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Breakthrough Tracker record

A formalization-backed preprint claims Lehmer’s permutation conjecture

Verhoeff claims that every multiset-permutation neighbor-swap graph admits Lehmer’s imperfect Hamiltonian traversal, proving the equivalent non-stutter Hamiltonicity formulation and identifying the exact path-only exceptions.

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Stable ID
math-lehmer-neighbor-swap-conjecture-2026
Revision
math-lehmer-neighbor-swap-conjecture-2026.v1
Field
Mathematics and theoretical computer science · Enumerative combinatorics, graph Hamiltonicity and Gray codes
Evidence
Tier 1 · Peer reviewed: No
Record state
Provisional · Formalization-backed provisional resolution
Last checked

AI role

The manuscript attributes the hypercube representation and spanning-matching-tree proof to Claude Opus 5.5, major Lean packages to Aristotle, and project-side integration to DeepSeek V41 Flash. The human author reports directing and checking the work.

Record details

Problem or result
Lehmer’s 1965 permutation conjecture for neighbor-swap graphs
Authors
Tom Verhoeff
Institutions
Department of Mathematics and Computer Science, Eindhoven University of Technology
Result date
Manuscript dated September 30; first-version preprint submitted October 1, 2026

Why it matters

The claim resolves a 1965 permutation-generation conjecture recorded as an unsolved Knuth problem and provides executable and kernel-checked artifacts.

Limits

This reconciliation addition predates the prior cutoff but was absent from the dataset. The paper is not peer reviewed, the large formal package lacks an outside semantic audit, and its port from Lean and Mathlib 4.28.0 to a stable toolchain was still in progress.

Sources

  1. Primary and artifacts: Verhoeff, proof and ancillary Lean and Python files
  2. Institutional metadata: Eindhoven University of Technology research record

Correction and revision history

  1. 2026-10-05 — Added after primary-source, scope, status, AI-role and limitation review.

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