Why We Charge at the Riemann Hypothesis in 2026

"We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills." —— John F. Kennedy, 1962

"We choose to go to the moon in this decade and launch other endeavors, not because they are easy, but because they are hard; because this goal will gather and measure our finest energies and skills." —— John F. Kennedy, 1962

"Ask, and it will be given to you; seek, and you will find; knock, and it will be opened to you." —— Matthew 7:7

"Ask, and it will be given to you; seek, and you will find; knock, and the door will be opened to you." —— Matthew 7:7

Introduction: A Singularity in History

In 1859, Bernhard Riemann published a paper of only eight pages, On the Number of Primes Less Than a Given Quantity (《论小于给定数的素数个数》). In this paper, he casually mentioned a proposition, then added: "Certainly one would wish for a strict proof here; I have after some fleeting futile attempts provisionally set aside the search for this."

That setting aside has lasted one hundred and sixty-six years.

During those one hundred and sixty-six years, countless geniuses were captivated. Hardy proved there are infinitely many zeros on the critical line, but "infinitely many" is not "all"; Selberg proved a positive proportion of zeros lie on the line, but "proportion" is not "certainty"; computers verified the first ten trillion zeros all lie on the line, but "verification" is not "proof." The Riemann Hypothesis still stands, like an ice-covered summit, surveying all who attempt to climb it.

Yet today, March 2026, we stand at an unprecedented historical singularity. Not because humanity has suddenly become smarter, but because we possess infinite computational power for thought—a new cognitive paradigm capable of shuttling infinitely between form and intuition, logic and imagination, at the granularity of tokens.

At the start of 2026, the mathematical world was already experiencing undercurrents. Fields Medalist Terry Tao预言 that AI will autonomously conquer 1% to 2% of Erdős problems. Google DeepMind's Gemini model produced "elegant insights" in algebraic geometry that even Stanford professors marveled at. Elon Musk's Grok 4.20 achieved a square-root-level leap on the Bellman function problem in just five minutes. These are not science fiction; they are reality正在发生.

This article will systematically阐述 why this particular moment is the optimal window for charging at the Riemann Hypothesis. We will展开 from five dimensions: theoretical breakthroughs, AI capability leaps, formalization infrastructure, methodological innovation, and team configuration. But before beginning, I must make an honest declaration:

This article does not promise to prove the Riemann Hypothesis within 18 months.

What we promise is: during these 18 months, we will build a reproducible, scalable, inheritable "human-machine collaborative mathematical research infrastructure." Whether or not we ultimately reach the summit, the massive Lean 4 formalization library, the complete dataset of exploration failures, and the Turn-Lang meta-language architecture left behind by this system will all成为 permanent assets of human intellect. This is also part of the grand plan of the Phaenarete Project.

This is not a speculation on "results," but a strategic investment in the "process of pushing the boundaries of cognition."

Chapter One Historical Coordinates: One Hundred and Sixty-Six Years of the Riemann Hypothesis

Before deeply analyzing the window, we must first calibrate our historical coordinates.

In 1737, Euler discovered the prime product formula, opening analytic research on prime distribution. In 1859, Riemann analytically continued the $\zeta$ function to the entire complex plane, proposing the conjecture concerning the real parts of non-trivial zeros. In 1885, Stieltjes claimed to have proven the Riemann Hypothesis, but this was not recognized. In 1896, Hadamard and de la Vallée Poussin independently proved the Prime Number Theorem, a weak form of the Riemann Hypothesis, yet it already consumed forty years. In 1914, Hardy proved infinitely many zeros lie on the critical line. In 1942, Selberg proved a positive proportion of zeros lie on the line. In 1972, the chance encounter between Montgomery and Dyson revealed the astonishing connection between zero statistics and random matrix theory. In 1974, Deligne proved the Weil conjectures, paving the way for a geometric form of the Riemann Hypothesis. In 2000, the Clay Mathematics Institute listed it as one of the seven Millennium Prize Problems. In 2020, Rodgers and Tao proved the de Bruijn–Newman constant is non-negative, making the Riemann Hypothesis equivalent to a numerical equality. In 2022, Yitang Zhang (张益唐) achieved a polynomial-scale breakthrough on the Landau–Siegel zero problem. In 2024, Maynard and Guth used decoupling techniques to improve the zero density estimate that had stood unchanged for eighty years.

Every step consumed decades; every step melted a portion of this iceberg. But as of 2025, we still have not seen the full picture.

Why? Because the Riemann Hypothesis touches the deepest structures of mathematics—it connects the discrete distribution of primes with the continuous analysis of the $\zeta$ function, connects the statistical regularities of random matrices with the spectral distributions of quantum chaos, connects the trace formulas of noncommutative geometry with the functoriality of the Langlands program. This is a multi-dimensional, cross-domain giant puzzle that cannot possibly be completed by a single mind with a single tool.

Therefore, what we need is not another genius, but an entirely new cognitive paradigm.

Chapter Two Theoretical Window: Old Tools and New Perspectives

2.1 Guth–Maynard: The Power of Decoupling Techniques

In 2024, James Maynard and Larry Guth collaborated to introduce decoupling techniques from harmonic analysis into the study of Dirichlet sums. They achieved a fine decomposition of the frequencies of long Dirichlet polynomials, advancing the key exponent in zero density estimates from $3/5$ to $13/25$ .

The significance of this advance lies not in the specific numerical improvement, but in the methodological demonstration: a classical problem that no one could shake for eighty years was reopened by a set of tools originally not intended for this field. Maynard and Guth did not invent new mathematics; they merely examined old tools from a new perspective.

But we must honestly confront: from $13/25$ to $1/2$ , a theoretical chasm still横亘. In analytic number theory, approaching the critical line is often not a linear parameter optimization process, but involves essential theoretical barriers. We深知 that the contribution of decoupling techniques near the critical line exhibits diminishing marginal returns; therefore our focus is not on stacking compute for blind attempts, but on using AI to search for "nonlinear突变 points" of this tool under specific weight functions.

Phase goal (18 months): In a formalized system, leveraging PrimeClaw's search capability, explore whether the zero density bound can be advanced from $13/25$ to $13/25 - \varepsilon$ . Even a marginal improvement would be the second breakthrough in 80 years and would provide valuable data assets for subsequent research.

2.2 Yitang Zhang: The Second Miracle of Landau–Siegel

In 2022, Yitang Zhang (张益唐) published a 111-page paper proving that Dirichlet L-functions have no Landau–Siegel zero near $1$ , with a bound of $(\log \Delta)^{-2024}$ magnitude. This was the first advancement from exponential to polynomial scale on this problem since the 1930s.

Zhang's story is itself a metaphor. After completing his doctorate, he长期 remained obscure, teaching at the University of New Hampshire and even working at Subway. In 2013, at age 58, he proved a weak form of the Twin Prime Conjecture, shocking the world. In 2022, at age 67, he conquered the Landau–Siegel zero problem. His existence itself serves as a spiritual anchor.

But we must likewise be honest: extending Landau–Siegel techniques to complex zeros is目前看来 a "principle problem," not an "engineering problem." Zhang's proof strongly depends on the special algebraic properties of real zeros (the lower bound on $L(1,\chi)$ ); directly移植 this to complex zero distribution is几乎 a "category error" under existing frameworks.

So why do we still mention it? Because exploring "this path is impassable" itself constitutes valuable mathematics. Our goal is not to强行推广, but:

Use Lean 4 to completely formalize the core portions of Zhang's paper, converting it into machine-operable formalized assets.
Use AI to explore the specific mathematical structures where "extension is blocked," precisely刻画 the essential障碍 between real-zero techniques and complex zeros.
Mark these obstacles as "restricted zones" in the knowledge graph, informing future explorers: these paths have been tried; do not go there.

2.3 Geometric Langlands: The Dawn of Grand Unification

In February 2026, Sam Raskin, Dennis Gaitsgory, Nick Rozenblyum, and other mathematicians released an 800+ page proof of the geometric Langlands conjecture. This work打通 the deep logic from D-modules to stacks, revealing a unified framework of arithmetic geometry and harmonic analysis. The Riemann Hypothesis, as a core special case of the Langlands program, will directly benefit from this entirely new toolkit.

2.4 Spectral Embedding Conjecture: A New Approach Circumventing the Density Problem

In early 2026, an international team proposed the "spectral embedding" conjecture. The traditional Hilbert–Pólya approach is受阻 by the "density problem": the density of $\zeta$ function zeros is far higher than the energy level density of any known quantum system. The spectral embedding approach treats zeros as a subset embedded within a denser spectrum, selected by some mechanism.

The researchers constructed a family of supersymmetric quantum mechanical Hamiltonians, and through variational analysis found that the first several Riemann zeros can be recovered as approximate eigenvalues with high precision, with relative error below $8.5 \times 10^{-5}\%$ . This discovery circumvents the density problem and provides numerically verifiable predictions.

Transforming the problem into spectral embedding往往 only equivalently shifts the difficulty to "proving the existence and completeness of that specific operator family." Our goal: use exploration agents to generate variants of such conjectures, and use Lean for partial formalization verification—even if full proof is impossible, we can discover their logical consistency and potential value.

Chapter Three The AI Window: From Tool to Partner

If the theoretical window is a "hardware" upgrade, then the AI window is an "operating system" revolution. In 2025–2026, the leap in AI mathematical reasoning capability has made "human-machine collaboration" transition from science fiction to reality.

3.1 Aletheia and the FirstProof Challenge

In February 2026, Google DeepMind's mathematical research agent Aletheia autonomously solved 6 out of 10 research-grade problems in the inaugural FirstProof challenge. The proposers of these problems included Fields Medalists and full professors at top universities; Problem 7 had previously been listed as an open problem.

Aletheia's performance rests核心 on its "generate-verify decoupling" architecture and its "no bullshitting" design principle. When encountering problems it cannot solve, it directly outputs "No solution found" or remains silent within the time limit—sacrificing solving rate to换取 reliability of output. This design enables AI to evolve from a "competition contestant" to a "research collaborator."

Terry Tao commented: "AI has already become my junior co-author." The deep implication of this statement is: AI is no longer a tool, but a cognitive partner.

3.2 Lemmanaid: Neural-Symbolic Lemma Conjecture

Released in January 2026, the Lemmanaid tool is the first neural-symbolic lemma conjecture system. It uses fine-tuned large models to generate lemma templates, then employs symbolic methods to fill in details, discovering 50% and 28% of gold-standard reference lemmas on Isabelle's HOL library and the formal proof archive, outperforming pure neural network methods by 8-13 percentage points.

This progress proves: AI can already "conjecture" lemmas that mathematicians consider valuable. This is precisely the core capability needed by the "exploration agent" in our PrimeClaw system.

3.3 Aristotle and AEG: The First Practical Victory of Human-Machine Collaboration

Just days ago, we received令人振奋 news: Dr. Mingli Yuan's paper Arithmetic Expression Geometry I: Foundations has been completely formally verified by Harmonic's Aristotle mathematical hyperbrain system. It comprises 6 files and 37 theorems, covering all core content from arithmetic flow equations, arithmetic torsion, to contact structures. All theorems are fully proven, zero sorry, using only standard axioms.

This means:

AEG has become executable infrastructure. It is no longer geometric intuition on paper, but a formalized library callable by Lean 4. AEG provides not a simple solution, but an entirely new system of measures.
New tools for the three pillars. AEG's operator package (such as $\bar{\partial}_{\text{AEG}}$ and $\Delta_H + i\mu\lambda\partial_a$ ) can be embedded into PrimeClaw for exploring spectral problems.
The human-machine collaboration paradigm has been validated in practice. Dr. Yuan provided geometric intuition (though he did not participate in the collaboration), Aristotle completed the formalization labor—truth was born while he slept. This is the perfect预演 of the "Socratic midwifery" we承诺.

3.4 From "Problem Solving" to "Exploration": A Pragmatic Curriculum Learning Path

Despite AI's rapid capability advances, we must清醒 recognize: there is a巨大的 gap between Lean 4's "verification of known theorems" and "exploratory completion." Expecting it to autonomously收敛 a complete proof on RH-level problems is过度乐观. Therefore, we have designed a curriculum learning path:

First 6 months (warm-up period): Thoroughly understand and formalize the foundational tools of analytic number theory (circle method, sieve methods, Dirichlet polynomial estimates). Build a Lean 4 library containing 500+ core lemmas. The AEG formalization library will serve as a starting point and model.
Middle 6 months (assault period): Target the key techniques in the Guth–Maynard paper, using AI for parameter optimization, with the goal of advancing the zero density bound to $13/25 - \varepsilon$ .
Last 6 months (expansion period): Extend to the spectral embedding conjecture and noncommutative geometry paths, generating verifiable conjecture variants.

Chapter Four Infrastructure Window: Turn-Lang and the Future of Formalization

4.1 Crossing "Proof Islands"

Current formalization tools suffer from a fundamental flaw of incompatibility in foundational logic. A theorem proved in ZFC may need to be reproved in HoTT. This causes "islandization" of mathematical knowledge.

Turn-Lang is designed as a "foundation-independent type theory," treating mathematical foundations themselves as algebraic structures. When abstract logic needs to be compiled to ZFC, a functor maps the meta-logical category to the ZFC category; when compiling to HoTT, another functor maps to the HoTT category. Functors must preserve structure: propositions map to propositions, proofs map to proofs, composition preserves composition.

The functorial interpretation system automatically generates migration proofs from one interpretation to another. This resolves the most棘手 "proof island" problem in formal verification.

4.2 Confidence-Bounded Types: Demarcating Boundaries for Uncertainty

The probabilistic output of large language models cannot be processed by traditional type systems. Turn-Lang introduces confidence-bounded types:

Level	Type	Confidence	Verification Method
1	Proven type	1.0	Static type checking + formal proof
2	Fuzzy type	[c1, c2]	Runtime oracle verification
3	Experimental type	None	Heuristic/unverified

For "fuzzy types," the semantic mapping method in Lean 4 is尤为关键. We will leverage Lean's tactic framework to挂载 runtime oracles. The compiler generates JSON Schema based on the target type, and the oracle校验 the LLM output. This ensures cognitive type safety: even values from probabilistic models have their structural correctness guaranteed by the type system.

4.3 Lean4Lean: Formalization of Metatheory

In January 2026, Mario Carneiro of Chalmers University of Technology reported on the Lean4Lean project at the POPL conference—formalizing Lean's metatheory using Lean itself. The project provides the first complete type checker beyond the C++ reference implementation, running only 20%-50% slower, sufficient to verify the entire Mathlib library.

The significance of this progress is: the reliability of the formalization system itself is being formally verified. Turn-Lang will借鉴 Lean4Lean's experience, ensuring its functorial interpretation system's metatheory also withstands scrutiny.

4.4 Turn-Lang's Delivery Commitments

Turn-Lang is not an appendage of the Phaenarete Project; it is the permanent infrastructure we leave for the future of mathematics. We承诺:

Within 12 months: Release the Turn-Lang core compiler prototype, supporting functorial mapping from the meta-language to ZFC and CIC.
Within 18 months: Complete the runtime oracle implementation for confidence-bounded types, and open-source all code.

Chapter Five Methodological Innovation: Socratic Midwifery (苏格拉底助产术)

5.1 From "Machine Problem Solving" to "Human-Machine Co-Nurturing"

The Phaenarete Project拒绝 treating AI as a "problem-solving machine," instead proposing the "Socratic midwifery" paradigm. The core of this model is: human mathematicians provide high-dimensional intuition, define problem boundaries, and evaluate the "elegance" and "significance" of proofs; AI systems则 perform large-scale analogy, lemma generation, logical search, and real-time verification within constraints.

Phaenarete (法埃娜蕾特), Socrates' mother, was a midwife. Socrates said he did the same—helping people "bring forth" their own truth, rather than灌输 ready-made answers. The Phaenarete Project takes this as its name and as its foundation: we do not produce truth; we merely midwife it.

5.2 PrimeClaw: A Four-Agent Collaborative System

To realize this paradigm, we have developed the core technology platform—PrimeClaw. This system coordinates four specialized AI agents:

Knowledge Agent: Utilizing semantic graph technology to实时 track and index海量 literature, providing the system with the latest "battlefield map."
Exploration Agent: Based on Monte Carlo tree search and policy networks, attempting different proof paths, generating大胆 conjectures and simulations.
Proof Agent: Converting exploration paths into strict mathematical logic, attempting to generate proof scripts in the Lean 4 environment.
Verification Agent: As the final sentinel, using the formal verification kernel for line-by-line checking, ensuring "zero-defect" logic.

5.3 Bridging the "Semantic Gap": The Auto-Formalization Intermediary Layer

This is the most关键环节 in the PrimeClaw architecture. The high-dimensional intuition discovered by the Explorer agent via MCTS is typically non-formalized; if directly handed to the Prover, it极易陷入 error feedback loops in Lean 4.

Our solution is to introduce a dedicated auto-formalization intermediary layer. It aims to跨 the "semantic singularity" between non-formalized intuition and formalized logic, preventing proof paths from陷入 combinatorial explosion or invalid search under Lean 4's strict type checking. This intermediary layer (a fine-tuned model base) has the sole responsibility of translating pseudocode into valid Lean 4 Tactics. This forms the closed loop of conjecture → translation → verification → feedback.

5.4 Three Pillars: An Honest Progress Schedule

Based on current progress, we have sketched a possible proof blueprint composed of three pillars. But more importantly, we have set deliverable phase goals for each pillar:

Pillar One: Arithmetic Spectral Constraint (Connes Path)

Theoretical risk: High. The bridge between Connes's noncommutative geometry framework and classical analytic number theory has尚未 been fully established.
18-month goal: Complete Lean 4 formalization of key lemmas in Connes's early papers, construct a knowledge graph of "arithmetic spectral constraint," and明确 the theoretical障碍 on this path. AEG's arithmetic torsion and contact structures may为此 provide new geometric language.

Pillar Two: Density Bound Coupling (Guth–Maynard + Yitang Zhang Path)

Theoretical risk: Medium-high. Extending Landau–Siegel techniques to complex zeros is a principle-level problem that cannot be强行突破.
18-month goal: ① Complete formalization of the core estimates in the Guth–Maynard paper; ② Explore whether the zero density bound can be advanced from $13/25$ to $13/25 - \varepsilon$ ; ③ Formalize key portions of Zhang's Landau–Siegel paper, precisely刻画 the mathematical structures where extension is blocked. AEG's "cumulative exchange space" and weighted area integrals may为 understanding these estimates提供 new perspectives.

Pillar Three: Statistical Universality (Random Matrix Path)

Theoretical risk: Medium. The equivalence between GUE statistics and RH has尚未 been rigorously proven.
18-month goal: ① Construct a large-scale dataset of $\zeta$ function zero distributions; ② Use interpretability tools to analyze anomalous patterns in zero statistics; ③ Attempt to prove a weak form of "GUE ⇒ RH." AEG's horizontal holomorphic fields and twisted harmonicity may与 the spectral statistics of random matrices存在 deep connections.

Progress for each pillar will be publicly reported every quarter, including successful paths and failed attempts. This is our highest诚意 toward "open science."

Chapter Six Team, Resources, and Open Science

6.1 Cross-Generational Convergence of Intellect

Our team covers the complete闭环 from classical analytic number theory to frontier AI architecture:

Role	Member	Background and Contribution
Project Lead	Liangzhi (良之)	Project founder, Turn-Lang co-founder and architect of the midwifery framework
Chief Mathematician	Travor Liu	Stanford University doctoral student, independent arXiv paper author
Strategic Advisor	Academician Jianya Liu (刘建亚)	CAS academician, chief of the national key R&D program on the Riemann Hypothesis
Special Advisor	Professor Yuri Matiyasevich	Russian Academy of Sciences academician, solver of Hilbert's Tenth Problem
Mathematics Advisor	Academician Binyong Sun (孙斌勇)	CAS academician, L-function and representation theory expert
Honorary Advisor	Professor Yitang Zhang (张益唐)	Pioneer of the Twin Prime Conjecture, solver of the Landau–Siegel zero problem

(Note: We hereby cite and thank Dr. Mingli Yuan. He is the founder of AEG theory, whose work has been completely formalized by Aristotle, and we are deeply inspired by his theory in this project.)

This is the first time in history—a solver of a Hilbert problem, two academicians, a young genius, a cross-disciplinary architect, and a geometer—have collectively focused on another Hilbert problem.

6.2 Team Governance: Reconstructing Expert Feedback Mechanisms

Addressing the criticism that "academician weekly evaluation" is impractical, we have designed a layered review mechanism:

Daily: Turn-Lang's confidence type system automatically筛选 out 90% of invalid exploration paths.
Weekly: Establish "young verifier" positions for preliminary manual review of high-value paths筛选 by AI.
Monthly: Only when the system打通 links with potential for重大突破 are they提交 to the academician team for strategic oversight.
Quarterly: Full advisory team reviews progress and调整 the next phase's exploration focus.

6.3 Personnel Expansion: Lean 4 Engineer Team

Travor Liu's role is定位 as "chief number theory architect," responsible for inputting mathematical intuition and判断 key paths. But he cannot simultaneously serve as the main force for Lean 4 code implementation.

Therefore, we will allocate专项 from the budget to hire 2-3 full-time senior Lean 4 formalization engineers, whose responsibilities are:

Efficiently converting the mathematical insights of Travor, Academician Liu's team, and Dr. Yuan into high-quality Lean 4 code.
Maintaining and optimizing PrimeClaw's proof agent and verification agent.
Training community contributors, expanding the talent pool for formalized mathematics.

6.4 Funding Guarantees and Open Science Commitments

Project funding:

Turn-Lang project funding: 15 million RMB
AI for Math Fund 2026: 1 million USD
Compute resources: sufficient

18-month funding allocation table:

Expenditure Category	Proportion	Purpose Description
Compute costs	45%	MCTS inference cluster, cloud computing resources
Personnel costs	35%	3 full-time Lean 4 engineers + 2 young verifiers' salaries
Community and bounty pool	15%	Bounty for global hackers to formalize small lemmas
Operations and travel	5%	Academic exchanges, advisory meetings

Open science commitments:

Regardless of fund approval outcomes, core research proceeds如期.
All code will be open-sourced under the MIT license.
All datasets will be publicly accessible.
All research outputs will be shared via preprints.
All failed paths will also be记录 on record, with periodic release of The Dead Ends We Walked reports.
AEG's formalization library will serve as the project's first open-source module, immediately open to the community.

This is not betting on "success," but betting on "process." Because even if the Riemann Hypothesis is ultimately not proven within this timeframe, the tools, methods, talents, and communities built along the way will all成为 permanent assets of human intellect.

Chapter Seven Risks and the Fourth Mathematical Crisis

7.1 Three Core Challenges and Responses

First, residual hallucination and the semantic gap.

Response: Confidence-bounded types + auto-formalization intermediary layer + young verifier weekly screening. Three layers of filtering reduce hallucination risk to the minimum. Aristotle's successful formalization of AEG proves the feasibility of the auto-formalization intermediary layer on成熟 theories has been验证.

Second, computational economics.

Response: Use confidence control to precisely引导 resources toward the most promising directions; the budget明确 allocates 45% to compute, ensuring 18 months of continuous operation.

Third, theoretical path uncertainty.

Response: If RH's final proof requires an entirely new branch of mathematics, our strategy is: maintain the exploration agent's freedom of exploration, allowing it to "get lost" in unexplored mathematical territory; rely on human mathematicians' intuition to判断 which "lost paths" are worth continuing. Meanwhile, our delivery targets have already been解耦 from "proving RH"—even if there is no fundamental theoretical breakthrough after 18 months, the formalization library, failure datasets, and Turn-Lang infrastructure still constitute a successful delivery.

7.2 Positioning Within the Fourth Mathematical Crisis

As machine-generated proofs逐渐 exceed the scope of human understanding, the mathematical community is facing the fourth crisis of "truth without understanding" (无理解的真理). In January 2026, Gemini produced a proof in algebraic geometry that a Stanford professor评价 as: "If I had come up with this myself, I would brag about it for the rest of my life."

The Phaenarete Project's response is: confidence-bounded types and the midwifery paradigm. We neither对抗 the "incomprehensibility" of machine proofs, nor盲目 follow them. We demarcate boundaries for uncertainty through the type system, and preserve human understanding space through human-machine dialogue.

Aristotle's formalization of AEG is precisely a successful demonstration of "comprehensible collaboration": Dr. Yuan's geometric intuition was完整保留 within the proof structure, and the formalization process itself produced no black-box steps that humans cannot追踪.

Conclusion: Greatness Speaks for Itself (伟大，无需多言)

Standing at the historical threshold of 2026, the Phaenarete Project is not only a belated response to Hilbert's 1900 challenge, but also a巨大飞跃 in the evolution of human intelligence.

The 2026 mathematical community's wager has already begun: the Riemann Hypothesis versus P vs NP, which will humanity conquer first? Most observers believe the Riemann Hypothesis has approximately a 70% probability, because it is a "capping工程," while P vs NP is a "creation工程." Humans are better at finishing a building already constructed than at conjuring a world from nothing.

But we must再次 emphasize:

We do not promise to prove the Riemann Hypothesis within 18 months.

What we promise is:

An analytic number theory Lean 4 formalization library containing 500+ core lemmas
An open-source, scalable PrimeClaw multi-agent collaborative system
A Turn-Lang meta-language compiler prototype capable of跨-foundation proof migration
Quarterly reports详细记录 successful and failed paths
A battle-tested human-machine collaborative mathematical research team
An active open-source formalized mathematics community
Formalized assets of frontier theories such as AEG, already验证 and integrated by Aristotle

The mathematical summit still隐没 in clouds and mist, but we are no longer climbers with bare hands. We wield infinite computational power of thought, carrying loyalty to truth and reverence for the unknown, just as Socrates midwifed truth within souls through dialogue, we will use PrimeClaw's intelligent collaboration to midwife this century's most profound mathematical mystery.

Phaenarete (法埃娜蕾特)—the name of midwifery, also the name of virtue: the acuity that penetrates to essence, the purity loyal to truth, the fearlessness of courageous exploration, the humility of审慎 boundaries, the openness包容 pluralism, the warmth关怀 community, the vision超越 the individual. These radiant virtues will guide us forward through the mist, making every knock on the door closer to the light.

Whether the Riemann Hypothesis is ultimately conquered or not, we are already on the road, and we will leave permanent assets on the road—formalized knowledge libraries, failure datasets, open-source infrastructure, and a ladder that successive generations can继续 to climb.

Per aspera ad astra. Through hardships, to the stars. And the stars are not far away—they reside in every process of intellectual midwifery.

Appendix: The Phaenarete Project's North Star Proposition

To ensure that each agent in the PrimeClaw system possesses a unified formalization target, we have made a底层 axiomatic declaration of the strong form of the Riemann Hypothesis in Lean 4. This code symbolizes the engineering starting point of the project—even though the Zeta function is目前 a black box to the system, our logical architecture is entirely transparent.

import Mathlib.Analysis.Complex.Basic
open Complex
/-!
# Phaenarete Project: Riemann Hypothesis
-/
/--
Placeholder for Riemann zeta function - until Mathlib is properly configured.
-/
axiom riemannZeta : ℂ → ℂ
/--
Riemann Hypothesis (conjecture):
Every non-trivial zero of the Riemann zeta function in the critical strip
lies on the critical line Re(s) = 1/2.
-/
def RiemannHypothesis : Prop :=
  ∀ s : ℂ, riemannZeta s = 0 → 0 < s.re → s.re < 1 → s.re = 1 / 2
#check RiemannHypothesis

Personal views; criticism welcome

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