What does this research mean for the field?

The paper establishes new lower bounds for the no-four-coplanar problem in the cube for specific sizes, including a(12) ≥ 31, and provides publicly accessible, certified configurations while correcting previous false novelty claims. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Determine maximal point sets in an n x n x n grid cube containing no four coplanar points (OEIS A280537) and provide certified, publicly accessible witness configurations.

September 5, 2026Open Access

Nineteen certified configurations, eighteen independent bounds for the no-four-coplanar problem in the cube, and what they do not establish (OEIS A280537)

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Authors

AKAleksei KudriashovNational Heritage Institute

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Implication

Computational search reports certified lower bounds for the no-four-coplanar problem in grid cubes, highlighting four genuine improvements over prior records.

Key Points

Determine maximal point sets in an n x n x n grid cube containing no four coplanar points (OEIS A280537) and provide certified, publicly accessible witness configurations.
Conducted combinatorial searches constrained to the cyclically invariant subspace and across 33 conjugacy classes of the cube group.
Verified all reported configurations using two independent programs that share no code.
Calibrated and benchmarked results against historical contest data and monotone closures where a(n+1) >= a(n).
Established four genuine lower bound improvements: a(12) >= 31, a(21) >= 47, a(22) >= 49, and a(27) >= 56.
Exhausted the cyclically invariant subspace to prove exact maximum bounds of 23 at n = 9 and 26 at n = 10 (1.52e9 nodes explored in 8.93 core-hours).
Identified that four symmetry classes are incompatible with the problem, resolving optimal rigid configurations across remaining strata for n = 7, 8, and 9.

Cite This Study

Aleksei Kudriashov (2026) studied this question.

synapsesocial.com/papers/6a9bd4726b95aff0620ec1cf https://doi.org/10.5281/zenodo.22272241

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Implication

Computational search reports certified lower bounds for the no-four-coplanar problem in grid cubes, highlighting four genuine improvements over prior records.

Nineteen certified configurations, eighteen independent bounds for the no-four-coplanar problem in the cube, and what they do not establish (OEIS A280537)

Discussion

Member takes

Implication

Key Points

Cite This Study

Nineteen certified configurations, eighteen independent bounds for the no-four-coplanar problem in the cube, and what they do not establish (OEIS A280537)

Discussion

Member takes

Implication

Key Points

Cite This Study

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