1 citations · 4 across the 9 of their papers we have counts for
9 papers
Multiplicative Turing Ensembles, Pareto's Law, and Creativity
Alexander Kolpakov, Aidan Rocke
We study integer-valued multiplicative dynamics driven by i.i.d. prime multipliers and connect their macroscopic statistics to universal codelengths. We introduce the Multiplicativ…
Lagrangians, Renormalization, and Quantization in Prefix Coding
Alexander Kolpakov, Aidan Rocke
We develop a statistical mechanics framework for prefix coding based on variational principles, renormalization, and quantization. A Lagrangian formulation of entropy-optimal encod…
Benford's Law from Turing Ensembles and Integer Partitions
Alexander Kolpakov, Aidan Rocke
We develop two complementary generative mechanisms that explain when and why Benford's first-digit law arises. First, a probabilistic Turing machine (PTM) ensemble induces a geomet…
Machine Learning of the Prime Distribution
Alexander Kolpakov, Aidan Rocke
In the present work we use maximum entropy methods to derive several theorems in probabilistic number theory, including a version of the Hardy-Ramanujan Theorem. We also provide a…
The Information Geometry of UMAP
Alexander Kolpakov, Aidan Rocke
In this note we highlight some connections of UMAP to the basic principles of Information Geometry. Originally, UMAP was derived from Category Theory observations. However, we posi…
On the impossibility of discovering a formula for primes using AI
Alexander Kolpakov, Aidan Rocke
The present work explores the theoretical limits of Machine Learning (ML) within the framework of Kolmogorov's theory of Algorithmic Probability, which clarifies the notion of entr…
An information-theoretic proof of the Erdős-Kac theorem
Aidan Rocke
In this article we show that the Erdős-Kac theorem, which informally states that the number of prime divisors of very large integers converges to a normal distribution, has an eleg…
All the information in the integers is in the primes
Aidan Rocke
Building upon the work of Chebyshev, Shannon and Kontoyiannis, it may be demonstrated that Chebyshev's asymptotic result: \begin{equation} \ln N \sim \sum_{p \leq N} \frac{1}{p} \c…
An information-theoretic upper bound on prime gaps
Aidan Rocke
Within the setting of rare event modelling, the method of level sets allows us to define an equivalence relation over rare events with distinct rates of entropy production. This me…