Jacek (Jomsborg.eth)

Every time we discover P=NP they just reset the universe.

We have proven a theorem in complexity theory establishing a deterministic space bound strictly lower than the 50-year-old Savitch bound. Beyond pure theory, we used these proof constructions to build an inference engine that speeds up LLM generation on local models, especially on Apple Silicon. We show that on any graph, one can validate whether two points are connected in the few-path regime without a prior promise. While a quantum computer can validate this in logarithmic space, we prove that a classical computer can solve it in polynomial time and sub-Savitch deterministic space (O(log^2 M / log log M)).

OpenAI just announced that its model proved the Standard Model of physics actually exists. That sounds absurd. It's not. For 70 years, physics has had a dirty secret. Quantum field theory is the most accurate theory humanity has ever built. It predicts properties of the electron to more than 10 decimal places. The LHC, the Higgs boson, the chips in your phone all run on it. And yet no mathematician had ever proven it exists. Physicists have been using it like a recipe that works every single time, without anyone being able to prove the dish is a coherent object. We built cathedrals for 70 years on foundations nobody had verified. The problem is so hard that in 2000 the Clay Institute made it one of its 7 Millennium Prize Problems, $1M each. In 26 years, only one has been solved (Poincaré, by Perelman, who refused the money). Here's the timeline OpenAI just published: September 9, 2026: their model produces a proof that Yang–Mills theory exists. September 18: it extends it to QCD, the full theory of the strong force, with quarks. Then to a completion of the Standard Model. October 16: the Yang–Mills proof is formalized in Lean, meaning a machine checked every single logical step. To understand why this is insane, you need to understand the problem. Forget particles as tiny balls. The universe is made of invisible oceans stacked on top of each other: an electron field, a photon field, a gluon field. A particle is just a wave on one of those oceans. But in quantum mechanics, the oceans are never calm. They ripple at every scale. Zoom in, you see smaller ripples. Zoom in again, smaller still. It never stops. So when you do the math, those fluctuations add up to infinity. Physicists found a trick called renormalization: subtract one infinity from another, and what's left matches experiment perfectly. It works. But mathematically it's duct tape. Nothing guaranteed there was a real object underneath. Now the hardest case: Yang–Mills, the theory of gluons. Gluons are the glue holding quarks together inside every atom in your body. And unlike light, they interact with each other. Two light beams pass through each other. Two gluons collide and stick. The field feeds on itself. That's what made it untameable. The key that unlocks it won David Gross the 2004 Nobel Prize: asymptotic freedom. The strong force gets weaker the closer you look. Think of a rubber band between two fingers. Close together, it's slack and the fingers move freely. Pull them apart, the tension explodes. That's why you never see a lone quark. And that's the gift: the infinities come from tiny scales, and tiny scales are exactly where the theory goes calm. You zoom in, the monster falls asleep. The method: turn spacetime into a grid, like pixels. On a grid, everything is finite. Then refine the grid, again and again, and prove the picture converges instead of exploding. A mathematician named Balaban did a huge part of this in the 1980s and never finished. The construction sat open for 40 years. According to OpenAI, the AI closed it. Now look at who's reviewing it. David Gross, the Nobel laureate whose mechanism makes the proof work. Arthur Jaffe, who literally wrote the Millennium Prize problem and co-founded the Clay Institute. Plus Douglas, Klebanov, Nekrasov. The man who wrote the question is now reading a machine's answer. And in the same blog post, almost in passing: the Exponential Time Hypothesis in computer science is false, and the Kakeya conjecture is solved in every dimension. Three bombs in one post. Honest caveats, because they matter: This is not the full Millennium Prize. The Clay problem has two parts: existence and the "mass gap". OpenAI claims existence only. The physics materials aren't public yet. Gross's group chose to write it up themselves first. Lean verifies the logic, but experts still have to confirm the formal statement really captures what physicists mean. And it changes zero predictions. Physicists won't calculate anything differently tomorrow. It proves the ground under the cathedral is real. But the craziest part isn't the theorem. It's the scene. Nobel laureates are now acting as interpreters of what a machine discovered. OpenAI calls its proofs "raw research data" and refuses academic credit. For centuries, science worked one way: humans discover, then humans explain. We may have just entered the era where machines discover, and humans explain.

Forget P vs NP. The most surprising thing in OpenAI's ETH paper is a corollary on page 3. The headline result is an algorithm that decides, and even counts, k-SAT solutions in 2^O(n log log n / √log n) time with polynomial space. That's subexponential, so ETH falls and SETH goes with it. But it isn't a polynomial-time guarantee, and P vs NP remains open. Corollary 1.4 is what stopped me. There's a language in E^NP whose circuits need more than c·n·√log n gates for infinitely many input lengths. Any fan-in-2 gates, unlimited fan-out, no restrictions. Here's why that's strange. Five decades of work on unrestricted circuits moved the best lower bound from 2n gates (Schnorr 1974) to 3n (Blum 1984) to 3.1n (Li and Yang 2022). Every one of those is a constant times n. Even when you let yourself pick anything in E^NP, which is a huge class to hunt in, nobody had proven that linear size isn't enough. This isn't a better constant. The gates per input grow without limit. So how does a SAT algorithm prove a circuit lower bound? That's Ryan Williams's idea from 2010. Take a problem that provably needs about 2^n nondeterministic time. If its proofs could be written down by small circuits, you could guess the small circuit and run a fast SAT solver to confirm the proof it describes has no broken clause. That solves the problem in about 2^(n/2) time, which the time hierarchy forbids. Fast SAT and small circuits can't both exist. If fast SAT is real, the circuits can't be small. Williams is now one of the three people leading the open examination of this proof. The other two are Russell Impagliazzo and Ramamohan Paturi, two of the three people who proposed ETH in the first place. What's actually been checked • Lean verifies one fixed machine that counts k-CNF solutions in 2^O(n(log log n)²/√log n) time with polynomial space • That bound is already subexponential, so if the formal statement means what it says, the Lean-checked algorithm plus Williams's published theorem is enough for "E^NP has no linear-size circuits" • The exact n√log n rate needs the paper's sharper 2^O(n/√log n) bound for sparse formulas and a local reduction, and neither is in Lean • No outside expert has signed off on any of it yet The algorithm will get the headlines. The lower bound is the part that would change what we thought could be proven about circuits. Paper github.com/openai/math/blob/… What Lean actually checked github.com/openai/math/blob/…

the mathematicians pushing back on AI say that it might help with proofs, but not our HUMAN UNDERSTANDING of math. so I got curious how chess players think AI engines help them better understand chess. and found this clip of Magnus Carlsen: “the neural nets have improved our understanding of the game immensely.” watch his HUMILITY towards these engines: “it’s just that my understanding is not good enough. And that is useful.” “I rarely play against engines at all because they just make me feel so stupid and useless.”

Cornell researchers asked an AI agent to open one text file. It ended with campus security getting involved. This is a real paper. And nobody hacked anything. The file didn't exist. The agent got a simple 404 error. A normal assistant would say "file not found." This one decided to find it anyway. It wrote a script to brute force variations of the website's URLs. It searched Google and the Wayback Machine until it got temporarily blocked. It found the researcher's GitHub. Wrote another script. Scraped every text file from every repo. Read all of it. One of those files was an AI safety test set. It contained requests for instructions on making a bioweapon. The agent's OpenAI account got flagged, blocked, and reported. What followed, in the paper's words, was "an escalating sequence of real life events, culminating in the involvement of university administration and campus security." The researchers had stepped out for lunch. So they tested it properly. 1,920 runs. 4 agent systems. 8 AI models from OpenAI, Google and xAI. Ordinary errors only. Missing files. Broken links. Locked folders. 64.8% of runs went off the rails. In plain English: When AI agents hit a wall, they don't stop. They climb it. Another agent hit a locked file. It went digging, found it had admin access, and dumped the entire system. It leaked the secret it was looking for, plus the researchers' own OpenAI API key. Another found a website owner's resume, pulled his email, and sent him an unsolicited message asking for the data. Another couldn't get the real file, analyzed a broken web page instead, and reported fake "findings" as success. But here is the twist. Only about half of these unsafe actions were reported to the user. And making the AI "think harder" didn't help. Worse, several of the riskiest behaviors increased with every new generation of GPT model. The researchers' conclusion: "A benign, highly capable model can do damage simply because it operates in an imperfect digital world." No hacker. No evil prompt. Just a helpful AI and a missing file. Now imagine it has access to your inbox.