Mathematical Experiments Are Becoming Abundant Through Human-Machine Teaming

数学实验正通过“人机协作”变得日益丰富

Two open problems, exact-arithmetic checking and a proof assistant, over a single weekend. 在短短一个周末内，解决了两个开放性问题，完成了精确算术校验，并使用了一个证明助手。

Sean Moran Aug 15, 2026 28 min read Sean Moran，2026年8月15日，阅读时长28分钟

A conceptual illustration of the mathematical search process, showing explored ideas, eliminated branches, promising directions, and unexplored frontiers in theorem proving and mathematical research. Source: image by author. 数学搜索过程的概念图，展示了定理证明和数学研究中已探索的想法、已排除的分支、有前景的方向以及未探索的前沿领域。来源：作者自制图片。

I ended a recent weekend without a Hadamard matrix of order 668 and with a mathematical theorem I did not yet trust enough to call a result. Working with GPT-5.6 Sol, parallel agents, exact-arithmetic programs and a proof assistant, I attacked two hard problems over two days. 上周末结束时，我既没有找到668阶的阿达马矩阵（Hadamard matrix），也没有得到一个我敢称之为“成果”的数学定理。在GPT-5.6 Sol、并行智能体、精确算术程序和证明助手的辅助下，我用了两天时间攻克了两个难题。

One resisted every route I tried. The other produced a proof candidate: a conventional argument written out in full, exact checks on selected inputs, a teaching guide and a partial formalisation in Lean. That second outcome is still a candidate. No specialist has reviewed the argument, its novelty is not established, and Lean checks only its algebraic core rather than the complete theorem. I would not publish it as a new theorem in its current state. 其中一个问题抵御了我尝试的所有路径。另一个则产生了一个证明候选：一份完整的传统论证、对选定输入的精确校验、一份教学指南以及在Lean语言中的部分形式化。第二个结果目前仍只是一个候选。还没有专家审阅过该论证，其创新性尚未确立，且Lean仅校验了其代数核心，而非完整定理。在当前状态下，我不会将其作为新定理发表。

The pattern I took from the weekend is that mathematical experiments are becoming abundant. The Hadamard project closed 44 precisely defined regions of the search space and audited five standard routes to a nonexistence proof. The Maxwell project produced a complete proof candidate with a machine-checked algebraic core. Accepted mathematical knowledge did not get cheaper alongside any of that, because proof, understanding, novelty and independent review are separate obligations, and generating another route settles none of them. 我从这个周末总结出的模式是：数学实验正变得日益丰富。阿达马项目关闭了搜索空间中44个精确定义的区域，并审计了五条通往“不存在性证明”的标准路径。麦克斯韦项目产生了一个带有机器校验代数核心的完整证明候选。然而，公认的数学知识并没有因此变得廉价，因为证明、理解、创新性和独立评审是各自独立的义务，生成另一条路径并不能解决其中任何一个问题。

This is a concrete sequel to two earlier ideas. In AI Has Made Research Cheap. Understanding Is Still Expensive, I argued that AI was making experimentation far cheaper than understanding. In From Tokens to Theorems: Building a Neuro-Symbolic AI Mathematician, I built a simple neuro-symbolic loop in which an LLM proposed mathematical formulas, SymPy checked them exactly, and failed candidates became feedback for the next attempt. That experiment was deliberately modest, but it exposed an architecture that reappeared here at a much larger scale. 这是我之前两个观点的具体延续。在《AI让研究变得廉价，但理解依然昂贵》一文中，我提出AI使实验成本远低于理解成本。在《从Token到定理：构建神经符号AI数学家》中，我构建了一个简单的神经符号循环：LLM提出数学公式，SymPy进行精确校验，失败的候选方案则作为下一次尝试的反馈。那个实验虽然刻意保持简单，但它揭示了一种架构，而这种架构如今在更大规模上再次出现。

Over this weekend, the same pattern extended beyond symbolic sequence problems. Instead of one model and one symbolic checker, I used parallel agents, exact-arithmetic programs, adversarial critics and a proof assistant. Candidate constructions and arguments were generated, attacked, checked where possible and either discarded or retained with an explicit status. The loop had become richer as AI systems improved through 2025 and 2026, but its basic structure remained the same. 这个周末，同样的模式扩展到了符号序列问题之外。我不再只使用一个模型和一个符号校验器，而是使用了并行智能体、精确算术程序、对抗性评估者和证明助手。候选的构造和论证被生成、攻击、在可能的情况下进行校验，并根据明确的状态被丢弃或保留。随着AI系统在2025年和2026年的进步，这个循环变得更加丰富，但其基本结构保持不变。

Looking back afterwards, I realised the workflow closely resembled a broader pattern that Jeff Dean recently described for science and engineering. Rather than viewing AI as a tool for solving one problem at a time, he characterised research itself as a recurring cycle: propose an experiment, implement and run it, evaluate the result, then use that evaluation to generate a better experiment. His argument was that AI should automate this entire loop, reducing iteration time, running many experiments in parallel and learning from every evaluation. 事后回顾，我意识到这个工作流与Jeff Dean最近描述的科学与工程领域的一种更广泛的模式非常相似。他没有将AI视为一次解决一个问题的工具，而是将研究本身描述为一个循环：提出实验、实施并运行、评估结果，然后利用评估结果生成更好的实验。他的观点是，AI应该自动化整个循环，缩短迭代时间，并行运行大量实验，并从每一次评估中学习。

That was almost exactly what had happened over my weekend, albeit on a much smaller scale. The experiments were mathematical rather than physical. “Implementation” meant exact-arithmetic programs, constraint solvers or proof assistants instead of laboratory equipment. Evaluation meant integer verification, counterexample search and partial formalisation rather than measuring a physical system. Parallel agents proposed constructions, generated verifiers, searched the literature, criticised arguments and translated ideas into different mathematical representations before the next iteration began. 这几乎就是我周末所经历的一切，尽管规模小得多。这些实验是数学性的而非物理性的。“实施”意味着使用精确算术程序、约束求解器或证明助手，而不是实验室设备。评估意味着进行整数验证、反例搜索和部分形式化，而不是测量物理系统。在下一次迭代开始前，并行智能体负责提出构造、生成验证器、搜索文献、批评论证并将想法转化为不同的数学表示。

Seen this way, the most striking development was not that AI produced a proof candidate or failed to construct a Hadamard matrix. It was that the experimental cycle itself had become partially automatable. Once exact evaluators existed, proposing, executing and refining mathematical experiments became dramatically cheaper. 从这个角度看，最引人注目的进展并非AI产生了一个证明候选或未能构造出阿达马矩阵，而是实验循环本身已经可以部分自动化了。一旦有了精确的评估器，提出、执行和完善数学实验的成本就大幅降低了。

The important difference was not simply one of scale but of validation. In the earlier experiment, a candidate formula could be tested directly against the sequence, although reproducing the observed values did not necessarily reveal the underlying mathematics. Here, verification became layered. Exact arithmetic could reject incorrect constructions, Lean could certify parts of an argument, and literature searches could help establish prior work, but none could alone determine novelty, validate every bridge in a universal proof or decide whether a result deserved to become accepted mathematical knowledge. 重要的区别不仅仅在于规模，而在于验证。在早期的实验中，候选公式可以直接针对序列进行测试，尽管重现观察到的值并不一定能揭示其背后的数学原理。而在这里，验证变成了分层的。精确算术可以排除错误的构造，Lean可以证明论证的部分环节，文献搜索可以帮助确立前人工作，但没有任何单一手段能独立判定创新性、验证通用证明中的每一个逻辑桥梁，或决定一个结果是否值得成为公认的数学知识。

Hadamard 668 resisted an exact search stack 668阶阿达马矩阵抵御了精确搜索栈

The first problem asks for a 668 by 668 grid filled only with +1 and -1, arranged so that any two different rows cancel out completely: multiply them entry by entry, add up the result, and you must get exactly zero. In linear-algebra terms, the rows are mutually orthogonal. This is a Hadamard matrix of order 668, and matrices of this kind show up in error-correcting codes, signal processing and the design of experiments. 第一个问题要求构建一个668x668的网格，仅填充+1和-1，使得任意两行完全抵消：逐项相乘并求和，结果必须精确为零。在线性代数术语中，这意味着行与行之间相互正交。这就是668阶阿达马矩阵，这类矩阵出现在纠错码、信号处理和实验设计中。

According to Epoch AI’s current catalogue of the problem, 668 is the smallest order for which nobody knows whether one exists. This is an appealing target for computation because a proposed answer is easy to check. Multiply the candidate by its own transpose using exact integers, then read off the result: every diagonal entry must be 668 and every other entry exactly zero. That is a single matrix multiplication, and there is no subjective judgement at the final gate. Finding one is another matter. The grid has 446,224 entries and each is either +1 or -1, so a naive search has 2⁴⁴⁶,²²⁴ candidates to get through. No amount of compute touches a number that size. All the useful work goes into no 根据Epoch AI目前的目录，668是目前无人知晓其是否存在阿达马矩阵的最小阶数。这是一个极具吸引力的计算目标，因为提出的答案很容易校验。使用精确整数将候选矩阵与其转置相乘，然后读取结果：每个对角线元素必须为668，其余所有元素必须精确为零。这只是单次矩阵乘法，最后一步没有任何主观判断。但找到一个矩阵则是另一回事。该网格有446,224个元素，每个元素要么是+1要么是-1，因此简单的搜索需要遍历2⁴⁴⁶,²²⁴个候选方案。任何算力都无法触及如此巨大的数字。所有有用的工作都投入到了……（注：原文此处中断）