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Finite cylinder approximation of ruled sets
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Abstract
We approximate compact ruled families of segments by finitely many cylinders with square intercept tiles and perpendicular-base area at most their integral projection cost plus any positive error. The velocity field is C1, and its differential has opposite real eigenvalues whose magnitudes are strictly below the inverse segment half-length; both eigenvalues may vanish. The result includes square-zero differentials with unrestricted shear and fields that pass between the two regimes. It holds for arbitrary compact label sets.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Ricardo Pinho, An explicit finite cylinder cover of the regular tetrahedron saving $3.86\times10^{-4}$ below the half-area bound (hexagon:2610.00304v2, v1 17 Oct 2026, v2 18 Oct 2026). Gives 4,096 triangular cylinders with normalized cost approximately 0.499613830783714 and exact rational certificates; it cites this paper among the four family preprints but builds mainly on the angular-sector construction of the companion paper. [AI-generated per source] [family-level]
Formalization
- OpenAI math release, Lean scope notes for family 100: Cylinder coverings below the half-area bound (lean/docs/100.md, 2026). States that the formalization approximates compact ruled families, including irregular compact label sets, by finite cylinder covers up to any positive error under the $C^1$ trace-zero/determinant and segment-length hypotheses. It covers square-zero, nilpotent, product-cell and logarithmic cases (Comparator file
RuledCovering.lean). [family-level]
Possibly relevant literature (found by the agent, not checked)
- K. Bezdek, A. E. Litvak, Packing convex bodies by cylinders (arXiv:1507.05115; Discrete Comput. Geom. 55 (2016) 725–738). Packing counterpart of the Bezdek–Litvak covering bound $A_{\min}(K)/3$ that the paper cites as motivation.
Digests and expositions
- Latent Seminar, Finite cylinder coverings below the tetrahedral half-area bound (6 October 2026). Explains the four manuscripts' constructions, approximation hypotheses and two area normalizations, while expressly disclaiming independent verification of the complete proofs. [family-level]
Elsewhere
- Hexagon: An explicit finite cylinder cover of the regular tetrahedron saving $3.86\times10^{-4}$ below the half-area bound (AI-generated; human understanding: some)
- Emergent Mind: family 100
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Finite-cylinder-approximation-of-ruled-sets-September-27-2026,
author = {{OpenAI}},
title = {{Finite cylinder approximation of ruled sets}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026/main.pdf}{OAI:Finite-cylinder-approximation-of-ruled-sets-September-27-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/100.md. The Lean statements have not been checked by people against the paper's claims.
Family 100: Cylinder coverings below the half-area bound
Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.
Other papers in this family:
- Finite angular cylinder covers below the half-area bound · September 27, 2026
- Slope-field perturbations of the two-cylinder covering · September 27, 2026
- Finite triangular approximation of radial sweeps · September 27, 2026