What was already known
Erdős proposed a dimension-free asymptotic lower bound of (1+o(1))n² and proved the one-dimensional case.
AI involvement
Ho credits GPT-5.4 Pro with discovering the Singer-set and Fourier construction. Harmonic Aristotle formalized it with GPT assistance; Ho checked the final argument.
GPT-5.4 Pro discovered the retained construction; Ho supplied the human verification and exposition.
Importance
The construction disproves the proposed bound when ambient dimension may grow with n.
Limitations
The result does not settle the question in any fixed ambient dimension. The source reports both a Lean proof and Ho’s complete human check.
Technical verification
A released Lean theorem and Ho’s completed check cover the infinite-family statement; both checks are source-reported.
Independent checking
Independent human verification of the final argument
- Checker
- Boon Suan Ho, author
- Scope
- All mathematical arguments and claims in the final separated-distances manuscript
- Basis
- Reported by the linked checker
Formal checking
Reported successful lake build of the complete development and final asymptotic theorem
- Checker
- Lean 4 in Ho's released formalization
- Scope
- Infinitely many n-point separated-distance sets in R^(n²-n) with asymptotic diameter constant 1-1/π²
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
No named human explanation of the model-discovered mechanism is established; Few is the evidence default.
Correction and publication history
The high-dimensional asymptotic counterexample has a different scope from the fixed-dimensional Erdős 670 construction.
Read this published revision