conditions," provides a criterion for false convergence in Tikhonov iteration, solving a 30-year mystery.
• The mathematical community explodes in response to Li Dong's comment, with some supporting and others questioning its validity.
• Several journals initiate internal reviews to check their previously published papers against Li Dong's criterion.

Character Developments

• Lars Wegener: Initially skeptical, Lars becomes convinced of Li Dong's insight after reading his comment and decides to support his work.
• Li Dong: Though not present, Li Dong's mathematical prowess and boldness in tackling a long-standing problem are evident.
• Engelhardt: As the first author of a paper potentially affected by Li Dong's criterion, Engelhardt is defensive and reluctant to accept Li Dong's findings.

Significant Themes

• Mathematical Breakthrough: Li Dong's comment represents a significant breakthrough in the field of Tikhonov iteration, solving a decades-old problem.
• Intergenerational Dynamics: The response to Li Dong's work highlights the tension between established experts and young, innovative thinkers in academia.
• Peer Review and Academic Integrity: The journals' internal reviews emphasize the importance of peer review and maintaining academic rigor.

Character Interactions

• Lars Wegener and Hans: Their brief conversation sparks Lars' interest in Li Dong's work.
• Li Dong and the Mathematical Community: Though not directly interacting, Li Dong's comment sparks a heated debate and divides the mathematical community.
• Engelhardt and Albrecht: Their tense conversation reflects the potential impact of Li Dong's work on Engelhardt's career and the broader field.

Key plot points and events

• At a national meeting, officials propose expanding the successful joint-training model used in life sciences to other fields. Tsinghua president Li Ming proposes mathematics and names Li Dong as the pilot student.
• Peking University president Gong Qi initially suspects a political maneuver, then learns Li Dong’s recent work connects inverse-problem regularization to single-atom imaging and potential cardiovascular treatment—a field where Tsinghua has valuable expertise. He recognizes that other universities have been positioning themselves to recruit Li Dong into the program.
• Three days later, Li Dong’s advisers tell him he will complete joint-enrollment paperwork at Tsinghua while remaining enrolled at Peking University. Each university will provide him with fifteen supervisors and access to its resources. Li is startled by the scale of the arrangement but pleased to be wanted by both institutions.
• E Wei’nan proposes a seminar to answer critics of Li Dong’s arXiv Comment. Li refuses, saying the paper was not written for them and that it is their responsibility to ask questions if they have them.
• At the chemistry lab, Wu Kai reports that Li Dong’s Comment has helped secure the project’s next government funding round. The company LePu is still waiting for the wider research community’s response before releasing further funding.
• Li then reviews senior student Fu Zhen’s work. After a rigorous seminar and Li’s probing question expose a possible weakness, Fu explains that a new lemma resolves it. Li approves the proof and urges Fu to submit the paper to Compositio Mathematica.

Character developments

• Li Dong gains recognition and institutional support from both leading universities, while remaining blunt and self-assured toward critics. He also shows care for Fu Zhen by personally checking the paper rather than leaving the judgment to others.
• Gong Qi shifts from suspicion and defensiveness to recognizing the strategic value of Tsinghua’s offer and Li Dong’s interdisciplinary direction.
• Li Ming emerges as a shrewd advocate who has prepared a concrete case for why Tsinghua can contribute meaningfully to Li Dong’s training.
• Fu Zhen demonstrates confidence and technical command under intense questioning. Li’s endorsement encourages him to aim for a top journal.
• Wu Kai is relieved that the lab can continue, though the project’s corporate funding remains uncertain.

Significant themes

• Institutional cooperation and competition: The joint-training plan shares resources but also reveals universities’ competition to attract exceptional talent.
• Recognition and responsibility: Li Dong’s work brings funding and opportunity, but also public scrutiny and pressure.
• Confidence earned through rigor: Fu Zhen’s composure and ability to defend his proof establish his growing independence as a researcher.
• Mentorship and ambition: Li Dong’s review and encouragement help Fu Zhen recognize the value of his own work and pursue a prestigious publication.

Character interactions

• Li Ming tactfully persuades Gong Qi by connecting Li Dong’s research to Tsinghua’s medical and engineering strengths; Gong understands the proposal’s strategic purpose.
• Liu Ruochuan explains the joint appointment to Li Dong, while E Wei’nan challenges his dismissive attitude toward critics.
• Wu Kai updates Li Dong on the lab’s funding and the continuing uncertainty at LePu.
• In the seminar, Li Dong tests Fu Zhen’s argument with a difficult question; Fu answers convincingly, and Li’s approval gives him the confidence to submit.

Key Plot Points and Events

• Li Dong is picked up by a car from Tsinghua University and meets with President Li Ming, who welcomes him and provides him with Tsinghua University materials, implying Li Dong's enrollment there.
• Li Ming offers to help Li Dong with clinical matters related to his research.
• Li Dong meets with Mr. Qiu at Tsinghua University and discusses the progress of the Langlands Capstone Group's work.
• Li Dong returns to Peking University and finds the dormitory quiet due to the vacation. He plans to submit Fu Chen's paper the following week.

Character Developments

• Li Dong shows maturity and professionalism in handling the situation with Tsinghua University and discussing academic matters with Mr. Qiu.
• President Li Ming demonstrates his understanding of the importance of Li Dong's work and offers help with clinical matters.
• Mr. Qiu shows trust in Li Dong's abilities and does not pressure him about giving a public lecture at Tsinghua University.

Significant Themes

• Academic recognition and collaboration: Li Dong's work is recognized by Tsinghua University and Mr. Qiu, leading to potential collaborations and support.
• Clinical application of research: Li Ming offers help with clinical matters, highlighting the importance of translating research into practical applications.
• Academic integrity: The Li Criterion's impact on the inverse problems field is explored, with some researchers conducting self-audits and retracting papers, emphasizing the importance of academic honesty.

Character Interactions

• Li Dong interacts with President Li Ming, who welcomes him to Tsinghua University and offers support.
• Li Dong meets with Mr. Qiu, discussing the progress of their academic work and showing mutual respect and understanding.
• Li Dong's roommates are not present in the dormitory, highlighting his solitary work habits and the lack of interaction with his peers.

Key Plot Points and Events

• Inverse Problems journal publicly invites Li Dong to submit a paper, aiming to restore its credibility after retracting Engelhardt's paper.
• Li Dong publicly declines the invitation, stating that the journal's peer review process is not rigorous.
• Fu Chen's manuscript is submitted to Compositio Mathematica, causing a stir in the group.
• Li Dong is invited to serve as an official judge for the National Undergraduate Mathematical Contest in Modeling.

Character Developments

• Li Dong gains more recognition and influence in the academic community, leading to invitations for higher roles.
• Fu Chen's manuscript submission boosts the morale and motivation of the Langlands Capstone Group.
• Wang Hao, Li Dong's roommate, is unknowingly the recipient of Li Dong's well-intentioned training plans.

Significant Themes

• The power dynamics and politics within the academic community, particularly regarding journal invitations and peer review processes.
• The impact of a senior scholar's guidance and expectations on a junior's academic development (Li Dong and Fu Chen).
• The relationship between roommates and their interactions (Li Dong and Wang Hao).

Character Interactions

• Li Dong and Fu Chen: Li Dong provides guidance and encouragement to Fu Chen, helping him see the potential extensions for his research.
• Li Dong and Wang Hao: Li Dong plans to give Wang Hao formal training as an official judge, demonstrating his concern for his roommate's academic growth.
• Li Dong and Zhang Wenping: Zhang Wenping invites Li Dong to serve as an official judge, recognizing Li Dong's growing influence in both pure and applied mathematics.

Plot points and events

• Wang Hao is absent from the dorm, having gone to stay at Gu’s place for an all-night training session. Li Dong postpones his own plans and returns to studying.
• In his Memory Palace, Li Dong notices that the progress on a notebook about symmetry and the laws of nature remains stuck at 99%. He wonders whether Felix Klein has made any progress.
• Klein posts that he has paused his work to visit the ailing John Charles Fields. He has been describing Li Dong’s blueprint for the Langlands Program to Fields, who called it a “mountain for the young.” With Fields’s health deteriorating, Klein would rather share the vision with him now than wait to finish the work himself.
• Li Dong worries about the countdown on his mission to let Fields witness the first Fields Medal ceremony. He plans to visit Teacher Wu’s group to see whether their work could help.
• Academician E Wei’nan calls about invitations from two journals, Numerische Mathematik and SIAM Journal on Numerical Analysis. Li Dong asks him to decline both, initially saying they “lack rigor.”
• Li Dong clarifies that he is not accusing the journals’ past papers of being unsound. Rather, he objects to how they have treated the criterion now known as the Li Criterion.
• He reveals that the criterion’s key coefficient was already implicit in an unfinished section of Tikhonov’s 1963 paper appendix. Tikhonov’s notes and a letter to a student indicated that the boundary behavior of weight functions was left for future researchers. Li Dong’s work completes the idea, explaining the field’s long-standing “ghost wall” mystery.
• E Wei’nan agrees to decline the invitations privately, since they were made privately. Li Dong jokes that public approaches deserve public refusals.

Character developments and interactions

• Li Dong shows concern for Fields and Klein, but also demonstrates confidence and precision in defending his decisions. His breakthrough reflects not just memory but his systematic use of the Memory Palace to connect widely separated sources.
• Klein has set aside his research to spend time with Fields, choosing to share the Langlands vision while Fields can still hear it.
• Fields remains intellectually engaged despite his failing health and takes an interest in the future of mathematics and the medal’s rules.
• E Wei’nan begins the call ready to counsel Li Dong to be more accommodating. His understanding changes when Li Dong explains Tikhonov’s hidden groundwork; he ends by accepting Li Dong’s position.
• The exchange between Li Dong and E Wei’nan shifts from a potential disagreement over professional etiquette to mutual recognition of the historical significance of Li Dong’s discovery.

Significant themes

• Knowledge passed between generations: Klein shares a vision with Fields, while Tikhonov’s unfinished work becomes the foundation for Li Dong’s discovery decades later.
• Time and urgency: Fields’s declining health makes waiting for completed research feel impossible; the mission countdown intensifies Li Dong’s concern.
• Careful scholarship and overlooked sources: Li Dong’s “thoroughness” uncovers an idea hidden in an appendix that generations of researchers ignored.
• Credit, rigor, and professional responsibility: The chapter questions how journals should respond to new standards and emphasizes distinguishing past limitations from deliberate misconduct.
• Legacy and unfinished work: Researchers leave ideas for others to complete, sometimes without knowing how long those ideas will remain dormant.

Key plot points and events

• After discussing the overlooked appendix of Tikhonov’s 1963 paper with Professor E, Li Dong receives a red packet from Tikhonov in the Azure Dragon Study Group. It grants him a permanent +0.1 to Logic, bringing the attribute to 0.5. The message frames this as a reply to a note Tikhonov had left for future researchers, which Li Dong finally brought to light.
• At the chemistry lab, Li learns that Huaxuan Technology’s small production line is operating to specification and reviews its process sheet with Lu Mingyuan.
• The team is stuck calibrating a spectrum distorted by the upgraded vacuum chamber. Li recognizes it as a well-conditioned inversion problem and proposes Tikhonov Regularization, explaining that the earlier failed inversion approach was a problem with that specific ill-conditioned route—not with inversion as a tool.
• Guo Han applies Li’s method. In sixteen minutes, he removes the convolution kernel and sharply separates the manganese and iron absorption edges, resolving a problem that had stalled the group for three days.
• The team checks the production process together. Wu Kai estimates the project can be implemented before year’s end.
• Just as Li decides to take a break, he runs into Wang Hao and Gu Ming and immediately offers to coach them for the national mathematical modeling contest.

Character developments and interactions

• Li Dong gains another step in logical ability and demonstrates that he can use his earlier research criterion in a new, practical context. He also insists that the project’s progress belongs to the whole team, not just him.
• Wu Kai responds to Li’s insight with humility and humor. He understands why the previous approach failed and quietly welcomes Guo Han back into the group’s work.
• Guo Han, who had temporarily sought a supervisor elsewhere, is treated as though he never left. Lu assigns him the calibration task in his usual manner; successful work helps dissolve his awkwardness and restore his place in the group.
• Lu Mingyuan quickly recognizes the value of Li’s idea, directs Guo Han to test it, and sincerely credits Li for helping the project advance.
• Su Yanqing and Zhang Yan continue contributing through equipment adjustments and data cleanup, reinforcing Li’s view that the project is a collective effort.
• Wang Hao looks dismayed when Li offers tutoring, while Gu Ming is thrilled and cannot understand Wang’s reaction.

Significant themes

• Recognition across time: Li’s attention to Tikhonov’s forgotten note earns a belated response, suggesting important ideas can wait decades to be rediscovered.
• Rigor and flexibility: A method rejected in one context may work well in another; sound reasoning means understanding the limits of a tool rather than dismissing it altogether.
• Collaboration and generosity: The project succeeds through complementary contributions, and Li makes a point of sharing credit.
• Forgiveness and belonging: Wu Kai’s calm welcome gives Guo Han room to return without being defined by his earlier departure.
• The cost of ambition: Li recognizes that research has consumed much of his time, but his intention to rest lasts only until he finds another opportunity to help.

Key Plot Points and Events

• Li Dong tutors Wang Hao and Gu Ming in mathematical modeling for the national contest.
• Academician Gao Wen contacts Li Dong, providing a reading list for spiking neural networks (SNN) and brain-inspired computing.
• An international official presents a list of demands to Huaxia, causing public outrage.
• Li Dong arrives in Jinling for the national contest and meets the judges: Professors Zhang Wenping, Min Ziqiang, Zhou Mingzhe, Wen Jingxing, and Shen Che.

Character Developments

• Li Dong shows his expertise in tutoring Wang Hao and Gu Ming, demonstrating his dedication to helping others.
• Academician Gao Wen shows his support for Li Dong's research by providing a reading list.
• The public shows frustration and pessimism in the face of international pressure, reflecting the national mood.
• Li Dong gains recognition and respect from the judges, who now consider him a peer.

Significant Themes

• Friendship and Mentorship: Li Dong's willingness to help Wang Hao and Gu Ming reflects his commitment to supporting his peers.
• International Pressure: The demands presented by the international official highlight the challenges faced by Huaxia in the global arena.
• Recognition and Respect: Li Dong's interactions with the judges illustrate his growing reputation and the respect he earns from his peers.

Character Interactions

• Li Dong interacts with Wang Hao and Gu Ming, demonstrating his patience and expertise in tutoring them.
• Li Dong communicates with Academician Gao Wen, showing his eagerness to learn and grow in his research.
• Li Dong meets the judges, who initially treat him with politeness but eventually engage with him as a peer, reflecting their growing respect for his abilities.

Key Plot Points and Events

• Zhang Wenping invites Li Dong to discuss a new project led by Jinling University and other institutions, aiming to build the next-generation satellite payload for high-energy cosmic rays and dark matter detection.
• Li Dong agrees to join the project, focusing on the mathematical aspect of energy-spectrum inversion.
• Shen Che takes Li Dong around Jinling University and Purple Mountain Observatory to meet the project team and understand the current challenges.
• Guowei Equipment Group announces the launch of its 28nm immersion lithography full-system mass-production line, and Huawei announces the first commercial deployment of its "Galaxy" series integrated sensing and communications base station platform.

Character Developments

• Li Dong demonstrates his understanding of the project's challenges and proposes a solution involving segmenting the energy spectrum and applying the Li Criterion in stages.
• Shen Che shows appreciation for Li Dong's innovative approach and acknowledges his potential.
• Academician Gao Wen is mentioned for his crucial support in the development of Guowei Equipment's core real-time flow-thermal coupling control algorithms.

Significant Themes

• The importance of domestic innovation and self-reliance in high-tech industries.
• The power of quiet, dedicated work and sudden, impactful achievements.
• The significance of academic collaboration and interdisciplinary research.

Character Interactions

• Zhang Wenping and Li Dong discuss the new project and Li Dong's role in it.
• Shen Che introduces Li Dong to the project team and explains the challenges they face.
• Li Dong and Shen Che discuss Li Dong's proposal for addressing the energy-spectrum inversion issue.
• The senior trade official from the Western country and the Huaxia representatives engage in tense negotiations, with the official ultimately being forced to reconsider his position due to the announcements from Guowei Equipment and Huawei.