{
  "abstract": "Forsythe's conjecture, published in 1968, asserts that for each restart length $s$, every exact-arithmetic restarted conjugate-gradient iteration on a real symmetric positive definite problem either terminates or has normalised residuals that converge separately along the even and odd restart subsequences. Apart from the classical steepest-descent case, this asymptotic question remained unresolved in full generality for nearly six decades. We give a complete classification by restart length in the original finite-dimensional setting and identify a sharp threshold. For $s=2$ and $s=3$, every problem either terminates or has convergent even and odd residual directions. For every $s\\ge4$, there is a diagonal positive definite counterexample of dimension $s+4$ which never terminates and whose even residual directions do not converge. Together with Akaike's theorem for $s=1$, this shows that the conjectured universal conclusion is true precisely for $s\\in\\{1,2,3\\}$ and false for every $s\\ge4$. The positive results follow from a degree-independent double-orthogonality identity and an analysis of the low-degree fixed-point sets. At restart length four, rational interval arithmetic and Sturm sequences certify a transverse Hopf point of the leading vector field of the rescaled squared-weight map. Analytic periodic-orbit and shadowing arguments yield the counterexample at restart length four, and degree elevation extends the construction to every larger restart length. The classification for all $s\\ge2$ is also formally verified in Lean.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04659",
  "author_total": 3,
  "authors": [
    {
      "name": "Matthew J. Colbrook"
    },
    {
      "name": "George Stepaniants"
    },
    {
      "name": "Alex Townsend"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aeq711gcef451.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04659",
  "fulltext_url": null,
  "id": "aeq711gcef451",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04659",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.04659",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aeq711gcef451.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.04659",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04659"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Complete Resolution of Forsythe's Conjecture for Restarted Conjugate Gradients",
  "title_text": "A Complete Resolution of Forsythe's Conjecture for Restarted Conjugate Gradients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/aeq711gcef451",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.5281/zenodo.22557261",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}