What was already known
Earlier SETH hardness reached dimensions of order 2^Θ(log* n), leaving the smaller superconstant range open.
AI involvement
ChatGPT 5.5 Pro generated the initial proof after the authors suggested a proof direction. Saha, Xu and Ye validated, edited and explained the complete argument.
ChatGPT generated the initial full proof and key OV-to-integer-OV reduction; humans supplied a proof-technique prompt and validated the result.
Importance
The conditional theorem closes the dimension gap for several fine-grained geometric problems.
Limitations
The lower bounds depend on OVH or SETH. Lean formalizes the key local embedding lemma under three algebraic-number-theory inputs, while the full reductions are author-checked.
Technical verification
The authors report validating the full proof by hand. The Lean check covers only the key embedding lemma and is conditional on three standard number-theory inputs.
Formal checking
Author validation and substantial mathematical editing of the proof
- Checker
- Saha, Xu and Ye
- Scope
- Final conditional hardness theorem and reductions
- Basis
- Reported by the linked checker
Lean formalization with three algebraic-number-theory assumptions
- Checker
- Aristotle/Lean 4, as reported by authors
- Scope
- Lemma 3.1, improved local OV-to-integer-OV embedding
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors give a detailed technical overview of the number-theoretic encoding and final reductions.
Correction and publication history
The conditional theorem closes the dimension gap for several fine-grained geometric problems.
Read this published revision