Breakthrough Tracker record
A 146-page preprint claims a complete classification of Forsythe's conjecture
Matthew Colbrook, George Stepaniants and Alex Townsend prove that Forsythe's restarted-conjugate-gradient conclusion holds exactly for restart lengths one, two and three, while diagonal positive-definite counterexamples exist for every restart length at least four.
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- Stable ID
math-forsythe-restarted-cg-classification-2026- Revision
math-forsythe-restarted-cg-classification-2026.v1- Field
- Mathematics · Numerical analysis, optimization and dynamical systems
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional complete-classification preprint
- Last checked
AI role
The authors describe several weeks of iterative work with GPT-5.6, including discovery of the route for lengths two and three and the oscillatory obstruction at length four. They checked and developed the final mathematics and accept responsibility.
Record details
- Problem or result
- Forsythe's 1968 asymptotic-direction conjecture for restarted conjugate gradients
- Authors
- Matthew J. Colbrook, George Stepaniants and Alex Townsend
- Institutions
- University of Cambridge; California Institute of Technology; Cornell University
- Result date
- Preprint submitted September 4, 2026
Why it matters
The work gives a sharp restart-length boundary for a nearly six-decade-old numerical-linear-algebra problem and combines analytic dynamics with exact rational interval and Sturm-sequence certification.
Limits
The 146-page first-version preprint has not been peer reviewed. Exact programs certify the finite restart-four premises, while the general higher-length construction is analytic; no independent specialist correctness assessment was located.
Sources
Correction and revision history
- 2026-09-07 — Added after primary-source, scope, status, AI-role and limitation review.
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