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The Generalized Vanishing Conjecture

Zhao's Generalized Vanishing Conjecture asks whether, for a differential operator with constant coefficients, Λm(Pm)=0\Lambda^m(P^m) = 0 for all large mm forces Λm(PmQ)=0\Lambda^m(P^m Q) = 0 for all large mm.

Result
Disproved(see note)
Status
Resolved
AI contribution
AI-assisted
Method
Construction
Field
Commutative algebra; differential operators
Posed by
Wenhua Zhao
Year posed
2007
Years open
19y
Solved
2026-08-05
Model
GPT-5.6 Sol
Vendor
OpenAI
Collaborators
Alexander Dvorsky, Roy van Rijn
Verification
Lean-verified
Publication
Preprint
Significance
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For n <= 2, we proof that the Generalized Vanishing Conjecture holds, and we have a failure at dimension 3 (including a counter-example).

This improves the original failure at dimension 5.

What the AI did

The initial paper (Alexander Dvorsky) used "as an interactive research assistant": the paper credits it with testing the homogenized five-variable construction, generating computational verification code and locating relevant literature, with all computations and references independently verified afterwards.

A second paper (Roy van Rijn) used OpenAI's Codex to close the conjecture completely. It proofs that the conjecture holds for n <= 2 and fails at dimension 3 (counter-example provided in the paper). Codex monitored/ran several experiments and the human steered the experiments.

Verification

The preprints have (as of yet) no independent review. The counterexample can be checked directly, the proof has additional Python calculations and Lean verification.

Sources

Changelog5 changes
  • Roy van Rijncleared Claim issue
  • Roy van Rijnchanged What the AI did from “The initial paper (Alexander Dvorsky) used "as an interactive research assistant": the pap…” to “The initial paper (Alexander Dvorsky) used "as an interactive research assistant": the pap…”
  • Roy van Rijncommented
  • Roy van Rijnchanged Solve date from “2026-08-07” to “2026-08-05”
  • CosmicWombat329changed Verification note from “A preprints have (as of yet) no independent review. The counterexample can be checked dire…” to “The preprints have (as of yet) no independent review. The counterexample can be checked di…”, also More links, Source name, What was actually shown, Collaborators, More links, Verification, Statement, What the AI did, Source URL, Verification note, Claim issue, Source name

Discussion1

Roy van Rijn01 Sep 2026, 09:25 UTC· edited

I've added my research (which moves the conjecture to fully closed):
https://zenodo.org/records/21803325

It has a (Lean verified) proof that the conjecture holds for n <=2, and we show a smallest failure at dimension 3.

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