What does this research mean for the field?

New certified lower bounds for the maximum number of points in an n×n×n grid with no four coplanar (OEIS A280537) are established for n up to 29, including a(9) ≥ 23 and a(29) ≥ 59. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Determine certified lower bounds and structural limits for the maximum number of points in an n×n×n grid where no four points share a common plane (OEIS A280537).

August 23, 2026Open Access

No four coplanar in the cube: nineteen certified configurations for n <= 29, and what they do not establish (OEIS A280537)

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Authors

AKAleksei KudriashovNational Heritage Institute

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Implication

Computational study demonstrates certified lower bounds for no-four-coplanar configurations in 3D grids up to n=29, indicating tighter constraints on spatial point distributions.

Key Points

Determine certified lower bounds and structural limits for the maximum number of points in an n×n×n grid where no four points share a common plane (OEIS A280537).
Constructed nineteen point configurations across grid sizes up to n = 29.
Employed dual, independent verification algorithms (exact 3×3 integer determinant calculations and scalar product checks against triple-derived plane normals) with corrupted witness pre-testing.
Conducted exhaustive search on the cyclically invariant subspace for n = 9 and n = 10 (1.52 × 10^9 nodes across 8.93 core-hours for n = 10).
Established eighteen new lower bounds for OEIS A280537, spanning a(9) ≥ 23 to a(29) ≥ 59.
Proved that the maximum point counts within the cyclically invariant subspace are exactly 23 at n = 9 and 26 at n = 10.
Showed that cyclic subspace exhaustion is insufficient to determine true global maxima, as evidenced by a(6) = 16 exceeding the symmetric maximum of 15.

Cite This Study

Aleksei Kudriashov (2026) studied this question.

synapsesocial.com/papers/6a8aade47677a34114446728 https://doi.org/10.5281/zenodo.22023079

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Implication

Computational study demonstrates certified lower bounds for no-four-coplanar configurations in 3D grids up to n=29, indicating tighter constraints on spatial point distributions.

No four coplanar in the cube: nineteen certified configurations for n <= 29, and what they do not establish (OEIS A280537)

Discussion

Member takes

Implication

Key Points

Cite This Study

No four coplanar in the cube: nineteen certified configurations for n <= 29, and what they do not establish (OEIS A280537)

Discussion

Member takes

Implication

Key Points

Cite This Study

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