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A power saving over Dey's bound for planar halving lines

Theorem 1.1: there are absolute $\varepsilon>0$ , $C$ , $n_0$ with $h(P)\le Cn^{4/3-\varepsilon}$ for every even $n\ge n_0$ and every $n$ -point planar set with no three collinear. Theorem 1.2 bounds the switches at every rank uniformly by $CN^{4/3-\varepsilon}$ , and Corollary 1.3 gives $O(n(k+1)^{1/3-\varepsilon_0})$ $k$ -sets for $1\le k\le n/2$ (also for levels in line arrangements). The constants and exponent are ineffective; no numerical value of $\varepsilon$ is given. It does not determine the order of $h(n)$ , which could still lie anywhere between $n e^{\Omega(\sqrt{\log n})}$ and $n^{4/3-\varepsilon}$ .

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: A power saving over Dey's bound for planar halving lines · Halving lines below n^(4/3)

Confidence: Not scored

Registry verification: lean checked · announcement · partial

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Unreleased internal OpenAI model
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VibeMathed record: A power saving over Dey's bound for planar halving lines

specialist registry · passed

No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorems 1.1 and 1.2 and Corollary 1.3 of the TeX source, read against the halving-line problem as cited. Lean: Comparator challenge HalvingLines, declaration OAI.PlanarHalving.power_bounds. This challenge is not in lean/formalization.yaml; its JSON config and solution module exist at the pinned commit. Its statement was read here: there exist $\varepsilon>0$ , $C$ , $n_0$ such that every even $n\ge n_0$ and every injective planar configuration with no three collinear has at most $Cn^{4/3-\varepsilon}$ unordered halving pairs; plus a uniform $Cn^{4/3-\varepsilon}$ bound on rank switches for generic configurations. That is the headline. The $k$ -sensitive corollary $O(n(k+1)^{1/3-\varepsilon_0})$ is not formalized. Not rebuilt here. The paper states the proof is nonquantitative.

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A power saving over Dey's bound for planar halving lines

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean Comparator challenge HalvingLines (OpenAI math release)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Dey 1998, Improved bounds for planar k-sets and related problems

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean Comparator challenge HalvingLines (OpenAI math release)

lean artifact · pending

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This event attributed to Unreleased internal OpenAI model

A power saving over Dey's bound for planar halving lines evidence for this event

Dey 1998, Improved bounds for planar k-sets and related problems evidence for this event

VibeMathed record: A power saving over Dey's bound for planar halving lines evidence for this event

A power saving over Dey's bound for planar halving lines parent of this event

Lean Comparator challenge HalvingLines (OpenAI math release) evidence for this event

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