What was already known
Ordinary Benjamini–Hochberg control was known under independence and positive regression dependence. Its validity for correlated two-sided Gaussian tests had remained conjectural.
AI involvement
Edgar Dobriban reports giving GPT-5.6 Pro the problem definition alone. The model produced the counterexample, analytic proof and certificate code; Dobriban checked and edited the result.
The source says the model produced the counterexample and proof from only the problem definition, with later human checking and editing.
Importance
Refutes the conjectured general FDR guarantee with a concrete asymptotic counterexample.
Limitations
The theorem concerns the specified dependence model and asymptotic N, not every Gaussian dependence structure. Dobriban’s complete review and the exact interval certificate are reported by the paper; neither was rerun here.
Technical verification
The author reports a complete check of the analytic argument and associated certificate, and the final strict inequality is certified by outward-rounded Arb interval arithmetic over the specified finite grids. These are source-reported checks.
Formal checking
Careful check of the entire argument and numerical certificate; source-reported.
- Checker
- Edgar Dobriban, paper author
- Scope
- Theorem 1's FDR > 0.0104 counterexample in the specified Gaussian factor model for all sufficiently large N
- Basis
- Reported by the linked checker
Outward-rounded ball arithmetic with rational inputs and strict interval comparisons; source-reported.
- Checker
- Arb interval arithmetic through the Appendix B Python certificate, as reported by Dobriban
- Scope
- Strict finite inequalities yielding the certified FDR lower bound 0.010416829... on the specified grids
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Dobriban explains why the opposed common-factor movements of the null and signal blocks create the failure, and lays out the threshold and certificate argument in Sections 1 and 3–6.
Correction and publication history
Refutes the conjectured general FDR guarantee with a concrete asymptotic counterexample.
Read this published revision