OpenAI Math Breakthrough, Anthropic IPO, and Reasoning Model Tutorial

OpenAI Releases Three Fundamental Science Results in a Row, Math Conjecture Proof Sparks Controversy

OpenAI recently announced research results in three fields—mathematics, computer science, and physics—covering harmonic analysis, computational complexity, and quantum field theory. The company candidly admitted that these manuscripts are "more like raw research data" and require scrutiny by external experts before they can be transformed into scientific understanding.

The math result targets the local smoothing conjecture proposed in 1991, which concerns the focusing of waves and posits that the most extreme focusing can only be brief. OpenAI claims its model produced a proof in all dimensions, and on that basis resolved the Kakeya conjecture and the Stein restriction conjecture, with the proof resting on a new auxiliary partial differential equation. The AI output was generated on October 18 and formalized in Lean the next day.

The computer science result is more radical: OpenAI says its system proved that the Exponential Time Hypothesis (ETH) is false. ETH is a long-standing conjecture about the computational complexity of Boolean formula satisfiability that is widely believed to hold, and hundreds of papers use it to derive hardness conclusions. The proof was found on October 16 and formalization-verified the next day. OpenAI stressed that much follow-up research is still needed to obtain faster algorithms, and that this work will be overseen by three algorithms and complexity experts, two of whom are the original proposers of ETH.

The physics result concerns the rigorous mathematical foundations of quantum field theory. OpenAI says the model provided existence proofs for certain interacting quantum field theories, including quantum chromodynamics and a unified completion of the Standard Model, with the existence proof for pure Yang-Mills theory as a sub-problem—this falls under the Clay Mathematics Institute's Millennium Prize Problems. An expert panel led by a Nobel laureate will oversee it, and it has decided not to release the AI artifacts for now, writing its own explanation first.

The three results adopted three different release strategies: the math result released the AI output directly, the computer science result was released with an expert panel formed, and for the physics result the panel decides whether the materials are released. The collaboration follows principles of transparency, not seeking academic credit, recognizing contributions, and voluntary and unpaid participation by independent participants.

It should be noted that this conclusion is still under expert review and community scrutiny. OpenAI pledged to release all raw data for independent community verification and agreed to fund a conference. The natural-language proofs and Lean proofs have been published in a public GitHub repository.

The Exponential Time Hypothesis (ETH) is a core conjecture in computational complexity theory about the Boolean formula satisfiability problem (SAT), systematically proposed by Impagliazzo and Paturi in 1999. It asserts that there is no algorithm that can solve general $n$-variable SAT in $O(2^{\delta n})$ time (for some fixed $\delta < 1$), i.e., that in the worst case SAT cannot be significantly faster than brute-force enumeration. ETH is stronger than P ≠ NP and is the cornerstone of "fine-grained complexity"; a large number of papers use it as a condition to derive exact lower bounds for various problems (such as edit distance and orthogonal vectors). If ETH were overturned, it would mean an exponential speedup algorithm exists, which would cascade to overturn hundreds of conditional conclusions—so if OpenAI's claim holds, its impact would be extremely broad.

Lean is an interactive theorem prover and functional programming language led by Microsoft Research, based on dependent type theory (a variant of the Calculus of Constructions). Proofs written by users in Lean are checked line by line by the kernel; as long as they pass, logical correctness is guaranteed (provided the axioms are trustworthy). In recent years, Lean and the mathlib community have driven the "formalized mathematics" movement, actively advocated by Terence Tao and others. The article's statement that formalization was "completed the next day" is significant: natural-language proofs may have gaps, but Lean verification provides a machine-checkable certificate of correctness—though it only guarantees that the chain of reasoning is sound, not that the statement itself is exactly the conjecture people intended to prove. This is also one of the focal points of controversy.

Anthropic Files for IPO: Losses Narrow but Governance Structure Raises Concerns

According to TechCrunch, Anthropic has officially filed for an IPO, planning to list under the ticker NT. The filing discloses details of its finances and business.

Revenue in the first half of 2026 was $16.3 billion, compared with $1.2 billion in the same period last year. Operating loss was $693 million, down from $2.1 billion a year earlier. Total losses were $80.7 billion, mainly from non-cash adjustments related to convertible notes. The company says it may not achieve or maintain profitability in the future, and the size of the offering and issue price have not been disclosed.

On enterprise business scale, as of the end of September, more than 11,900 customers spent over $100,000 annually, more than 1,900 spent over $1 million annually, and about 250 spent over $10 million annually. Net revenue retention at the end of Q3 was about 900%. Among customers spending over $100,000 annually, more than half use both the platform and first-party applications, compared with about 30% at the end of last year.

Anthropic relies on AWS, Google Cloud, and Microsoft for co-selling, contracting directly with customers and paying fees to the cloud providers. But the filing acknowledges that these cloud partners increasingly see it as a competitor, which could make it harder to obtain the compute it needs—Amazon, Google, and Microsoft may prioritize their own needs. The filing also flags the risk of customer concentration.

The governance structure is also a focus of attention: seven co-founders form a Founder LLC holding 50.1% of voting power, which the company says is to protect its AI safety mission. A long-term benefit trust established in 2023 has the right to appoint a majority of directors. The report says this means voting power is disproportionate to the founders' economic ownership, and public shareholders will have very limited actual control.

SpaceX Secures Massive GPU Financing; AI Business Revenue Exceeds Half of Total Revenue

SpaceX disclosed a financing commitment of about $40 billion, with funds used for AI chips and data center equipment. TechCrunch calls it the largest GPU financing commitment.

An entity managed by Apollo buys the chips and servers and leases them to SpaceX for about 5 to 5.25 years, starting from the delivery of each batch of equipment. The deal is expected to close before the end of the year, with the lease starting in the first half of 2027. Apollo says this will help deliver Nvidia's next-generation Vera Rubin chips.

In Q3, SpaceX's AI business revenue was $7.6 billion, more than half of total revenue of $14.6 billion. The business includes the Grok chatbot, the Cursor programming tool, and cloud compute. Capital expenditure was nearly $20 billion, compared with $6.9 billion a year earlier. The CFO said Q4 spending is expected to be $35 billion to $40 billion, and that the payback period for AI infrastructure investment is less than a year. Musk said on X that more than 200 new cloud providers will face elimination and consolidation.

Open-Source Models Fall Short on Tool-Calling Accuracy; Constrained Decoding Is the Solution

In an engineering article, cloud provider Ubicloud found that open-source large models perform far below benchmark scores on actual tool calling. They deployed Kimi K2.6 with vLLM version 0.28 and found that only 92% of calls had the correct format—meaning two failures out of every 25 calls. Failed calls pollute the context with erroneous information and increase costs, which is especially pronounced in multi-turn agent scenarios.

The solution is constrained decoding: guaranteeing correct format from the generation stage. In version 0.31, vLLM made it a server-side switch; the engine masks illegal tokens at each model stage, allowing only compliant output. The article says this turns accuracy from "best effort" into a "runtime guarantee," and that after precomputing masks the additional overhead is negligible.

However, vLLM later changed the default behavior to off, which dropped Ubicloud's accuracy back to 91%. The reason is that vLLM needs to be compatible with a large number of models, and some models error out when it is enabled—for example, grammar conversion bugs and conflicts with speculative decoding. Ubicloud focuses on serving Kimi and DeepSeek^Ad, so it actively enables it, and in tests tool-calling accuracy rose from 92% to 100%.

Constrained decoding is a class of techniques that force output during generation to satisfy a given formal grammar (such as JSON Schema, regular expressions, or context-free grammars). A typical approach: at each decoding step, a precomputed token mask blocks tokens that would lead to an illegal prefix, and sampling occurs only within the legal set. Common implementations include Outlines, Guidance, XGrammar, and vLLM's guided decoding. The cost is that it may distort the model's probability distribution and reduce semantic quality, and it may conflict with optimizations such as speculative decoding and parallel sampling; masks need to be precomputed and cached for the vocabulary, hence the article's claim that the additional overhead is negligible.

8 Videos, 11 Hours: Build a Reasoning Model from Scratch Tutorial

A well-known blogger released a video course on building a reasoning model from scratch—8 videos, over 11 hours, covering evaluation, inference-time scaling, reinforcement learning, and distillation, all implemented in Python and PyTorch. The course starts from a pre-trained small Qwen 3 model and is accompanied by a new book of the same name.

The blogger believes a reasoning model is a regular large model with reinforcement learning post-training added, trained with verifiable rewards, and can also be combined with inference-time scaling techniques. He compares the reasoning model to an engine, while the agent shell provides tools and context management. He says most current large models are themselves reasoning models or come with a reasoning mode.

On hardware requirements, the early text generation and evaluation examples can run on a CPU, while later training experiments are best done with a GPU. For developers who want to deeply understand the internal mechanisms of reasoning models, this course offers a complete path from theory to code.

One-line Summary

OpenAI claims to have disproved ETH and two other scientific results pending review, plus Anthropic IPO, SpaceX GPU financing, and more.

Paragraph Summary

OpenAI announced results in math, computer science, and physics: claiming a full-dimensional proof of the local smoothing conjecture and resolving the Kakeya and Stein restriction conjectures, proving the Exponential Time Hypothesis (ETH) false, and providing existence proofs for interacting quantum field theories such as QCD, touching on Millennium Prize Problems. The three results used different release strategies, all require expert scrutiny, and remain under review and community scrutiny. Anthropic filed for an IPO with H1 2026 revenue of $16.3 billion and narrowing losses, but a founder LLC holding 50.1% of voting power raises governance concerns. SpaceX secured about $40 billion in GPU financing commitments, and Q3 AI business revenue of $7.6 billion exceeded half of total revenue. Ubicloud found open-source models achieved only 92% correct tool-call format, which constrained decoding can raise to 100%. An 8-video, 11-plus-hour reasoning model building course was also released.