Research Log // January 2026

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 map $T(n)$ as a single operation. But mechanically, it is a competition between two opposing forces:

1. The Generator ($\mathcal{G}$): A linear force operating in the polynomial ring $\mathbb{GF}(2)[x]$. This force tries to build a perfect fractal structure (the Sierpinski Triangle).
2. The Dissipator ($\mathcal{D}$): The "Carry Propagation" inherent to Integer arithmetic. It's a non-linear force that "leaks" information from low-order bits to high-order bits.

The Collatz map is essentially a "Leaky Field." It tries to behave like a Linear Feedback Shift Register (LFSR), but the arithmetic carries act as entropy, burning the structure down.

Visualizing the Ghost

To test this, I built the Comparator module in the Base-1 dashboard ui. It runs two universes side-by-side:

1. Universe A (Physical): The standard Collatz orbit in $\mathbb{Z}$.
2. Universe B (Ideal): The perfect Galois orbit in $\mathbb{GF}(2)[x]$, where carries don't exist.

When I subtracted the two ($A \oplus B$), I didn't get random noise. I found a ghost.

This confirms that the underlying driver of the Collatz map is a linear fractal generator. The chaotic behavior we see is simply the "arithmetic friction" of the binary carry operations destroying that fractal.

Measuring the "Burn Velocity"

If the carries are "burning" the information, can we measure the temperature?

I used the engine's science_mode to inject Mersenne Numbers ($N = 2^k - 1$) into the system. In the Base-1 physics model, Mersenne numbers have the maximum-possible bit density, maximum carry propagation. (In binary, a Mersenne number consists entirely of 1's)

The result was a stunningly linear decay. The "Burn Velocity" of the carries is constant for a given density. This implies that the stopping time isn't random; it's a function of the Bit Density ($\rho$).

The Criticality Threshold

This led to the final experiment: The Criticality Scanner.

The results of the numerical experiments indicates there is a specific phase transition point where the "Burn Rate" of the carries overwhelms the "Growth Rate" of the multiplication (measured as $\log_2 3 \approx 1.585$ bits).

The scanner detected a phase transition at a bit-density of $\rho \approx 0.029$. Above this density, the "friction" of carry propagation scales super-linearly, overwhelming the growth force and causing the orbit to collapse.

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The Engine Under the Hood

This visualization is only possible because I wrote the Base-1 codebase. It implements:

• Science Mode: A high-performance wrapper for arbitrary-precision integer physics.
• Galois Factory: A polynomial engine that handles Finite Fields $\mathbb{F}_{p^n}$ and the isomorphism to LFSRs.
• Matter Protocol: A strictly typed system to ensure mass conservation during arithmetic operations.

The Collatz Conjecture isn't just a math problem. It's a fluid dynamics problem. And thanks to Base-1, we can finally watch it flow.

View Base-1 on GitHub →