GPT-5.6 used a prompt to close a 30-year gap in convex optimization
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在数学领域取得的一项重大突破中，一位来自 UC Berkeley 的研究人员利用 GPT-5.6 Sol Pro 解决了一个自 1996 年以来一直未解的凸优化问题。该研究聚焦于确定性零阶凸优化的 oracle 复杂度，即确定在最坏情况下，对于凸且 1-Lipschitz 的函数，找到一个 ε- 最优点所需的查询次数。虽然 Protasov 之前已给出阶为 d^2 的上界，但长期以来下界一直不得而知，使我们对这一基本问题存在一个线性的理解缺口。
该作者有应用数学背景，曾花近一年时间尝试用传统方法和早期的 AI 模型来弥补这一差距，但均未成功。受到 OpenAI 最近利用 AI 证明 Cycle Double Cover Conjecture 的启发，该研究人员精心编写了长达十页的 prompt，为模型提供了清晰的问题描述、相关的现有研究，以及为证明不存在比 Protasov 模型更优算法所需的对抗性预言机策略的方法论。
在约 148 分钟的连续运行后，模型给出了一个成功弥合该复杂度差距的证明。作者对该论证进行了人工核验，并且至关重要的是，使用 Lean 对该证明进行了形式化验证，确保数学陈述在逻辑上经得起检验。该结果在 Lean 中的成功验证凸显出一个日益明显的趋势：AI 辅助的证明越来越多地接受严格的机器检验以保证可靠性。
作者在反思人工智能对学术界的影响时指出，尽管这一成果并不一定在凸几何中引入根本性的新技术，但它证明了现代 AI 在解决那些可用现有工具达到的问题上非常高效。作者并不认为这会使数学家过时；相反，这将促使该行业将精力从低难度或中等难度的问题转向那些真正需要新颖方法和人类创造性洞见的领域。
能够攻克长期悬而未决问题的 AI 工具的出现，引发了关于数学研究与教育未来的激烈讨论。该领域的一些人担心，AI 处理复杂证明的便捷性可能会削弱学生的积极性或淡化传统学术贡献的价值；另一些人则认为这些进步是推动科学发展的强大加速器，AI 可以承担证明过程中繁琐的劳动，从而使研究者有更多精力去追求更高层次的理解与复杂、创造性的综合工作。
In a significant breakthrough for the field of mathematics, a researcher at UC Berkeley utilized GPT-5.6 Sol Pro to solve a complex problem in convex optimization that had remained open since 1996. The study focused on the oracle complexity of deterministic zeroth-order convex optimization, specifically determining the worst-case number of queries needed to find an epsilon-optimal point for convex, 1-Lipschitz functions. While previous work by Protasov had established an upper bound of order d-squared, a definitive lower bound had long eluded researchers, leaving a linear gap in our understanding of this fundamental problem.
The author, who has a background in applied mathematics, spent nearly a year attempting to resolve this gap through traditional methods and previous iterations of AI models without success. Inspired by OpenAI's recent success in using AI to prove the Cycle Double Cover Conjecture, the researcher crafted an extensive, ten-page prompt. This prompt provided the model with a clear problem description, existing relevant research, and a methodology for the adversarial oracle strategy required to demonstrate that no algorithm could perform better than Protasov's existing model.
After approximately 148 minutes of uninterrupted work, the model generated a proof that successfully closed the complexity gap. The author personally verified the argument and, crucially, formally confirmed the proof using Lean, a proof assistant software that ensures mathematical statements are logically sound. The successful verification of this result in Lean highlights a growing trend in the mathematical community, where AI-assisted proofs are increasingly subjected to rigorous, machine-checked validation to ensure reliability.
Reflecting on the implications for artificial intelligence in academia, the author suggests that while this result does not necessarily create fundamentally new techniques in convex geometry, it demonstrates that modern AI is highly effective at resolving problems that are reachable with existing tools. The researcher does not believe this will make mathematicians obsolete. Instead, they argue it will shift the focus of the profession away from low-hanging or medium-difficulty problems and toward areas that require truly novel approaches and creative human insight.
The emergence of AI tools capable of solving long-standing open problems has sparked intense discussion regarding the future of mathematical research and education. Some in the field worry that the ease with which AI can handle complex proofs may discourage students or devalue traditional academic contributions. Others view these advancements as a powerful acceleration of scientific progress, noting that AI can handle the nitty-gritty labor of proof development, potentially freeing researchers to pursue higher-level understanding and complex, creative synthesis.
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在优化复杂性领域，证明非平凡的下界一向很难，因为这要求对所有可能的算法给出约束，而不是只分析某一具体算法。最近的结果表明，对于凸且梯度满足 Lipschitz 条件的函数，可以得到 Ω(d^2) 的下界，这与三十年前的算法上界相匹配，也凸显了现有数学技术所能达到的极限。
一个常见的误解是把现代的非凸目标与旧文献中关于凸且梯度为 Lipschitz 的函数的结论混为一谈。现代 AI 研究更多聚焦于那些经典优化理论难以处理的非凸问题。
重大的研究成果往往需要大量准备：多年调查以及篇幅长且针对性强的复杂 Prompt 等。这样的成果并没有使人类专家过时，反而表明工具能够自动化处理低到中等难度的任务，从而让专家更快地找到解决思路。
软件开发与数学在对可维护性与规划的依赖上本质不同。 AI 可能能处理孤立且定义明确的数学证明，但在软件开发中往往会生成错综复杂的"Slop"代码。在长期可持续的项目中，人类的规划能力与架构纪律仍然不可或缺。
AI 的"廉价"和"高效"具有主观性。一次推理会话可能只花费几十美元，但它依赖于庞大的基础设施、先前人类生成的数据以及大量用于搭建框架与验证的专家时间，因此解决问题的实际总成本仍然很高。
目前在战略性地使用 AI 时，更需要的是 "Harness engineering"，而不是简单的 prompt engineering 。那些理解领域概念、会设计需求并能进行严格验证的专家最能从这些工具中获益，因为他们能引导 AI 代理完成规划、测试与改进等复杂流程。
写作与研究本质上是为了提炼个人的心理模型，而不仅仅是产出文本。即便大型语言模型能吸收或复制知识，人的内在综合过程对于理解并规范个人工作仍是不可替代的。
Lean 等形式化验证系统的出现至关重要：它们使得数学证明可以不受来源（无论是人类还是 AI）影响而被信赖，为防止生成式文本中常见的"幻觉"问题提供了严谨基础。
技术进步常因缺乏立竿见影的可见益处（例如治愈复杂疾病）而遭到质疑。但科学发现相互关联，数学工具和预测建模（如 AlphaFold）方面的突破最终会扩散到生物学、医学等领域，带来更广泛的改进。
社会对 AI 的焦虑源自对人类劳动价值被贬低的担忧。虽然机器可能在特定任务上超越人类，但当专业能力转向协同与高层战略时，这种转型会带来个人如何维持生计的重大社会经济问题。
讨论的核心是：在 AI 能解决定义明确（虽复杂）问题的时代，人类专业角色正在演变。各方普遍认为，AI 确实能自动化技术性劳动、处理某类优化问题和数学证明，但它无法替代人类在选题、架构规划和深层概念理解上的作用。参与者强调，要取得成功需要高水平的领域知识来有效"驾驭"AI，这意味着未来的工作将从手工执行转向流程工程与严格验证。 • Proving nontrivial lower bounds in optimization complexity is historically difficult because it requires constraining all possible algorithms rather than just measuring a specific one. The recent proof establishes an Omega(d^2) lower bound for convex, Lipschitz functions, matching a 30-year-old algorithm and highlighting that current AI models can solve problems attainable with existing mathematical techniques.
• A common misconception conflates the modern, nonconvex objectives of AI with the older literature on convex, gradient-Lipschitz functions. Modern AI research is primarily focused on the non-convex problems that classical optimization theory historically struggled to address.
• Significant research results often require extensive preparation, including years of prior investigation and complex, domain-specific prompts spanning multiple pages. Such results do not render human experts obsolete but rather demonstrate that tools now allow specialists to navigate toward solutions faster by automating lower-to-medium-hanging fruit.
• Software development and mathematics are inherently different in their reliance on maintainability and planning. While AI can tackle isolated, well-defined mathematical proofs, it often produces convoluted "slop" code in software, where human-level planning and architectural discipline remain essential for sustainable, long-term project management.
• The "cheapness" and "efficiency" of AI are subjective. While a single inference session might cost twenty dollars, it leverages massive infrastructure, prior human-generated data, and significant expert time for framing and verification, meaning the true cost of solving a problem remains substantial.
• Strategic use of AI currently requires "harness engineering" rather than simple prompt engineering. Experts who understand domain concepts, requirements design, and rigorous validation are the most empowered by these tools, as they can direct AI agents through complex processes of planning, testing, and refinement.
• Writing and research are fundamentally about refining an individual's mental model, not just producing an output. Even if an LLM can ingest or replicate knowledge, the internal process of synthesis remains a uniquely human necessity for understanding and legislating one's own work.
• The emergence of formal verification systems like Lean is critical, as it allows mathematical proofs to be trusted regardless of whether they were generated by humans or AI. This provides a rigorous foundation that prevents the "hallucination" problems often seen in purely generative text.
• Technological advancement is often criticized for a perceived lack of immediate, tangible benefits, such as curing complex diseases. However, scientific discovery is deeply interconnected; breakthroughs in mathematical tools and predictive modeling (like AlphaFold) eventually propagate into broader improvements in biology, medicine, and beyond.
• Societal anxiety regarding AI stems from a fear that human labor is becoming devalued. While machines may eventually outperform humans in specific tasks, the transition presents a massive socioeconomic challenge regarding how individuals will sustain themselves as the nature of professional expertise shifts toward orchestration and high-level strategy.
The discussion centers on the evolving role of human expertise in an era where AI can solve well-defined, albeit complex, problems. A consensus emerges that while AI can automate technical labor and tackle specific classes of optimization and mathematical proofs, it does not replace the human need for problem selection, architectural planning, and deep conceptual understanding. Participants emphasize that successful outcomes require a high level of domain mastery to "pilot" AI effectively, suggesting that the future of work involves a shift from manual execution to process engineering and rigorous validation.