{
  "abstract": "We investigate which metric information can be retained when an initial metric space is realized in a complete algebra satisfying quantitative laws. Using free metric realizations, we express preservation of designated distances as constraints on the permitted errors. In a fixed weighted-path presentation, the associated feasible error budgets form a closed, convex, upward-closed region with finite-path certificates of failure, and free term distances depend continuously on the budgets. For an idempotent binary operation with positive coordinate weights $a+b=1$, we establish sharp associativity-defect bounds and an exact formula for the generator distances in the free realization with associativity tolerance $α$: $\\min\\{d_X(x,y),α/(ab)\\}$. This yields explicit injectivity and isometry thresholds. We also give a necessary diameter bound when both idempotence and associativity are approximate, and a countably presented example whose free distance is continuous but not Lipschitz at zero.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24927",
  "author_total": 1,
  "authors": [
    {
      "name": "Daksh Pandey"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7zcn6498pn8r2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24927",
  "fulltext_url": null,
  "id": "7zcn6498pn8r2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24927",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7223037337",
  "pdf_url": "https://arxiv.org/pdf/2610.24927",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7zcn6498pn8r2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24927",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24927"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Metric preservation under approximate algebraic laws",
  "title_text": "Metric preservation under approximate algebraic laws",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/7zcn6498pn8r2",
  "venue": "arXiv",
  "versions": []
}