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March 7, 2026Open Access

IV. Arithmetic obstruction to mixed orbits in the 2-adic Collatz dynamics

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Authors

MBMiguel Cerdá Bennassar

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Overview

Rules out indefinite mixed orbits in 2-adic dynamics, indicating all orbits converge to 1.

Key Points

  • To investigate the possibility of orbits in the 2-adic Collatz dynamics visiting both valuation classes indefinitely without converging to 1.
  • Analyzed the 2-adic structure of orbits within the Collatz dynamics.
  • Utilized a global 2-adic budget argument to assess bit consumption in orbits.
  • Considered cylindrical structures and local affine relations to ensure disjoint blocks.
  • Proved that all orbits originating in the valuation-2 class ultimately converge to 1.
  • Showed that excursions in both classes require non-regenerating bits, leading to contradictions with the finite parameters.

Cite This Study

Miguel Cerdá Bennassar (2026) studied this question.

synapsesocial.com/papers/69abc1d75af8044f7a4eaea3https://doi.org/10.5281/zenodo.18879168
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Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

  1. 1III. Arithmetic obstruction to indefinite survival in the 2-adic Collatz dynamics2026
  2. 2VIII. Return map, rigid regime, and invariance gap in the 2-adic Collatz dynamics2026
  3. 3VI. Structural reduction of the Collatz conjecture: segments, portals and 2-adic survival sets2026
  4. 4II. Cylinder collision, bit non-reuse and effective non-degeneracy in the 2-adic Collatz dynamics2026
  5. 5V. The function φ and the extension of the 2-adic budget argument to arbitrary k_0 in the Collatz dynamics2026