Maximum number of pairwise equidistant lines in $\mathbb{R}^3$

It is proved that no eight affine lines in $\mathbb{R}^3$ have a common positive pairwise distance. Together with the known construction of seven such lines by Bozóki, Lee and Rónyai, this establishes the exact maximum of seven. The upper bound combines a five-line obstruction from hyperbola secants with exact verification over 135 uniform rank-three oriented-matroid types. It settles the full stated maximum-cardinality question; it does not classify all seven-line configurations or address cylinders of unequal radii.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 30, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: site-confirmed. Publication: announcement. AI contribution: ai-co-developed. 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: Maximum number of pairwise equidistant lines in $\mathbb{R}^3$ · Seven equidistant lines

Confidence: Not scored

Registry verification: site confirmed · announcement · resolved

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Attribution

VibeMathed
registry · event recorded by

GPT-6 Astra Pro
model · ai model contributor · OpenAI

Claude Fable 5.1
model · ai model contributor · Anthropic

Mingchang Liu
human · human collaborator

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Maximum number of pairwise equidistant lines in $\mathbb{R}^3$

specialist registry · passed

Rebuilt by this site on 6 October 2026. The Lean development added on 4 October (commit 065e5e2; Lean 4.32.1 with mathlib, about 6,760 lines) was built from a clean clone with no warnings, passed a fresh leanchecker kernel replay, and #print axioms gives only propext, Classical.choice and Quot.sound for thm_1_1 and cor_1_2_upper. No sorry, axiom declarations, native_decide or implemented_by; finite steps use decide +kernel. The statement was read: no eight affine lines in R^3 have pairwise distance 1, with lines as affine subspaces of one-dimensional direction and distance the infimum of explicit Euclidean point distances; the corollary covers any n >= 8 and any d > 0. No general-position, non-parallel or catalogue hypothesis. The proof formalises the parallel and coplanar direction lemma, the four-line circuit condition and the five-line obstruction, and replaces Finschi's catalogue with a self-contained kernel-checked enumeration. Separately, on 4 October the Python verifier at 9c37327 was re-run here and passed (135 classes, 1,027 metric certificates). Not formalised: the seven-line construction of Bozoki, Lee and Ronyai, which is published. The statement audit is this site's alone.

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Maximum number of pairwise equidistant lines in $\mathbb{R}^3$

witness · pending

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

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Exact verification code and original certificates

code · pending

code link supplied by the VibeMathed registry.

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Hofer (2026): the previous upper bound of nine

paper · pending

paper link supplied by the VibeMathed registry.