{
  "abstract": "We study the positive odd integers under the three affine operations A(x) = (2x-1)/3, C(x) = (4x-1)/3, V(x) = 4x+1, where A is legal on inputs 2 (mod 3), C on inputs 1 (mod 3), and V is always legal. These operations are inverse pieces of the accelerated Collatz map T(x) = (3x+1)/2^(v_2(3x+1)) in the precise sense that T (A(x)) = x, T (C(x)) = x, and T (V (x)) = T (x). The first point is uniqueness. The images of A, C, V are separated modulo 8. This gives a single-valued parent map P. Thus every point has at most one ancestry chain, and a collision of two affine closures is an ancestry relation. The counting argument is based on an exact change of coordinates. We can show that an AC-pull has binary mass 2^{-a(w) + 2c(w)), while a V-run of length t has mass 4^{-t}. The map V supplies the required ternary sampling: one V-cycle modulo 3^k visits every residue class exactly once.After truncating the AC-depth at K(n) = ⌊ 1/2 log_3(n)⌋, the discarded mass is o(1), and the retained paths require only O(K(n)) extra binary headroom. Hence the affine closure of 1 occupies density one in the odd integers. Finally, every fixed root has positive affine-closure density after removing at most one finite cycle. Density one then forces a collision between the tree originating at 1 and the tree originating at x, and the parent map turns this collision into x is in the tree originating at 1. Since x is arbitrary, one obtains that the tree originating at 1 is the set of all positive odd integers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "name": "Anonymous Author"
    },
    {
      "name": "Marie Graff"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dmae9w4v3pm4v.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "dmae9w4v3pm4v",
  "issns": [],
  "landing_url": "https://hal.science/hal-05748420v1",
  "language": "en",
  "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": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-co"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05748420/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dmae9w4v3pm4v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:34Z",
      "native_id": "hal-05748420",
      "source": "hal",
      "url": "https://hal.science/hal-05748420v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-co",
      "subfield": "combinatorics"
    }
  ],
  "title": "Reachability and uniqueness of the affine Collatz inversion",
  "title_text": "Reachability and uniqueness of the affine Collatz inversion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:34Z",
  "url": "https://research.draconitic.com/paper/dmae9w4v3pm4v",
  "venue": "HAL",
  "versions": []
}