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AI推翻87年数学难题？困扰数学界的雅可比猜想，可能被三行公式证伪了_我的网站

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    Drivers queue at a petrol station to refuel in Tehran, on August 25, 2026. Photo: VCG
As military strikes and diplomatic negotiations have both failed to bring an early end to the roughly six‑month‑old US-Iran conflict, Washington rolled out new sanctions against Tehran on Monday local time intended to further strangle its economy. Iran, for its part, has vowed a tougher pushback.
In a news conference on Monday, Treasury Secretary Scott Bessent threatened new sanctions on countries refusing to cut economic ties with Iran, following recent rhetoric from Washington threatening an "economic D-Day" for Iran. 
Meanwhile, the Office of Foreign Assets Control (OFAC) under the US Treasury announced secondary sanctions on nearly 60 entities, individuals, and vessels in multiple jurisdictions which, according to the US' claims, enable Iran's activities, including so-called nuclear and missile technology procurement, cyber operations, and oil-revenue generation networks.
According to Reuters, those sanctions targeted businesses in China, the UAE, Singapore and several other countries, including a cooking-oil refinery in France, but did not include any of the Chinese financial institutions suspected of facilitating Iran's oil trade.
In response to US Treasury Secretary Scott Bessent laying out plans for economic asphyxiation of Iran, expanding Washington's secondary sanction threats and warning of dire consequences for countries that do not join the campaign, Chinese Foreign Ministry spokesperson Lin Jian said at a press briefing on Tuesday that China has made clear on many occasions its firm opposition to illicit unilateral sanctions that have no basis in international law or the authorization of the UN Security Council. 
"Economic warfare and maximum pressure provide no solution. On the contrary, they only serve to fuel tensions and lead to risk spillover, which will disrupt the global economic and financial order, and harm the legitimate rights and interests of other countries," the spokesperson said. 
The pressing task is to facilitate the deescalation of the situation and return to dialogue and negotiation as soon as possible, Lin said, adding that China will do everything necessary to firmly safeguard its rights and interests, Lin added. Lin also said China's cooperation with Iran is conducted within the framework of international law, thus should not be disrupted.
When asked whether China has received any such timeline, as the US stated it would give countries a timeline on when to shut down the actions that it has identified, and how China intends to respond to the relevant announcements by the US Treasury involving China-based entities and individuals, Chinese Foreign Ministry spokesperson Lin Jian stated on Tuesday that China is closely following the developments, and will do everything necessary to firmly safeguard our rights and interests.
In May this year, China's Ministry of Commerce issued a blocking ban prohibiting any recognition, enforcement or compliance with US sanctions imposed on five Chinese companies on the grounds of their alleged involvement in Iranian petroleum transactions.
Relying on its financial hegemony and long‑arm jurisdiction, the US enforces unilateral sanctions with no basis in international law. Such measures severely undermine fundamental consensus for international trade and erode the existing international order, Zhou Mi, senior research fellow at the Chinese Academy of International Trade and Economic Cooperation, told the Global Times on Tuesday.
In essence, Washington is imposing its Iran‑focused policies to enterprises across the globe, compelling third‑party actors to fall in line with its diplomatic objectives, Zhou noted.
The latest sanctions would affect a broader range of countries, industries and firms. They will not only draw wider opposition from the international community, but also boomerang on the interests of US‑owned enterprises, he added. 
Fewer levers left 
"We are launching an economic onslaught against Iran's financial connections around the globe," Bessent claimed at a press conference. "Our objective is to sever every economic lifeline" that sustains the country, he said, according to a Treasury release. 
The US Treasury Department said in a release that the new sectoral sanctions determinations target five critical sectors - digital assets, technology, gold, aviation and shipping - that the US claimed Iran has relied on to support its economy.
US Treasury claimed that "any entity that facilitates money laundering or sanctions evasion on behalf of Iran risks being cut off from the US financial system."
OFAC also suspended several general licenses that previously authorized certain remittance payments to Iran and Iranian access to the US cultural and academic system. The office also issued additional guidance on the so-called sanctions risks of bowing to Iranian demands related to shipping in the Strait of Hormuz.
Mohammad Mokhber, an aide to Supreme Leader Mojtaba Khamenei, said in an X post on Monday that Iran's response to US sanctions and pressure will be "more decisive than before."
Iranian Economy Minister Ali Madanizadeh said that Tehran "is and was ready," with a two-year plan to counter Washington's new sanctions. He told state television that "it seems they wished to suffer yet another defeat," per an AFP report. 
Iranian Parliament Speaker Mohammad Bagher Ghalibaf responded in an X post that "the Americans know no one will believe their nonsense," noting that the US is not in a position economically to restrict its relations with other countries. 
Alan Eyre, a distinguished diplomatic fellow at the Middle East Institute, told NBC News that after six months of combat, the US government "has few remaining levers it can use to convince Iran's major trading partners to materially reduce their economic ties."
While Washington's maximum‑pressure campaign may deliver short‑term shocks to Iran's economy, it can hardly "strangle" the country completely, said Liu Zhongmin, a professor at the Middle East Studies Institute of Shanghai International Studies University.
He added that Iran's economic resilience cannot be underestimated. Having borne decades of crippling sanctions from the US and Western nations, Tehran has built substantial experience in sanction‑busting and developed coping mechanisms ranging from basic‑industry self‑reliance and regional trade networks to oil‑for‑goods swaps and non‑US dollar‑denominated trade. Furthermore, the Strait of Hormuz remains a source of strategic leverage for Iran to push back against US pressure.
The expert said that Tehran and Washington are encountering "a battle of wills," and Iran believes the economic costs the US would suffer from a Hormuz blockade would outweigh the damage inflicted on Iran by sanctions.
Washington's "economic strangulation" policy risks ending in failure unless it yields short‑term outcomes, particularly securing the opening of the Strait of Hormuz, Liu added. 
Self-created dilemma
As the US ramps up pressure on Iran, Americans are growing averse to a protracted war.
According to a Reuters/Ipsos poll that closed on Monday, US ‌public approval of the war with Iran fell to its lowest level since the conflict's early days, with just 31 percent of Americans supporting US military action.
On June 18, the US and Iran signed an MoU, under which the two were scheduled to hold negotiations within a period of 60 days, which ended on August 17, to reach a final agreement. However, the talks' fate remains unclear following renewed escalation between the two countries.
Iranian political analyst Mostafa Khoshcheschm told Al Jazeera that the new US sanctions intended to create an "economic onslaught" were a "political show" of intimidation. 
Liu said that by resorting to an "economic strangulation" tactic after miring itself in a fruitless military quagmire over Iran, the US has laid bare its practical predicament and strategic dilemma, and seemingly running out of cards.
Citing sources, CNN reported earlier this month that the US military has exhausted nearly 80 percent of its interceptors for a key missile defense system as senior US military commanders are warning that the Pentagon's munitions stockpile is "dangerously low."
Faced with depleted ammunition stockpiles and insufficient military‑industrial capacity, and amid growing domestic criticism of its Middle East policy, the White House has opted for tough economic measures against Iran, Liu said. "Less costly than military coercion, such moves help Washington strike a tough posture before voters and shun political risks from further military escalation," said the expert.
There is, however, an increasingly obvious paradox in Washington's economic warfare against Iran: the tighter the sanctions, the tighter global energy supplies become, and the higher costs borne by American consumers and businesses, the expert added. 
。    悬而未决 87 年的经典数学难题——雅可比猜想（Jacobian Conjecture），可能被一个只有三行公式的反例推翻了。         7 月 20 日，数学家、Anthropic 研究人员 Levent Alpöge 在社交平台 X 公布了一个三元多项式映射。它满足了雅可比猜想要求的前提条件，却违背了猜想的结论，因此构成了一个可能推翻整个猜想的反例。         更引人关注的是，Alpöge 在帖子中将这一反例归功于 Anthropic 最新发布的 AI 模型 Claude Fable 5。他还表示，是朋友 Akhil 首先向模型提出这一问题，而 Fable 5 则在他们观看世界杯决赛期间持续搜索，最终找到了这一反例构造。         （来源：X）          消息发布后，数学界迅速展开核验。斯坦福大学数学系助理教授 Jared Duker Lichtman 转发称这一结果“相当惊人”；MathOverflow 很快收录了相关讨论，不少研究者还利用 Wolfram Alpha、SymPy 等工具进行了独立计算。目前，最关键的两项验证：雅可比行列式是否恒为 −2，以及三个不同点是否确实映射到同一点，均已通过公开计算验证。         如果后续没有发现新的漏洞，这将意味着：1939 年提出的雅可比猜想在三维情形下不成立，并可进一步推出其在所有三维及以上维数下均不成立；而二维情形仍然保持开放。

二 |          被称为“民科坟场”的雅可比猜想          雅可比猜想由德国数学家奥特－海因里希·凯勒（Ott-Heinrich Keller）于 1939 年提出，讨论的是一个看似简单、却困扰数学界近九十年的问题：什么时候，一个多项式映射一定能够"倒推"回原来的函数？          图｜海因里希·凯勒（左）（来源：Wikipedia）          理解这个猜想，可以先想象一张橡胶膜。

三 | 如果局部看，每一小块都没有被撕裂或压扁，人们很自然会问：整张膜会不会在别的地方折叠起来，让两个不同的位置重合？          数学里的情况与此类似。

四 | 对于多变量多项式映射，如果它在每一个局部都满足一种“没有折叠”的条件，也就是雅可比行列式始终等于同一个非零常数。那么它是否一定在整体上也是一一对应的？它的逆映射又是否仍然是一个多项式？          这就是雅可比猜想。它的陈述只有短短几句话，却异常难以证明。不少数学研究者戏称它为“crank graveyard”——“民科坟场”。

五 | 许多人被它吸引，最终却陷入漫长而徒劳的推导之中。         著名华人数学家张益唐，就曾把整个博士阶段投入到这个问题。1991 年，他在普渡大学完成博士论文《雅可比猜想与域扩张的次数》，导师莫宗坚是国际上研究雅可比猜想最重要的学者之一。张益唐的工作推进了这一研究方向，却并没有解决猜想本身。后来，他离开代数几何，转向解析数论，并最终因证明有界素数间隔而闻名世界；雅可比猜想，则依然留在开放问题名单中。         1998 年，菲尔兹奖得主 Stephen Smale 将其列入著名的《下一个世纪的数学问题》（Mathematical Problems for the Next Century）第 16 题，认为它的解决将对数学发展产生重要影响。         过去近 90 年里，一维情形早已证明成立，二维至今仍未解决，而三维及更高维也始终没有答案。由于这一猜想要求所有维数都成立，因此，只要在任意一个维数找到反例，整个猜想就会被推翻。         三个点，击穿一个猜想          而此次，Alpöge 公布的是一个从三维复空间到三维复空间的多项式映射：          直接计算可知，这个映射的雅可比行列式恒等于 −2。由于 −2 是一个非零常数，因此它完全满足雅可比猜想的前提条件。

六 |          然而，当把三个不同的点          分别代入这一映射时，得到的结果却都是同一个点：          这足以构成反例。原因很简单：如果多个不同的输入对应同一个输出，那么映射就不是单射。换句话说，当已知输出时，无法唯一确定原来的输入，因此它不可能拥有逆映射，更不可能拥有由多项式组成的逆映射。这与雅可比猜想的结论正面冲突。         这个反例还具有极强的可核查性。许多重大数学成果往往需要专家花费数月甚至数年时间，逐页检查上百页证明；而这个结果的核心，仅涉及两项有限的代数计算：一是验证雅可比行列式恒等于非零常数，二是验证三个不同的点确实映射到了同一个点。         截至目前，这两项计算均已得到公开复核。研究者 Zihan Zhang 给出了独立的 SymPy 验证脚本，并指出，只需将其中一个输出坐标乘以 −1/2，就可以把雅可比行列式规范化为猜想中更常采用的常数 1，而这不会改变反例的本质。         如果这一反例最终成立，它不仅推翻三维情形，也意味着所有更高维情形都不再成立。因为只需在三维反例后增加若干保持不变的坐标，就可以直接构造出任意更高维的反例。不过，二维雅可比猜想仍然是一个独立的开放问题          值得补充的是，Levent Alpöge 并不是从 AI 领域半路进入数学，而是一位长期从事数论与算术几何研究的职业数学家。近年来，他才加入 Anthropic，从事人工智能研究。         他本科毕业于哈佛大学，主修数学，并获得物理学硕士学位；随后在剑桥大学完成数学高级研究课程。2020 年，他在普林斯顿大学获得数学博士学位，导师是菲尔兹奖得主 Manjul Bhargava。         图｜Levent Alpöge（来源：Quanta Magazine）          大型语言模型兴起后，Alpöge 开始转向人工智能。他曾在个人主页写道，GPT-4 让他认为人工智能是“人类创造过的最有意思的东西”，因此重新回到计算机科学领域，希望增加这场变化产生积极结果的可能性。         根据目前公开的信息，这次发现的大致过程是：Akhil 提出问题，Alpöge 将其交给 Claude Fable 5 进行探索，模型最终给出了具体的反例构造；随后，Alpöge 公开了这一构造，数学界再通过人工推导以及 Wolfram Alpha、SymPy 等工具进行了独立验证。

七 |          这一过程与 AI 替数学家写出一份传统意义上的数学证明有所不同。

八 | 对于否定一个猜想而言，并不需要解释所有可能情况，只需要找到一个满足前提、却违反结论的反例即可。这样的任务天然适合计算探索：模型可以不断尝试构造候选映射、调整参数、检查雅可比条件，再寻找是否会出现不同点映射到同一点的情况。         如果这一反例最终得到确认，它带来的意义将十分重大。因为它展示了一种此前极少出现的研究模式：不是 AI 帮助数学家验证已经存在的思路，而是AI参与寻找一个人类 87 年来始终没有找到的数学对象。         从 AlphaGeometry 解决竞赛题，到 GPT、Gemini 参与证明具体命题，再到这一次可能找到推翻经典猜想的反例，大模型参与数学研究的角色正在发生变化。它开始探索的，不再只是“答案”，而是过去只有数学家才会进入的未知搜索空间。         参考链接：          1.https://x.com/leventalpoge          2.https://mathoverflow.net/questions/513387/galois-structure-of-the-new-counterexample-to-the-jacobian-conjecture-an-explic          3.https://www.leventalpoge.com/          4.https://dataspace.princeton.edu/          5.https://www.ams.org/prizes-awards/ams-morgan-prize          运营/排版：何晨龙          注：封面由 AI 辅助生成。

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