Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 38 min ago
PreprintCC BYZenodo (CERN European Organization for Nuclear Research)

Gao's dyadic density-one coalescence conjecture

  • David Leen
Open at sourceDOICite

Abstract

We give a short proof of the density-one assertion posed by Gao for consecutive Collatz starts, independent of the first public proof, due to Said Duran (September 2026). Let C be the ordinary map and let H(n, m) count halving branches among the first m ordinary steps. Write Gk(n) when n and n+1 have, by time k, a common endpoint, equal halving counts, and have stayed above one up to that endpoint. If d̄k = 2−k·#{1 ≤ n < 2k − 1 : Gk(n)}, then d̄k → 1. A uniform absorption estimate for the 2-adic pair process gives the quantitative estimate 1 − d̄k = O(k−1/2), and the exact leading constant is determined in a companion paper. The argument concerns synchronization of two paths; it proves neither termination of every path nor the Collatz conjecture. Companion papers: Exact rates in Collatz synchronization; Quantitative synchronization of Collatz trajectories; The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits.

Related

  1. The Local Dyadic Conjecture
    Mahtab Alghasi, Bertrand Guenin, Levent Tunçel
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsOptimisation & controlarXiv
  2. A new class of smooth algebraic varieties with the density property and a counterexample to the Tóth-Varolin Conjecture
    Sushil Gorai
    arXivPreprintarXiv licenceFull textAlgebra & number theoryAnalysis & PDEsarXiv
  3. Consecutive integers coalesce with density one under the Collatz map: a formally verified proof of a conjecture of Gao
    Omar Javier Said Duran
    Zenodo (CERN European Organization for Nuclear Research)PreprintCC BY9 citedMathematical physicsOpenAlex
  4. Coalescence and fluctuations of O'Connellunicodex2013Yor polymers via the free-energy correlation profile
    Firas Rassoul-Agha, Xiao Shen, Ruixuan Zhang +1
    arXivPreprintCC BY 4.0Full textMathematical physicsProbabilityarXiv
  5. First pair coalescence among three geodesics
    Arjun Krishnan, Firas Rassoul-Agha, Timo Seppäläinen
    arXivPreprintarXiv licenceFull textProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@misc{leen2026gaos,
  title = {{Gao's dyadic density-one coalescence conjecture}},
  author = {David Leen},
  howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23268761},
  url = {https://doi.org/10.5281/zenodo.23268761},
  note = {CC BY},
}
RIS
TY  - UNPB
TI  - Gao's dyadic density-one coalescence conjecture
AU  - David Leen
PY  - 2026
DA  - 2026/10/09
PB  - Zenodo (CERN European Organization for Nuclear Research)
DO  - 10.5281/zenodo.23268761
UR  - https://doi.org/10.5281/zenodo.23268761
AB  - We give a short proof of the density-one assertion posed by Gao for consecutive Collatz starts, independent of the first public proof, due to Said Duran (September 2026). Let C be the ordinary map and let H(n, m) count halving branches among the first m ordinary steps. Write Gk(n) when n and n+1 have, by time k, a common endpoint, equal halving counts, and have stayed above one up to that endpoint. If d̄k = 2−k·#{1 ≤ n < 2k − 1 : Gk(n)}, then d̄k → 1. A uniform absorption estimate for the 2-adic pair process gives the quantitative estimate 1 − d̄k = O(k−1/2), and the exact leading constant is determined in a companion paper. The argument concerns synchronization of two paths; it proves neither termination of every path nor the Collatz conjecture. Companion papers: Exact rates in Collatz synchronization; Quantitative synchronization of Collatz trajectories; The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits.
ER  -