On 2026-11-06, Physical Superintelligence (PSI, @psipbc) said its internal research system, running non-public Anthropic model instances, produced a complete proof of the Aaronson–Ambainis conjecture (open since 2008) on day three, accepted by Lean, then improved the result twice. The research page and Lean 4 formalization repo are public. Vendor wording; no journal peer review yet; the README still asks humans to check that the formal statement matches the printed conjecture.
Key Takeaways
- ✓Sources: PSI research page 2026-11-06 + @psipbc original post + GitHub physical-superintelligence/aaronson-ambainis
- ✓Claim: complete proof on day three, accepted by Lean; two later improvements also Lean-accepted (vendor wording)
- ✓Implication: if the conjecture holds, quantum machines get at most polynomial speedups on random unstructured inputs; PSI now calls it a theorem
- ✓Checkability: public Lean 4.33/Mathlib repo; four main theorems with no sorry/extra axioms; README still leaves statement fidelity to human readers
- ✓Limits: Anthropic model unnamed, no journal peer review, paper prose not machine-checked — cite as vendor claim
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Core Background & Industry Pain Points
Open since 2008, the Aaronson–Ambainis conjecture asks whether quantum computers can be exponentially faster than classical ones on random, unstructured inputs. On 2026-11-06, PSI said its internal research system, using non-public Anthropic model instances, found a complete proof on day three that Lean accepted. The research page and Lean repo went public the same day.
Architecture Highlights & Internals
PSI says the run used its own infrastructure. The proof follows a Lovett–Zhang plan, replaces one conjectured inequality, and improved the result twice. Public artifacts include a paper and a Lean 4.33 / Mathlib formalization (Apache 2.0). David Turturean led the system and paper correctness; advisor Jim Halverson audited the proof.
Authoritative Benchmarks & Measured Scores
Vendor wording: Lean accepted the proof; leanchecker / lean4lean kernel replays are documented. No journal peer review yet. The README still asks humans to confirm PrintedConjecture matches the printed conjecture. The Anthropic model is unnamed, so only the output is checkable. PSI quotes Stanford special-case authors calling it a "huge result" (relayed wording).
Developer Hands-on Guide
Read https://psi.inc/research/aaronson-ambainis/ and github.com/physical-superintelligence/aaronson-ambainis; follow README make check / make replay. Cite as vendor claim without peer review, and separate "Lean accepted" from "accepted by the math community."
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