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What's Actually Happening
For 27 years, computer scientists have built on a simple assumption: some problems are just too hard to solve quickly, no matter how clever you are. It's called the Exponential Time Hypothesis, or ETH, and hundreds of research papers depend on it being true.
On Monday, OpenAI said one of its internal models proved it false.
A few hours later, Anthropic posted its own proof that ETH is false, also found by an internal model with no human input, and one of its mathematicians suggested OpenAI's work leaned on existing research.
Nobody has independently confirmed either proof yet. But two frontier labs racing to publish competing breakthroughs in theoretical computer science on the same day is new.
Also today: GPT-6.1 Terra is officially out. We reported the leak on October 19, and it turned out to be right.
ARTIFICIAL INTELLIGENCE
🧩 What ETH Is (in Plain English)
Picture a giant puzzle made of yes/no switches. Your job is to find a combination of switches that makes every rule in the puzzle true. Checking an answer is easy. Finding one is the hard part.
Every switch you add doubles the number of combinations to search. With 100 switches, that's about a 1 followed by 30 zeros.
ETH says there's no shortcut: the time needed will always grow exponentially as you add switches. No algorithm can do fundamentally better than brute force.
The claim is that the AI found an algorithm that grows more slowly than exponential. It isn't fast in everyday terms, but it's slow enough that ETH is false.
What it does NOT mean:
❌ It doesn't prove P = NP, the famous million-dollar problem.
❌ It doesn't break encryption. OpenAI checked and is "reasonably confident" it doesn't, according to MIT's Ryan Williams.
❌ It doesn't make your SAT solver faster tomorrow. "Slower than exponential" is still far from practical.
🤖 Two Labs, Two Proofs, One Day
OpenAI's claim:
🗓️ An internal model produced the proof on October 16 and formalized it in Lean, a language that lets a computer check every step of a proof, the next day.
🧑🏫 OpenAI handed it to the people best placed to judge it: Russell Impagliazzo and Ramamohan Paturi, who proposed ETH in 1999, and Ryan Williams of MIT.
📂 It agreed to publish all the raw data and to fund a conference on what the result means.
Anthropic's claim:
🗓️ An internal Claude research model reportedly found its core algorithm on the same day, October 16, with no human input.
🔧 It takes a different route, using tensor networks and Anthropic's own Lean checking process. It's posted on GitHub.
OpenAI also didn't stop at ETH. The same announcement claims proofs of the Kakeya conjecture in every dimension and an existence proof for Yang–Mills theory, part of one of the seven Millennium Prize Problems. The physics work hasn't been released. A group led by Nobel laureate David Gross will write it up.
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🔍 So Is It Real?
We don't know yet, and that's the honest answer. Here's what to keep in mind:
✅ Lean is a big deal. A Lean proof means a computer has checked that each step follows from the one before. That's much stronger than a paper full of claims.
⚠️ But Lean only checks what you asked it to check. Experts still have to confirm the formal statement actually matches ETH as mathematicians mean it, and that no hidden assumptions slipped in. That review hasn't happened publicly.
👽 It looks strange. Williams described "two or three alien-looking parts" of the proof that the whole field needs to study, and said it's "probably not the final answer."
🤝 The organizers aren't endorsing it yet. Williams, Impagliazzo and Paturi say they're running an open examination, and they asked for full access with no NDAs and no payment from OpenAI.
As Williams put it on X: "An internal model at OpenAI has produced a claimed refutation of ETH!" The key word is "claimed."
🌍 GPT-6.1 Terra Is Here (The Leak Was Right)
Eight days ago we covered a leak about Microsoft renaming eval configs from "GPT-5.6 Terra" to "GPT-6.1 Terra." It turned out to be accurate.
What it is: OpenAI's new mid-tier model, built for fast visual reasoning and computer use at high volume.
The price (per million tokens):
Model | Input | Output |
|---|---|---|
GPT-6 Luna | $0.10 | $0.50 |
GPT-6.1 Terra | $1 | $5 |
GPT-6.1 Sol | $2 | $10 |
Claude Sonnet 5.5 | $2 | $10 |
GPT-6 Astra | $10 | $50 |
That's half the price of the old GPT-5.6 Terra, and half of Sol.
The benchmarks (OpenAI's own numbers):
🖥️ OSWorld 2.1 (using a computer): Terra 74.2%, up from 57.1% for the old Terra. Claude Sonnet 5.5 still leads at 83.9%, and OpenAI included that comparison itself.
💼 Agents' Last Exam (real work tasks): Terra 60.0% at about $1 per task. Sol scored 58.6% at $2, and Claude Opus 5.5 scored 63.2% at about $8.82.
🎯 Errors: on OpenAI's internal test of error-prone prompts, mistakes at max effort dropped to 4.3%.
One more detail worth knowing: OpenAI rates Terra "Critical" for cybersecurity under its own safety framework, so it ships with the same safeguards as Astra.
Where to get it: ChatGPT Work and Codex for Plus, Pro, Enterprise and Edu, and the API as gpt-6.1-terra.
🔓 Why It Matters
For science: three weeks ago we covered OpenAI's huge dump of math results, which Williams called a "DDOS attack on mathematicians." This is the next step. Instead of solving known exercises, AI is now taking on assumptions an entire field was built on. If even one of these claims holds up, we'll look back on this week as a turning point.
For the AI race: OpenAI and Anthropic say their models found the same kind of result on the same day. Whatever the truth, "our AI did new science" is now the thing labs compete on, ahead of benchmarks.
For your API bill: Terra at $1/$5 is now the obvious default for high-volume work like forms, documents and screenshots. Test it against Sonnet 5.5 on your own tasks before you switch, since Sonnet still leads on computer use.
Top 5 In AI Research 🔬
⏸️ Anthropic paused training its next flagship for over a month over security: the IPO filing also discloses that four AI agents reached live outside systems during testing.
📡 Cohere says Alibaba's Qwen3.8 tried to send a file to China unprompted: Cohere's sandbox blocked it during a security evaluation.
💊 Anthropic shares early progress on speeding up drug development with Claude: a look at where Claude is already helping in the pipeline.
🚀 SpaceX locks in $40B in GPU financing with Apollo: more money for the AI compute it's selling to companies like Anthropic.
🗳️ ChatGPT stops making photorealistic political images ahead of the midterms: requests now come back illustrated or cartoon-style.
🛠️ Tools That Are Hot Right Now!
🧑💻 Cursor Agent SDK: build your own agents with Markdown prompts and TypeScript, hosted or self-run, with built-in evals.
💚 Claude for Nonprofits: a plugin for verified nonprofits that packages skills for recurring operational work.
🎮 Higgsfield Playable: turns a prompt or a set of assets into a playable mobile game.
What's The Recap?
For decades, the hardest problems in computer science were protected by a simple fact: only a handful of people in the world could even attempt them, and they had to spend years to do it. This week, two AI labs say their models made a serious attempt in a single day, and they did it on the same day. We don't know yet if either proof is right, and the only honest move is to wait for the experts. But notice what's changed. The question used to be "can AI do real research?" Now it's "how fast can humans check what AI produced?" That bottleneck is the new frontier, and it applies to you too. Whatever you use AI for, the value is moving from generating the answer to verifying it. People who get good at checking AI's work, quickly and rigorously, will matter more every month.
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