A new paper says a famous limit on computing is wrong. What would it take to believe it?
In October 2026, OpenAI posted a paper claiming to break one of computer science’s long-standing “speed limits.” This page starts with what that means in plain terms and goes deeper as you scroll. Stop wherever you like. If you make it to the bottom, there’s something there for you.
Start with the plain version ↓ Read the paper (PDF) ↗A puzzle, a speed limit, and a claim.
Flip the switches until every rule is green.
Each rule needs at least one of its three conditions to be true. Tap a rule to see which switches it watches.
This puzzle is a Boolean formula. See it as code.
# each switch is a variable: True (on) or False (off) x1, x2, x3, x4, x5, x6, x7, x8 = ... # each rule is an OR; "every rule green" is an AND rules = ((x1 or x2 or not x3) and (not x1 or x3) and (x4 or not x2 or x3) and (not x4 or x1 or x2) and (x5 or x6 or x7) and (not x6 or not x8) and (not x5 or x8 or x7) and (not x4 or x6 or x8) and (x4 or not x7 or x5)) # SAT: is there ANY setting that makes rules True? # #SAT: HOW MANY settings do? ← what the paper counts # always computable: try all 2**8 = 256. the question is how fast.
Why this puzzle matters: any program whose answer you can check quickly can be rewritten as one of these (the Cook–Levin theorem). So a faster way to solve or count SAT is a faster way to do a lot of things.
How many of the 256 settings make every rule green?
Pin switch 4, and the puzzle breaks in two.
Every rule that touches switches 1–3 also stays away from 5–8. Switch 4 is the only thing holding the two halves together.
That’s the whole idea, small. Counting every setting took 256 checks. Pinning one switch and counting each piece separately took 48, and got the same answer. Real puzzles rarely split this cleanly, which is why people bet in 2001 that no trick always works (ETH). The paper claims an algebraic version of this trick always does. That’s the claim the rest of this page checks.
Enjoyed figuring that out? That’s what our classes feel like: real questions, worked out by playing with them, with an AI as your lab partner. No math background needed.
See the classes ↓Two strands, growing toward one question.
Solvers have always found shortcuts in sub-problems on many puzzles. The paper claims the shortcut is always there. Its first step is a branching split (sparsification), and its engine is the algebra strand (§1.1.4, §1.1.6).
What it says. What it says it doesn’t.
Math is minified code, so each claim comes with its translation. Check them against the PDF.
- A preprint, credited only to “OpenAI”. Not peer-reviewed.
- No implementation, benchmark, or formal proof is reported in the paper.
- One corollary’s context cites another OpenAI preprint from September 2026, which is also unreviewed.
- We have not found an independent check of the proof. If you find one, that’s new evidence: update.
No shortcut escapes doubling.
# for EVERY algorithm, some puzzle forces steps(n) >= 2 ** (c * n) # some c > 0
A conjecture, never proved. Whole fields built “this is probably optimal” results on it.
Paper p.4 · [IP01]Counting, with a shrinking cost per switch.
L = log(n)
steps(n) = 2 ** (c * n * log(L) / sqrt(L))
# c depends on clause width k. Not given.Exact counts, deterministic, small memory. If true it also topples SETH, Gap-ETH, NSETH, and #ETH.
Paper p.2, p.4–5What it says it doesn’t do.
- ✕ P = NP. Shrinking isn’t polynomial.
- ✕ Free speedups for all of NP. The gain gets lost when you translate other problems into SAT (p.2).
- ✕ Unlimited rule size. Only for rules of a fixed size (p.5).
- ✕ Every hardness result. Some still stand on other assumptions (p.5).
“Subexponential” is not the same as “fast.”
The claimed cost has a hidden constant c that the paper never gives. Guess first, then move it.
On a 1,000-switch puzzle, is the new method faster than the best one we already had?
Hint: the headline says it beats the speed limit.
# what the chart plots def doublings_per_switch(n, c): L = log(n) return c * log(L) / sqrt(L) # brute force: 1.0
Natural logs, as in the paper. A polynomial factor is left out, and the paper doesn’t give it either. The sparse curve covers puzzles with a bounded number of rules per switch, which includes the hardest random ones.
Guess what the paper says, then see where it says it.
Each node is one claim, with a question about it. No math needed: every question can be guessed from common sense, and a wrong guess still gets you the answer. A full node means you’ve guessed it. A bright one means you called it. Speculation stays speculation however many arrows point at it.
What if it’s right? Where could it lead?
Everything in this section is speculation. That’s not an insult. Research directions usually start as stories about what might be true, and they become research when someone writes down the first step precisely enough to fail. Each rung and card below tries to do that: what it would take, the closest real precedent, and what would tell you it’s a dead end.
From here to P = NP, one rung at a time
Six places to point an agent
Six traps you can now spot.
Headlines like this one are built to trip everyone, researchers included. This page’s own first draft fell into several. Tick the ones that would have caught you last week, then add one we missed.
You just did research. Vibe Research is the class.
You guessed before you looked. You checked a number against the thing that produced it, sorted claims by their evidence, and learned to spot the traps. Vibe Research is a cohort that takes one question all the way through a research cycle with your own agent.
See Vibe Research →Three more ways to use what you just practised.
Each is a one-day intensive. Take them on their own, or before Vibe Research.
Critical Thinking and Creativity with AI
Use critical thinking to beat AI brain-drain, then put AI to work as a co-pilot, never a replacement. Set up your own assistant, make it interview you, and use it to reverse-engineer what makes great writing work. The AI does the labor; you keep the judgment.
See the class → You checked a number at its sourceData Activism: Turn Data Into Power
Go from “this is wrong” to a specific, data-backed ask delivered to the person who decides. Pull real public datasets and run real statistics without writing code, then package it into a one-pager and an honest infographic. Bring a number that can’t be spun away.
See the class → You read a title as a claimPractical Propaganda
“The Exponential Time Hypothesis is False” is a headline built to travel. Learn how persuasion is made, so you can spot it and make your own for causes you care about: persona targeting, AI-assisted campaign design, automated creative production. Bring your ideas.
See the class →