{
  "abstract": "We study simultaneous trajectories of the ordinary Collatz map. A balanced coupling of the shortcut trajectories gives a uniform absorption estimate of order √((1 + log(1 + |d|))/h) for offset d and shortcut horizon h. Its realization preserves both the ordinary clock and the number of halvings. Consequently Gao's finite-horizon dyadic coalescence proportions tend to one, as first proved publicly by Said Duran, and the balanced deficiency is Θ(k−1/2) at horizon k, under Gao's original requirement that merger occur before either trajectory reaches 1. The upper bound is deduced from the pair estimate; the matching lower bound comes from a converse clock dictionary from balanced ordinary merger to symbolic absorption. The leading constant is determined in a companion paper. For every prescribed L(n) = o(√(log n)) and every fixed log4 3 < c < 1, combining an earlier merger deadline with Korec's theorem gives a common value below nc for all starts n, n+1, …, n+L(n), at one ordinary time t ≤ 2 log2 n, for natural-density-almost every n. Combining adjacent-pair absorption with Tao's almost-bounded minimum theorem yields a simultaneous conclusion: for every prescribed L(n) = o(√(log n)) and every f(n) → ∞, logarithmic-density-almost every n admits one ordinary time t and one positive value y < f(n) shared by all starts n, n+1, …, n+L(n). Both arguments first prove early merger and then force the chosen descent witness to occur after merger. No convergence-to-one assumption is used, and the exceptional integer sets are not removed. The natural-density minimum law, proved in a companion paper, upgrades the arbitrary-threshold window statement to natural density for the attained minimum and its first ordinary clock. Companion papers: Brownian motion in Collatz orbits; Exact rates in Collatz synchronization; Gao's dyadic density-one coalescence conjecture; The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "David Leen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/61ww34dshs2r5.bib",
  "cited_by": 6,
  "doi": "10.5281/zenodo.23268754",
  "fulltext_url": null,
  "id": "61ww34dshs2r5",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.23268754",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221189363",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/61ww34dshs2r5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7221189363",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.23268754"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Quantitative synchronization of Collatz trajectories",
  "title_text": "Quantitative synchronization of Collatz trajectories",
  "type": "preprint",
  "updated_at": "2026-11-06T13:20:22Z",
  "url": "https://research.draconitic.com/paper/61ww34dshs2r5",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}