{
  "abstract": "A typical first-order ODE syllabus is often presented as a sequence of named methods, which can obscure the unity among them. We propose organizing a substantial part of the classical symbolic first-order syllabus around a single computational target: an exact representative. For a regular scalar equation represented by a one-form $ω$, the reduction problem is to find, on an appropriate coordinate patch, a coordinate change $Φ$ and a nonvanishing factor $μ$ such that $μΦ^*ω=dG$, so that solution curves are level sets $G=C$. This yields a two-floor architecture. The structural floor records the target, transformations, and domain restrictions; the operational floor executes substitutions, normalizations, and integrations. Linear and separable equations arise from the simplest one-variable integrating factors, while homogeneous, Bernoulli, and related classes become reduction paths to the same target. In an AI- and CAS-assisted environment, this shifts emphasis toward recognition, justification, auditing, and interpretation. The extended version also develops reverse construction, a structural exercise laboratory, comparisons of reduction paths, and a higher-order outlook through first integrals, operator factorization, and structural normalization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22962",
  "author_total": 1,
  "authors": [
    {
      "name": "Gabriel Ben-Simon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d53g92de401pa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22962",
  "fulltext_url": null,
  "id": "d53g92de401pa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22962",
  "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",
      "math.GM"
    ],
    [
      "openalex",
      "2611|Mathematical and Computational Methods"
    ]
  ],
  "openalex": "W7214044096",
  "pdf_url": "https://arxiv.org/pdf/2609.22962",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d53g92de401pa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22962",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22962"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Exactness as an Explanatory and Computational Target: A Reduction Architecture for First-Order ODEs, with a Higher-Order Outlook",
  "title_text": "Exactness as an Explanatory and Computational Target: A Reduction Architecture for First-Order ODEs, with a Higher-Order Outlook",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/d53g92de401pa",
  "venue": "arXiv",
  "versions": []
}