Benchmarks are the primary tool for assessing progress in artificial intelligence (AI), yet current practice evaluates models on isolated test suites and provides little guidance for reasoning about generality or autonomous self-improvement. Here we introduce a geometric framework in which all psychometric batteries for AI agents are treated as points in a structured moduli space, and agent performance is described by capability functionals over this space. First, we define an Autonomous AI (AAI) Scale, a Kardashev-style hierarchy of autonomy grounded in measurable performance on batteries spanning families of tasks (for example reasoning, planning, tool use and long-horizon control). Second, we construct a moduli space of batteries, identifying equivalence classes of benchmarks that are indistinguishable at the level of agent orderings and capability inferences. This geometry yields determinacy results: dense families of batteries suffice to certify performance on entire regions of task space. Third, we introduce a general Generator-Verifier-Updater (GVU) operator that subsumes reinforcement learning, self-play, debate and verifier-based fine-tuning as special cases, and we define a self-improvement coefficient $κ$ as the Lie derivative of a capability functional along the induced flow. A variance inequality on the combined noise of generation and verification provides sufficient conditions for $κ> 0$. Our results suggest that progress toward artificial general intelligence (AGI) is best understood as a flow on moduli of benchmarks, driven by GVU dynamics rather than by scores on individual leaderboards.

We extend the moduli-theoretic framework of psychometric batteries to the domain of dynamical systems. While previous work established the AAI capability score as a static functional on the space of agent representations, this paper formalizes the agent as a flow $ν_r$ parameterized by computational resource $r$, governed by a recursive Generator-Verifier-Updater (GVU) operator. We prove that this operator generates a vector field on the parameter manifold $Θ$, and we identify the coefficient of self-improvement $κ$ as the Lie derivative of the capability functional along this flow. The central contribution of this work is the derivation of the Variance Inequality, a spectral condition that is sufficient (under mild regularity) for the stability of self-improvement. We show that a sufficient condition for $κ> 0$ is that, up to curvature and step-size effects, the combined noise of generation and verification must be small enough. We then apply this formalism to unify the recent literature on Language Self-Play (LSP), Self-Correction, and Synthetic Data bootstrapping. We demonstrate that architectures such as STaR, SPIN, Reflexion, GANs and AlphaZero are specific topological realizations of the GVU operator that satisfy the Variance Inequality through filtration, adversarial discrimination, or grounding in formal systems.

We develop a moduli-theoretic view of psychometric test batteries for AI agents and connect it explicitly to the AAI score developed previously. First, we make precise the notion of an AAI functional on a battery and set out axioms that any reasonable autonomy/general intelligence score should satisfy. Second, we show that the composite index ('AAI-Index') defined previously is a special case of our AAI functional. Third, we introduce the notion of a cognitive core of an agent relative to a battery and define the associated AAI$_{\textrm{core}}$ score as the restriction of an AAI functional to that core. Finally, we use these notions to describe invariants of batteries under evaluation-preserving symmetries and outline how moduli of equivalent batteries are organized.

We propose a novel framework for learning hidden graph structures from data using geometric analysis and nonlinear dynamics. Our approach: (1) Defines discrete Sobolev spaces on graphs for scalar/vector fields, establishing key functional properties; (2) Introduces gauge-equivalent nonlinear Schrödinger and Landau--Lifshitz dynamics with provable stable stationary solutions smoothly dependent on input data and graph weights; (3) Develops a stochastic gradient algorithm over graph moduli spaces with sparsity regularization. Theoretically, we guarantee: topological correctness (homology recovery), metric convergence (Gromov--Hausdorff), and efficient search space utilization. Our dynamics-based model achieves stronger generalization bounds than standard neural networks, with complexity dependent on the data manifold's topology.

We introduce \texttt{cymyc}, a high-performance Python library for numerical investigation of the geometry of a large class of string compactification manifolds and their associated moduli spaces. We develop a well-defined geometric ansatz to numerically model tensor fields of arbitrary degree on a large class of Calabi-Yau manifolds. \texttt{cymyc} includes a machine learning component which incorporates this ansatz to model tensor fields of interest on these spaces by finding an approximate solution to the system of partial differential equations they should satisfy.

In this paper, we discuss how pure mathematics and theoretical physics can be applied to the study of language models. Using set theory and analysis, we formulate mathematically rigorous definitions of language models, and introduce the concept of the moduli space of distributions for a language model. We formulate a generalized distributional hypothesis using functional analysis and topology. We define the entropy function associated with a language model and show how it allows us to understand many interesting phenomena in languages. We argue that the zero points of the entropy function and the points where the entropy is close to 0 are the key obstacles for an LLM to approximate an intelligent language model, which explains why good LLMs need billions of parameters. Using the entropy function, we formulate a conjecture about AGI. Then, we show how thermodynamics gives us an immediate interpretation to language models. In particular we will define the concepts of partition function, internal energy and free energy for a language model, which offer insights into how language models work. Based on these results, we introduce a general concept of the geometrization of language models and define what is called the Boltzmann manifold. While the current LLMs are the special cases of the Boltzmann manifold.

In this work, we show that neural networks can be represented via the mathematical theory of quiver representations. More specifically, we prove that a neural network is a quiver representation with activation functions, a mathematical object that we represent using a {\em network quiver}. Also, we show that network quivers gently adapt to common neural network concepts such as fully-connected layers, convolution operations, residual connections, batch normalization, and pooling operations. We show that this mathematical representation is by no means an approximation of what neural networks are as it exactly matches reality. This interpretation is algebraic and can be studied with algebraic methods. We also provide a quiver representation model to understand how a neural network creates representations from the data. We show that a neural network saves the data as quiver representations, and maps it to a geometrical space called the {\em moduli space}, which is given in terms of the underlying oriented graph of the network. This results as a consequence of our defined objects and of understanding how the neural network computes a prediction in a combinatorial and algebraic way. Overall, representing neural networks through the quiver representation theory leads to 13 consequences that we believe are of great interest to better understand what neural networks are and how they work.

We present a pedagogical introduction to the recent advances in the computational geometry, physical implications, and data science of Calabi-Yau manifolds. Aimed at the beginning research student and using Calabi-Yau spaces as an exciting play-ground, we intend to teach some mathematics to the budding physicist, some physics to the budding mathematician, and some machine-learning to both. Based on various lecture series, colloquia and seminars given by the author in the past year, this writing is a very preliminary draft of a book to appear with Springer, by whose kind permission we post to ArXiv for comments and suggestions.

We propose a new statistical model suitable for machine learning of systems with long distance correlations such as natural languages. The model is based on directed acyclic graph decorated by multi-linear tensor maps in the vertices and vector spaces in the edges, called tensor network. Such tensor networks have been previously employed for effective numerical computation of the renormalization group flow on the space of effective quantum field theories and lattice models of statistical mechanics. We provide explicit algebro-geometric analysis of the parameter moduli space for tree graphs, discuss model properties and applications such as statistical translation.