{
  "abstract": "Infinite Numbers, introduced in preceding studies as functional expressions φ(ξ) of a distinguished infinity unit ξ, are here given a rigorous germ-based foundation within a Hardy-field setting. The real system is realized as an ordered non-Archimedean field canonically isomorphic to the chosen Hardy field, while complex Infinite Numbers arise by complexification. Equality is defined by eventual equality rather than asymptotic equivalence, thereby retaining lower-order and nonzero infinitesimal contributions. Within stated admissibility conditions, the principal arithmetic, differential, partially defined integral, and retained-transform operations of the preceding framework receive rigorous interpretations. The theory further establishes conditions relating discrete accumulation to asymptotic antiderivative representations and proves that the retained bilateral-transform family is injective on C(ℝ) along every fixed vertical line, yielding a well-defined inverse on its range. Two applications illustrate the framework. For a nonlinear boundary-value problem on the real line, retained finite symmetric transforms remain applicable despite the absence of a common classical bilateral region of convergence, leading to an explicit heteroclinic solution and nonzero infinitesimal endpoint corrections. A discrete-series limit illustrates the accumulation theorem and is independently verified by a classical estimate.",
  "abstract_shown": true,
  "affiliations": [
    "Applied Mathematics and Computers Laboratory (AMCL), Technical University of Crete, Akrotiri Campus, 73100 Chania, Greece"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Emmanuel Thalassinakis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/chh7y6jp7f5k9.bib",
  "cited_by": 0,
  "doi": "10.3390/math14193594",
  "fulltext_url": null,
  "id": "chh7y6jp7f5k9",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/19/3594",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2610|Mathematical and Theoretical Analysis"
    ]
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  "openalex": "W7220442747",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/chh7y6jp7f5k9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14193594",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/19/3594"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2610|Mathematical and Theoretical Analysis",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Infinite Numbers: A Rigorous Germ-Based Foundation in a Non-Archimedean Field",
  "title_text": "Infinite Numbers: A Rigorous Germ-Based Foundation in a Non-Archimedean Field",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:29Z",
  "url": "https://research.draconitic.com/paper/chh7y6jp7f5k9",
  "venue": "Mathematics",
  "versions": []
}