Published May 7, 2026
| Version v4
Working paper
Open
The Collatz Conjecture as Orbital Motion in Polar Phase Space: A Geometric Resolution
Authors/Creators
Description
The Collatz conjecture has remained unsolved for over eighty-five years. We demonstrate that this is because the problem has been studied exclusively through its linear formulation. The classical 3n+1map is not fundamental — it is a linear Cartesian projection of a deeper quadratic, orbital dynamics taking place in polar phase space. By reinterpreting the Collatz process using concentric orbital shells generated by a five-layer resonance score, we show that every positive integer follows a deterministic trajectory through discrete radial levels toward a central attractor. The return to 1 is therefore a geometric necessity rather than a probabilistic accident. This polar reframing resolves the conjecture and reveals the linear 3n+1 rule as the shadow of a richer geometric structure.
Files
collatz2.pdf
Files
(405.4 kB)
| Name | Size | Download all |
|---|---|---|
|
md5:59bf9317212d686e8db61bf4a8194ac5
|
405.4 kB | Preview Download |