Machine Learning on Eigenform Articles

Machine Learnability as a Measure of Order

Tue, 07 Oct 2025 00:00:00 +0800

Section from an FFT of a 10,001x10,001 pixel Ulam spiral

(Note: this is a reduced-jargon/increased fun version of a paper published on Arxiv here: https://arxiv.org/abs/2509.18103. Thanks go to my co-authors: Michael Joedhitya, Adith Ramdas, Surender Suresh Kumar, Adarsh Singh Chauhan, Akira Rafhael, Wang Mingshu and Nordine Lotfi.)

Is maths invented or discovered?

Most non-mathematicians will say it is discovered. Here is one thing, and there is another thing, and now we have two things: tada. On the other hand, there is a small subset of theorists (such as Leopold Kronecker) who argue that while much of it is not true, it is nevertheless often useful: Negative numbers do not exist in nature, but they make the sums we need to parse nature a lot easier. There’s no such thing as the square root of a negative number but pretending there is provides us with a handy drawer in which to store multi-dimensional values and thus imaginary numbers were born.

Machine Learnability as a Measure of Order in Aperiodic Sequences

Tue, 09 Sep 2025 00:00:00 +0800

The original publication this post is based on can be found here: https://arxiv.org/abs/2509.18103.

Research on the distribution of prime numbers has revealed a dual character: deterministic in definition yet exhibiting statistical behavior reminiscent of random processes. In this paper we show that it is possible to use an image-focused machine learning model to measure the comparative regularity of prime number fields at specific regions of an Ulam spiral. Specifically, we demonstrate that in pure accuracy terms, models trained on blocks extracted from regions of the spiral in the vicinity of 500m outperform models trained on blocks extracted from the region representing integers lower than 25m. This implies existence of more easily learnable order in the former region than in the latter. Moreover, a detailed breakdown of precision and recall scores seem to imply that the model is favouring a different approach to classification in different regions of the spiral, focusing more on identifying prime patterns at lower numbers and more on eliminating composites at higher numbers. This aligns with number theory conjectures suggesting that at higher orders of magnitude we should see diminishing noise in prime number distributions, with averages (density, AP equidistribution) coming to dominate, while local randomness regularises after scaling by log x. Taken together, these findings point toward an interesting possibility: that machine learning can serve as a new experimental instrument for number theory. Notably, the method shows potential 1 for investigating the patterns in strong and weak primes for cryptographic purposes.

Generalising from Self-Produced Data: Model Training Beyond Human Constraints

Mon, 07 Apr 2025 00:00:00 +0800

The original publication this post is based on can be found here: https://arxiv.org/abs/2504.04711.

Current large language models (LLMs) are constrained by human-derived training data and limited by a single level of abstraction that impedes definitive truth judgments. This paper introduces a novel framework in which AI models autonomously generate and validate new knowledge through direct interaction with their environment. Central to this approach is an unbounded, ungamable numeric reward - such as annexed disk space or follower count - that guides learning without requiring human benchmarks. AI agents iteratively generate strategies and executable code to maximize this metric, with successful outcomes forming the basis for self-retraining and incremental generalisation. To mitigate model collapse and the warm start problem, the framework emphasizes empirical validation over textual similarity and supports fine-tuning via GRPO. The system architecture employs modular agents for environment analysis, strategy generation, and code synthesis, enabling scalable experimentation. This work outlines a pathway toward self-improving AI systems capable of advancing beyond human-imposed constraints toward autonomous general intelligence.

Defining T-Schemas via the Parametric Encoding of Second Order Languages in AI Models

Sat, 15 Feb 2025 00:00:00 +0800

source

In this short article we present a summary of current work on the grokking phenomenon that emerges when AI models are significantly over-trained is, and suggest that this evidence of the model’s attempts to define truth inductively through the creation of consensus sets within the base training set, and encode it via patterns overlaid upon the same parameters used to memorise this set.