arXiv subjects

Aidan Rocke

Publications and source records attributed to Aidan Rocke.

8 recordsLinked to original sources

arXiv2026

Multiplicative Turing Ensembles, Pareto's Law, and Creativity

We study integer-valued multiplicative dynamics driven by i.i.d. prime multipliers and connect their macroscopic statistics to universal codelengths. We introduce the Multiplicative Turing Ensemble (MTE) and show how it arises naturally -- though not uniquely -- from ensembles of probabilistic Turing machines. Our modeling principle is variational: taking Elias' Omega codelength as an energy and imposing maximum entropy constraints yields a canonical Gibbs prior on integers and, by restriction, on primes. Under mild tail assumptions, this prior induces exponential tails for log-multipliers (up to slowly varying corrections), which in turn generate Pareto-type tails for additive gaps, with the survival exponent shifted by summation over primes. We also prove time-average laws for the Omega codelength along MTE trajectories. Empirically, Debian, PyPI, and CRAN package-size histograms have fitted Omega slopes well below the pure-Omega value $\log 2$, indicating heavier-than-pure-Omega tails within this energy scale. Taken together, the theory--data comparison suggests a qualitative split: machine-adapted regimes (Gibbs-aligned, finite first moment) exhibit clean averaging behavior, whereas human-generated complexity appears to sit beyond this regime, with tails heavy enough to produce an unbounded first moment, and therefore no averaging of the same kind.

cs.IT↗

arXiv2026

Lagrangians, Renormalization, and Quantization in Prefix Coding

We develop a statistical mechanics framework for prefix coding based on variational principles, renormalization, and quantization. A Lagrangian formulation of entropy-optimal encoding under the Kraft-McMillan constraint yields a Gibbs-type implied distribution and completeness of the optimal code. A renormalization operator acting on codeword distribution laws produces a coarse-graining flow whose fixed points have iterated-log structure; discrete quantizations of these fixed points include Elias' $ω$ code as a special case. Extending the theory to mixed discrete-continuous source laws, we show how continuous codelength functions can be quantized into countable prefix codes and derive resolution-adjusted entropy bounds together with Heisenberg-type and Boltzmann-type relations. This provides a unified and physically motivated view of universal coding, with Elias' $ω$ code as a guiding example.

cs.IT↗

arXiv2025

Benford's Law from Turing Ensembles and Integer Partitions

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 geometric law for codelength. Maximizing its entropy under a constraint on halting length yields Benford statistics. This model shows a phase transition with respect to the halt probability. Second, a constrained partition model (Einstein-solid combinatorics) recovers the same logarithmic profile as the maximum-entropy solution under a coarse-grained entropy-rate constraint, clarifying the role of non-ergodicity (ensemble vs. trajectory averages). We also perform numerical experiments that corroborate our conclusions.

cs.IT↗

arXiv2024

The Information Geometry of UMAP

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 posit that it also has a natural geometric interpretation.

cs.CG↗

arXiv2024

On the impossibility of discovering a formula for primes using AI

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 entropy as Expected Kolmogorov Complexity and formalizes other fundamental concepts such as Occam's razor via Levin's Universal Distribution. As a fundamental application, we develop Maximum Entropy methods that allow us to derive the Erdős-Kac Law and Hardy-Ramanujan theorem in Probabilistic Number Theory, and establish the impossibility of discovering a formula for primes using Machine Learning via the Prime Coding Theorem.

cs.CC↗

arXiv2024

An information-theoretic upper bound on prime gaps

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 method allows us to clarify the relation between the empirical density of primes and their source distribution which then allows us to address Cramér's conjecture, an open problem in probabilistic number theory. As a natural consequence, this analysis places strong epistemic limits on the application of machine learning to analyse the distribution of primes.

math.GM↗

arXiv2024

All the information in the integers is in the primes

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} \cdot \ln p \end{equation} has a natural information-theoretic interpretation as the information-content of a typical integer $Z \sim U([1,N])$, sampled under the maximum entropy distribution, is asymptotically equivalent to the information content of all primes $p \leq N$ weighted by their typical frequency. This result is established by showing that the correct information-theoretic interpretation of the entropy $\ln p$ may be deduced from the author's recent observation that the prime numbers have a uniform source distribution. Moreover, using Chaitin's theorem that almost all integers are algorithmically random we may extend this asymptotic result to the setting of algorithmic information theory and derive an upper-bound on the algorithmic information of a typical integer.

math.GM↗

arXiv2024

An information-theoretic proof of the Erdős-Kac theorem

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 elegant proof via Algorithmic Information Theory.

cs.IT↗