Algebra·OpenAI Math·Result 194·Lean formalisation listed
Lech’s multiplicity conjecture
One of the 372 results in OpenAI's public mathematics catalogue, filed under algebra: 1 preprint, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a proof of Lech’s multiplicity conjecture (algebra) in a preprint dated 23 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More, is on YouTube.
Chapter 32 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 10:45 · watch from there on YouTube →
What OpenAI claims
Proves e(R) ≤ e(S) for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic. (Lean)
Paper
Lech's multiplicity conjecture
OpenAI's abstract We prove Lech's multiplicity conjecture: Hilbert–Samuel multiplicity cannot decrease under a flat local homomorphism of nonzero Noetherian local rings. The result holds in arbitrary dimension, with no restrictions on the residue fields or characteristics.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Proves e(R) ≤ e(S) for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic. (Lean)
- Where is the proof?
- In the paper, Lech's multiplicity conjecture (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 32).
Also in
- Chapter 2 of 10 in Serre's Multiplicity Conjecture, Kaplansky Zero-Divisor Conjecture and 8 More · starts at 0:22 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · Compilation · 29:10 · chapter 32 from 10:45 · description, sources and chapters →
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