What was already known
Dick, Goda and Suzuki asked whether low-discrepancy digital nets could also have separation and covering radius of optimal order N^(−1/s) in dimensions s≥3. The authors described that combination as open.
AI involvement
The author says Claude Opus 5.5 and earlier Claude Fable 5.1 produced the constructions, proofs and Lean formalization under his direction; he selected the question, steered the project and reviewed the headline statement.
The author explicitly says Claude systems produced the constructions, proofs and Lean formalization while he chose the problem, directed the work and reviewed the headline statement.
Importance
Provides an explicit family in every dimension under stated base conditions, combining the digital-net property with separation and covering guarantees of the optimal order for fixed dimension.
Limitations
The source reports no independent human peer review. The headline theorem was checked against the manuscript by the responsible author. Comparator and two kernels reportedly accepted the four headline declarations, but the run was unsandboxed on macOS. The result requires the prime-factor condition on q and the extra q threshold for its separation bound. The Lean headline does not itself establish a general star-discrepancy formula or the classical lower bounds for arbitrary point sets.
Technical verification
The source reports that four headline declarations, including Theorem A and definition-fidelity specifications, were accepted by Lean and NanoDa kernels through Comparator. The comparator was run unsandboxed on macOS; no independent rerun is claimed.
Formal checking
The source reports four headline theorems accepted by the Lean and NanoDa kernels, with statement matching; the separate 616-theorem library run was transferred by comment-insensitive source comparison.
- Checker
- Lean 4 kernel and documented comparator/certificate checks run by the repository authors
- Scope
- For every dimension s≥3, base q≥2 whose prime factors are all at least s−1, and k≥1, the explicit construction gives a digital (0,sk,s)-net of N=q^(sk) points with covering radius at most √s·N^(−1/s). If q≥5a*_s+4 for the construction’s specified constant a*_s, distinct points are separated by at least (1−a*_s/q)·N^(−1/s); the theorem also bounds a*_s explicitly.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The author signed off the headline formal statement but reports no independent human proof review or documented human explanation of the AI constructions. Few is the evidence default.
Correction and publication history
Primary-source review of Theorem A, exact base/dimension conditions, AI construction attribution and source-reported formal check scope.
Read this published revision