What was already known
Ramachandra and Natarajan conjectured the 4/3 bound in 2025. It was proved for two and three elements and for some special functions and marginals.
AI involvement
The authors report one prove-or-disprove prompt to GPT-5.5 Pro, which returned the example. Ramachandra and Natarajan validated it and wrote exact primal and dual certificates.
The first-person account describes one task-setting prompt and an AI-returned complete counterexample, with human validation only after discovery.
Importance
The exact finite example disproves the proposed universal bound for nonnegative monotone submodular functions.
Limitations
The paper extends the failure to every size at least five; the four-element case remains open. The model searched with numerical linear programs, while the published refutation uses rational certificates.
Technical verification
The authors explicitly report validating the model output and provide complete exact rational primal/dual certificates for all five-element subsets. This is source-reported human checking of the finite counterexample, not a machine check of the final certificate.
Formal checking
Validation of model output using a feasible primal distribution and rational dual inequalities tabulated for all 32 subsets
- Checker
- Arjun Ramachandra and Karthik Natarajan, authors
- Scope
- The five-element coverage-function ratio is at least 640/479 > 4/3
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors explain the coverage function, marginals, primal distribution and dual upper bound that yield the strict ratio.
Correction and publication history
The exact finite example disproves the proposed universal bound for nonnegative monotone submodular functions.
Read this published revision