Visualizing the Collatz Conjecture as a Phase Transition
20 days ago
- #Phase Transition
- #Collatz Conjecture
- #Chaotic Dynamics
- The Collatz Conjecture is visualized as a phase transition, intersecting number theory and chaotic dynamics.
- A computational lab was built to separate arithmetic from algebra, revealing a hidden 'error map' of entropy.
- The 'Leaky Field' hypothesis suggests the Collatz problem behaves like a dynamical system similar to the Abelian sandpile model.
- The Collatz map is described as a competition between a linear fractal generator and a non-linear dissipator (carry propagation).
- Visual experiments compare standard Collatz orbits with ideal Galois orbits, showing underlying fractal structures disrupted by arithmetic carries.
- Mersenne numbers were used to measure 'Burn Velocity,' showing linear decay and constant carry propagation rates.
- A phase transition point was identified where carry propagation overwhelms multiplication growth, causing orbit collapse.
- The Base-1 codebase enables these visualizations with high-performance arbitrary-precision integer physics and polynomial engines.