A useful way to think about the Collatz map is to separate the orbit into two layers:

1. the parity pattern of the terms;
2. the arithmetic values that carry that pattern.

https://github.com/ratwolfzero/Collatz/tree/main/shadow_approach

If one records the orbit as a sequence of even/odd decisions, the resulting binary word can be viewed as a symbolic encoding of the dynamics. In the accelerated version of the map, where one removes all factors of 2 immediately after each odd step, the evolution of the values becomes closely tied to this parity word. At that symbolic level, the dynamics look simple: the orbit is encoded by a binary sequence, and the next state is determined by the next bit of that sequence.

This is a genuine simplification of the picture, but it is a simplification of the representation, not a solution of the problem.

I was curious how far PyDroid3 could be pushed. It turns out a transparent finite-element electrostatics solver using NumPy, SciPy and Matplotlib runs unchanged on my Samsung Galaxy S25. It solves sparse FEM systems locally and writes the same plots as on desktop

https://github.com/ratwolfzero/Capacitor_FEM/tree/main/Android

Abstract

This project introduces a compact three-parameter framework for characterizing the local geometry of polynomial roots. For each root, the framework records its position, its algebraic multiplicity, and a newly defined quantity called the characteristic deflection distance. This third parameter acts as a natural geometric scale: it measures how sharply or gradually the polynomial departs from zero in the immediate vicinity of the root, and it encodes the collective influence of all other roots through their distances from the one being analyzed.

The characteristic deflection distance generalizes the classical condition number of a simple root to roots of arbitrary multiplicity, and it allows direct geometric comparison across roots of different degrees. A key finding is that multiplicity alone does not determine geometric dominance — a lower-multiplicity root can have a larger spatial footprint than a higher-multiplicity one, depending on the global root configuration.

The framework extends naturally to polynomials over the complex numbers.

Global Visualization: Newton Flow and δ-Root Fields

To visualize the spatial competition between these triplets, we map the δ-Normalized Distance Field and the Newton Flow. This provides a "topographical" view of the polynomial's geometry where every root is evaluated against its own intrinsic scale.

Detailed explanations of math and computational impelmentations see:

GitHub