What was already known
Before White’s paper only the volume lower bound was known generally, while the Leader–Milićević–Tan question asked if f(d)≥2^d. Earlier constructed upper bounds included 82 at d=6 and an asymptotic constant 1.28; the question remains open from d=6 onward.
AI involvement
White chose the problem and questions and checked the work; he credits Claude and GPT-5.5 with the modular refinement and crossing lemma.
Specific retained mathematical ideas, including the decisive crossing lemma, are credited to the models, with White directing and checking.
Importance
Provides the first lower bounds beyond volume counting, settles the proposed 2^d threshold through d=5 and raises the d=6 bound past the profile-method limit of 57.
Limitations
The 2^d question remains open from d=6 onward. The d=4 and d=5 corollaries use the separately computed base value f(3)=11. The paper’s upper constructions have separate checking scope.
Technical verification
The assessed f(6)≥60 bound is reported as an unconditional Lean 4 theorem with standard axioms only. White also reports independent finite computation, but no separate human hand-check of the argument is documented. The f(4), f(5) corollaries assume f(3)≥11, supplied by separate exhaustive computation.
Formal checking
Lean theorem f6_ge_60_unconditional, no hypotheses or nonstandard axioms
- Checker
- Lean 4/mathlib formalization reported by Patrick White
- Scope
- Unconditional f(6)≥60 lower bound
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
White explains the line-rigidity, crossing and aggregation mechanism and why profile-only methods stop at 57.
Correction and publication history
Provides the first lower bounds beyond volume counting, settles the proposed 2^d threshold through d=5 and raises the d=6 bound past the profile-method limit of 57.
Read this published revision