{
  "abstract": "We study the density ρ(N,c) of distinct entries of the N×N multiplication table in the square-gap ((cN)², (cN+1)²], equivalently the density of integers near (cN)² possessing a divisor within a factor 1/c of their square root. Exact computation is carried to N = 10^8, an exact-integer Monte Carlo to N = 2^1024, and a continuum model of factorizations, anchored to the exact tiers at seven seam heights, to N = 2^{10^18}. All predictions were frozen before their data and every failure is logged. Three results emerge. First, square-centered windows are statistically indistinguishable from generic windows of the same magnitude, and the gap-to-gap coefficient of variation fits N^β with β = -0.440 ± 0.031, so the unproven localization of Ford's theorem to √x-scale windows is empirically a concentration statement with a square-root exponent. Second, the local decay of ρ attains Ford's shape -δ - (3/2)t^{-1} in t = log log x only from t ≈ 13–16, and the exponent δ = 0.086071 itself is never attained. At t = 39.5, the model height of quintillion-bit integers, the effective exponent still reads 0.077. The deficit follows a 1/t law with coefficient b = 0.333 ± 0.032 in the mature regime, a constant floor being rejected at Δχ² = 56.8. Third, an exactly computable ballot skeleton, Ford's order-statistics mechanism stripped of all divisor structure, reproduces δ, the t^{-3/2}, and the 1/t coefficient to within 1–2 standard errors. The lossy steps in the reduction from divisors to the ballot event therefore carry nothing at subleading order, and the divisor-specific contribution is localized to a transient dying near t ≈ 25. The subleading law is a measurement in the model, anchored to exact integers at the seams. It is not a theorem, and we state the theorem-shaped objects it calls for.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Michael Ross",
      "orcid": "0009-0001-3428-5337"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fhatxgtvdf68k.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23483059",
  "fulltext_url": null,
  "id": "fhatxgtvdf68k",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23483059",
  "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": null,
  "pdf_url": "https://zenodo.org/api/records/23483059/files/The_Multiplication_Table_Near_Perfect_Squares.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fhatxgtvdf68k.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:20:43Z",
      "native_id": "23483059",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23483059"
    }
  ],
  "subjects": [],
  "title": "The Multiplication Table Near Perfect Squares: Concentration, Typicality, and the Slow Approach to Ford's Exponent",
  "title_text": "The Multiplication Table Near Perfect Squares: Concentration, Typicality, and the Slow Approach to Ford's Exponent",
  "type": "preprint",
  "updated_at": "2026-11-02T12:20:43Z",
  "url": "https://research.draconitic.com/paper/fhatxgtvdf68k",
  "venue": "Zenodo",
  "versions": []
}