Visualizing the Collatz Conjecture as a Phase Transition

Summary

Research Log // January 2026 Visualizing the Collatz Conjecture as a Phase Transition The Ghost in the Machine. Paul Erdős famously said of the Collatz Conjecture ($3n+1$): "Mathematics is not yet ready for such problems." The problem lives at a jagged intersection where number theory crashes into chaotic dynamics. For the last week, I’ve been treating the Collatz iteration not as a sequence of numbers, but as a signal processing problem. I built a computational lab to decouple the "arithmetic" from the "algebra" and found a hidden "error map"—a ghostly structure of entropy that dictates why the numbers eventually collapse. You can run the experiment yourself below. The "Leaky Field" Hypothesis I built the Base-1 codebase in Python to simulate arithmetic operations as python string manipulation operations on unary strings. It seemed like maybe turning this engine toward the Collatz conjecture could yield some interesting data. Instead of dealing directly with the number $N$ in $3N+1$, I lift the problem to $P(2)=N$ where $P$ is a polynomial. This turns the Collatz problem into a dynamical system closely analogous to the Abelian sandpile model. Using the identity $3 = 1 + 2$ we have $$(1 + x)P(x) + 1= 3P(x) + 1$$ when $x=2$. The term $(1 + x)$ represents the sum of the identity operator ($1$) and the shift operator ($x$). A standard Galois Linear Feedback Shift Register (LFSR) implementation is defined by a characteristic polynomial $Q(x)$ over a finite field $\mathbb{F}_{q}$. The state evolution is given by multiplication by $x$ modulo $Q(x)$: $$S_{t+1}(x) = x \cdot S_t(x) \pmod{Q(x)}$$ This operation corresponds to a bit-shift with feedback. If $Q(x)$ is a primitive polynomial, the Galois LFSR generates a maximal-length sequence of length $2^n - 1$ before repeating. Unlike a standard LFSR, the Collatz Generator has no modulus $Q(x)$. The degree of the polynomial is allowed to increase indefinitely. $$\deg(P_{t+1}) = \deg(P_t) + 1$$ We usually think of the Collatz m...