Claude Produces First Complete Machine-Checked Proof of Fermat's Last Theorem
In what Imperial College mathematician Kevin Buzzard called an "extraordinary autoformalization achievement," Anthropic announced this week that Claude has produced the first complete computer-verified proof of Fermat's Last Theorem — and it did so in just 11 days, working largely autonomously.
The Lean formalization spans a staggering 13 million lines of code and proves 29,500 intermediate theorems, making it over five times the size of Mathlib, the largest existing library of formalized mathematics. The work followed a simplified version of Andrew Wiles's landmark 1995 proof, as outlined by Darmon, Diamond, and Taylor, and consumed roughly six billion output tokens from an internal Anthropic research model.
To be clear: Claude didn't discover Fermat's Last Theorem or replace Wiles's original mathematical insight. Formalization is the process of translating informal mathematical reasoning into rigorous, step-by-step code that a proof assistant like Lean can verify line by line. What makes this remarkable is the scale and speed — a task that the math community had considered a multi-year effort was completed in under two weeks.
The achievement signals a new era for AI in mathematics. If models can reliably formalize complex proofs, they could accelerate verification of new results, reduce errors in published mathematics, and open the door to AI-assisted discovery of entirely new theorems.
Source: Anthropic Research, SiliconANGLE
Apple's Foldable Moment: iPhone Ultra Expected at September 9 Event
With John Ternus now one week into his tenure as Apple's CEO — succeeding Tim Cook after 15 years — all eyes are on Apple's "Surprise and Shine" event on September 9. The headline attraction: Apple's first-ever foldable device, widely expected to be called the iPhone Ultra.
Leaks and renders point to a "passport-style" book-fold design with a 5.5-inch cover display and a 7.8-inch inner display at roughly a 4:3 ratio. Apple has reportedly been refining the hinge mechanism using liquid metal and ultra-thin glass to minimize the crease — long the Achilles' heel of foldable phones.
Beyond the foldable, Apple is expected to unveil the iPhone 18 Pro and iPhone 18 Pro Max, the Apple Watch Series 12 and Ultra 4, and potentially a new smart home hub dubbed the "HomePad." New AirPods with built-in cameras for AI-powered features are also rumored.
Ternus inherits the only major tech company without a frontier AI model of its own, making his strategic decisions on AI partnerships and on-device intelligence particularly consequential for Apple's next chapter.
Source: MacRumors, 9to5Mac, Al Jazeera
Sony and Warner Sue Anthropic in Multi-Billion-Dollar Copyright Battle
Sony Music Publishing and Warner Chappell Music have filed a 48-page complaint accusing Anthropic of "one of the largest and most blatant ongoing thefts of intellectual property in history." The two publishing giants allege that Anthropic engaged in a "brazen campaign of illegally torrenting, scraping and downloading copyrighted works on a massive scale" to train its Claude AI models.
The suit names approximately 20,000 copyrighted songs used without licensing or authorization, spanning works from The Beatles, Taylor Swift, Michael Jackson, and many more. Specific compositions cited include "Ain't No Mountain High Enough," "All I Want for Christmas Is You," "Eye of the Tiger," and "Paper Rings."
The publishers are seeking up to $150,000 per infringed work and $25,000 for each instance of removed copyright management information — potentially totaling billions of dollars. The complaint also references Anthropic's 2025 settlement of $1.5 billion with authors and publishers, which the plaintiffs cite as evidence that the company has continued infringing conduct despite prior legal consequences.
The lawsuit adds to mounting pressure on AI companies over training data practices and is one of the largest copyright actions targeting the AI industry to date.
Source: TechCrunch, Variety
TCS Commits $7.4 Billion to Build India's Largest AI Data Center Campus
Tata Consultancy Services subsidiary HyperVault has announced plans to invest up to 700 billion rupees ($7.4 billion) in a massive 1-gigawatt AI data center campus in Hyderabad, India. The project, which will be developed in phases across 264 acres, is expected to rank among India's largest AI infrastructure facilities at full buildout.
The campus is designed to serve AI companies and hyperscalers, supporting high-density GPU deployments for both training and inference workloads. TCS CEO K Krithivasan said Hyderabad offered "the scale, talent and ecosystem needed to serve global customers."
The investment follows a broader global trend of massive AI infrastructure spending. Earlier in the same week, Google committed up to $40 billion at a $350 billion valuation with 5GW of cloud capacity over five years, while Amazon doubled down with $25 billion in matching Trainium capacity. The race to build compute at scale shows no signs of slowing, as demand for AI training and inference capacity continues to outstrip supply.
HyperVault has said the project will leverage green energy and water-neutral design principles — an increasingly important consideration as the environmental footprint of AI data centers draws growing scrutiny.
Source: Bloomberg, The Next Web
Anthropic Ships Fable 5.1 with 75% Cache Cost Cut
Rounding out a momentous week for Anthropic, the company launched Claude Fable 5.1 on September 1 alongside its restricted-access twin, Mythos 5.1. Both models are generally available across the Claude API, AWS Bedrock, Google Cloud, and Microsoft Azure.
The headline pricing stays the same as Fable 5 — $10 per million input tokens and $50 per million output tokens — but the cache-read rate drops 75% to just $0.25 per million tokens, down from $1. For developers building applications with large context windows and repeated prompts, this represents a significant cost reduction.
On performance, Anthropic claims Fable 5.1 sets a new standard for coding and knowledge work, with a self-reported Terminal-Bench-Science score of 52.6, more than double Fable 5's 24.7. The model also features reduced false-positive safety restrictions, addressing a common developer complaint about overly cautious refusals.
Mythos 5.1, available only to vetted cybersecurity and life-sciences institutions, offers stronger capabilities in those domains that are normally constrained by production safeguards. The restricted release reflects growing industry caution around dual-use AI capabilities.
Source: TechCrunch, VentureBeat