X-Risk Weekly

Levent Alpöge shared the result on X, describing it as "a beautiful new geometric object" and crediting Claude for its role in the work, writing that "Claude really contains multitudes." Mathematicians have long known that S² and S⁶ are the only spheres that can carry an almost complex structure, but whether that structure on S⁶ could be made integrable, so that the sphere genuinely becomes a complex manifold, had remained open since the question was first posed by Heinz Hopf in the late 1940s.

The construction is intricate. According to accounts of the paper, the central object is built from a family of complex two-dimensional tori over a modular curve associated with the triangle group Δ(3,4,∞), completed at three special points with carefully chosen degenerations and monodromy. The resulting compact complex threefold is claimed to be diffeomorphic to S⁶, established through a topological calculation showing the manifold shares the same fundamental group and homology as the six-sphere, a property that matters because there are no exotic six-spheres, so those topological properties are central to identifying the resulting smooth manifold with the ordinary six-sphere. Alpöge has said Claude's Opus 5 model wrote out the full argument, reportedly running to more than 100 pages, though he has suggested the core construction can be reproduced from its opening pages. The claim has drawn both excitement and caution online. One widely shared post argued that if the result holds, "this could be one of the most important AI-assisted mathematical breakthroughs yet," while noting that the six-sphere problem has a long history of failed proofs, including a purported 2003 argument by Shiing-Shen Chern that was never published after flaws were found. Others on X have been more skeptical, treating the announcement as a first-party claim from the mathematician rather than an independently verified result, and questioning the extent of the AI's contribution versus human guidance.

The episode follows a separate case, reported roughly three weeks earlier, in which an unreleased Claude research model was said to have improved the proven proportion of Riemann zeta function zeros lying on the critical line, from 41.6% to just over 67%, orchestrating dozens of sub-agents and thousands of shell commands in the process. Together the two claims illustrate a pattern researchers have started describing as a shift toward "layered evidence" and formal verification tools such as Lean, rather than line-by-line human checking, as AI-generated mathematics grows too extensive for any single mathematician to fully audit by hand.

CNBC reported that Anthropic will rent around 460 megawatts of compute capacity at an Nscale data center development in West Virginia. The six-year agreement, centred on Nscale's Monarch campus, will draw on Nvidia's forthcoming Vera Rubin chip systems, with Blockspace noting that the commitment covers approximately 460 MW of power capacity and averages $7.5 billion of spending annually. The compute capacity is expected to start powering the AI lab's services in late 2027, according to a source cited by TechCrunch.

The deal extends a run of infrastructure agreements Anthropic has struck this year as it tries to keep pace with rival OpenAI. As TechCrunch reported, over the past eight months, Anthropic has aggressively scaled up its compute capacity in an effort to better compete with rivals, most notably OpenAI. Earlier deals include a computing arrangement with SpaceX, described in the same report: Anthropic revealed it had entered into a large computing deal with SpaceX, drawing computing capacity from two different SpaceX data centers and reportedly providing Anthropic with $1.25 billion worth of capacity each month. In April, Anthropic signed a deal to significantly expand its partnership with Amazon, gaining access to an additional 5 gigawatts of compute, and that same month expanded its relationship with Google and Broadcom. Anthropic has also separately signed a $10 billion six-year contract with Volta Infra Holdings using capacity in Norway, and a 20-year lease with TeraWulf in Kentucky valued at around $19 billion, according to Blockspace.

The scramble for capacity follows Anthropic's own account of strain on its systems. CNBC noted that Anthropic said earlier this year that growing demand for its Claude AI models and products has caused "inevitable strain" on its infrastructure, which impacted "reliability and performance" for its users, especially during peak hours. In November, Microsoft and Nvidia had already moved to secure a stake in that growth, investing a combined $15 billion in the company as part of a deal in which Anthropic committed to purchasing $30 billion of Azure compute capacity from Microsoft and contracted for additional compute capacity up to 1 gigawatt.

The Nscale deal arrives as both companies prepare for stock market listings. PYMNTS reported that Anthropic could aim to raise as much as $100 billion in its IPO and is targeting a valuation of about $2 trillion, while preparing to tell investors it anticipates potential revenues of more than $30 trillion. Nscale, founded in 2024 as a spinout from a mining infrastructure business, is separately pursuing its own listing; PYMNTS noted it was reported that Nscale aims to raise as much as $3 billion in an IPO that could take place as soon as September. Anthropic confidentially filed its IPO prospectus with the Securities and Exchange Commission in June, and has been engaging in preliminary meetings with prospective investors.

According to the Associated Press, the case adds "another wave of turbulence in the AI rivalry between China and the United States." The Keelung District Prosecutors' Office said eight of the nine defendants were charged with breach of trust and forging documents, while three also face embezzlement counts, and Bloomberg reported that the indictment marks Taiwan's "first known crackdown on the black-market trade of such accelerators."

Prosecutors allege the group falsified paperwork claiming that 130 Nvidia B300 servers manufactured by Super Micro would remain in Taiwan, when the intention was to move them to Chinese customers. Of those, 74 servers made it out of the island, with 50 transhipped through Indonesia, eight sent via Japan, and the rest exported directly; a further 56 were intercepted at Taiwan's border after being falsely declared for Japan. Prosecutors are seeking maximum five-year sentences for four of the nine defendants, including the Nvidia employee identified only by his surname, Chang (also rendered Zhang in some reports), whom they described as the "key figure" responsible for "authorizing the release of the B300 GPUs," adding that he "demonstrated a clearly poor attitude following the offense." He was detained the previous month and Bloomberg said he could not be reached for comment.

The Taiwan case runs parallel to a separate US prosecution: federal authorities charged Super Micro co-founder Yih-Shyan "Wally" Liaw and two others in March with diverting roughly $2.5 billion in Nvidia-equipped servers to China through a Southeast Asian intermediary, a case in which Liaw has pleaded not guilty and faces up to 20 years with trial set for November. Taiwanese prosecutors have said it remains too early to determine whether the two investigations are connected. Notably, violating US chip export restrictions is not itself a criminal offence under Taiwanese law, so prosecutors have had to rely on forgery, breach-of-trust and embezzlement statutes instead; a lawmaker from President Lai Ching-te's ruling party is reportedly drafting a Foreign Trade Act amendment that would create a dedicated ban on such shipments.

The financial incentive behind the scheme is stark. Gregory Allen, an analyst at the Center for Strategic and International Studies, has said that Nvidia GPUs retailing for $25,000 to $30,000 through authorised channels can fetch $40,000 or more on gray markets accessible to Chinese buyers, and has compared the resulting profit margins to those in narcotics trafficking. Neither Nvidia nor Super Micro has been charged as a corporate entity, and both companies have framed the matter as the conduct of individual employees rather than company policy.

The Ebola outbreak in the Democratic Republic of Congo has reached 5,515 confirmed cases and killed 2,642 people, according to the latest government figures cited by Al Jazeera, with 51 new cases detected in Ituri and North Kivu provinces. The case fatality rate has climbed to nearly 48 percent, meaning almost one in two confirmed infections ends in death, up sharply from about 20 percent in early June. Declared on 15 May, the epidemic is the DRC's 17th Ebola outbreak since 1976 and was designated a Public Health Emergency of International Concern by the World Health Organization, its highest alert level, also covering neighbouring Uganda. It is now considered the deadliest outbreak in the country's history, and the WHO has said it is on track to surpass the 2014-2016 West African epidemic that killed more than 11,000 people, which remains the deadliest Ebola outbreak on record. The strain driving the outbreak, Bundibugyo virus, has no approved vaccine or treatment, a fact Vatican News noted when reporting that over 2,500 people have died so far out of 5,300 confirmed cases. Speaking at the Angelus on Sunday, Pope Leo XIV said "In my prayers, I often remember the Democratic Republic of the Congo, particularly because of the spread of the Ebola epidemic, which is unfortunately claiming many lives," and called for international action, adding "I encourage a response from the international community that also involves local communities in prevention efforts, in order to save many human lives." The national fatality rate masks wide regional variation. NPR reported that while the case fatality rate is so far 47.4%, it is far worse in some places where response efforts are more challenging, such as in North Kivu province where the fatality rate is 70%. WHO Director-General Tedros Adhanom Ghebreyesus told a meeting of the agency that the outbreak "has spread rapidly and the risk of further national and international spread remains high," adding that "We must be frank: the epidemic is far from being under control." Contact tracing has nonetheless improved sharply, rising from 30 percent in June to more than 85 percent. Public health specialists warn the crisis could still worsen. Abdulsalami Nasidi, a consultant who helped establish the Africa Centres for Disease Control and Prevention, told Al Jazeera the outbreak was "getting out of hand" and warned "This is no longer just a national or regional issue; it is a global issue. If it spreads to neighbouring places with lower immunity, it will be a disaster." The WHO currently rates the global risk as low but the danger inside the DRC as very high. Response efforts have been complicated by armed conflict, displacement, attacks on health workers and facilities, and weak infrastructure across the six affected provinces, though Uganda and several health zones in Ituri and South Kivu have interrupted transmission, which the WHO cites as evidence that rapid detection and community cooperation can break the chain of infection. About 160 health workers have been infected in the outbreak and roughly 45 have died.