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Przemyslaw Chojecki

Publications and source records attributed to Przemyslaw Chojecki.

17 recordsLinked to original sources
arXiv2026

Bounded Representations by $x^2+y^2-z^2$

We prove that every sufficiently large integer $n$ can be written in the form $n=x^2+y^2-z^2$ with $\textrm{max}(x^2,y^2,z^2)\le n$. The proof converts the problem into finding a primitive binary quadratic form of positive discriminant $4n$ inside a fixed relatively compact open patch of the real hyperboloid $b^2-4ac=4n$. This is then supplied by Duke's theorem in the precise point-counting form deduced from the measure-theoretic duality of Einsiedler-Lindenstrauss-Michel-Venkatesh. A finite parity correction returns to the original ternary variables. This settles Erdős Problem 1148.

math.NT↗
arXiv2025

Mathematics and Coding are Universal AI Benchmarks

We study the special role of mathematics and coding inside the moduli space of psychometric batteries for AI agents. Building on the AAI framework and GVU dynamics from previous works, we define the Mathematics Fiber and show that, when paired with formal proof kernels (e.g. Lean, Coq), GVU flows on this fiber admit spectrally stable self-improvement regimes due to oracle-like verification. Our main technical result is a density theorem: under uniform tightness of agent outputs and a Lipschitz AAI functional, the subspace of batteries generated by mathematical theorem-proving and coding tasks is dense in the moduli space of batteries with respect to the evaluation metric. Coding alone is universal in this sense, while pure mathematics is not; its privilege is spectral rather than expressive. We interpret this as evidence that mathematics and coding provide ``universal coordinates'' for evaluation, and that formal mathematics is a natural ignition domain for recursive self-improvement in advanced AI agents.

cs.AI↗
arXiv2025

The Geometry of Benchmarks: A New Path Toward AGI

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.

cs.AI↗
arXiv2025

Self-Improving AI Agents through Self-Play

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.

cs.AI↗
arXiv2025

Psychometric Tests for AI Agents and Their Moduli Space

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.

cs.AI↗
arXiv2025

An Operational Kardashev-Style Scale for Autonomous AI - Towards AGI and Superintelligence

We propose a Kardashev-inspired yet operational Autonomous AI (AAI) Scale that measures the progression from fixed robotic process automation (AAI-0) to full artificial general intelligence (AAI-4) and beyond. Unlike narrative ladders, our scale is multi-axis and testable. We define ten capability axes (Autonomy, Generality, Planning, Memory/Persistence, Tool Economy, Self-Revision, Sociality/Coordination, Embodiment, World-Model Fidelity, Economic Throughput) aggregated by a composite AAI-Index (a weighted geometric mean). We introduce a measurable Self-Improvement Coefficient $κ$ (capability growth per unit of agent-initiated resources) and two closure properties (maintenance and expansion) that convert ``self-improving AI'' into falsifiable criteria. We specify OWA-Bench, an open-world agency benchmark suite that evaluates long-horizon, tool-using, persistent agents. We define level gates for AAI-0\ldots AAI-4 using thresholds on the axes, $κ$, and closure proofs. Synthetic experiments illustrate how present-day systems map onto the scale and how the delegability frontier (quality vs.\ autonomy) advances with self-improvement. We also prove a theorem that AAI-3 agent becomes AAI-5 over time with sufficient conditions, formalizing "baby AGI" becomes Superintelligence intuition.

cs.AI↗
arXiv2017

p-adic Jacquet-Langlands correspondence and patching

We describe two candidates for a local p-adic Jacquet-Langlands correspondence and using patching we show that they are in fact isomorphic. We then study locally algebraic vectors of the given correspondence.

math.NT↗
arXiv2016

An adjunction formula for the Emerton-Jacquet functor

The Emerton-Jacquet functor is a tool for studying locally analytic representations of p-adic Lie groups. It provides a way to access the theory of p-adic automorphic forms. Here we give an adjunction formula for the Emerton-Jacquet functor, relating it directly to locally analytic inductions, under a strict hypothesis that we call non-critical. We also further study the relationship to socles of principal series in the non-critical setting.

math.RT↗
arXiv2016

DeepAlgebra - an outline of a program

We outline a program in the area of formalization of mathematics to automate theorem proving in algebra and algebraic geometry. We propose a construction of a dictionary between automated theorem provers and (La)TeX exploiting syntactic parsers. We describe its application to a repository of human-written facts and definitions in algebraic geometry (The Stacks Project). We use deep learning techniques.

cs.AI↗
arXiv2016

Overconvergent modular forms and perfectoid Shimura curves

We give a new construction of overconvergent modular forms of arbitrary weights, defining them in terms of functions on certain affinoid subsets of Scholze's infinite-level modular curve. These affinoid subsets, and a certain canonical coordinate on them, play a role in our construction which is strongly analogous with the role of the upper half-plane and its coordinate `z' in the classical analytic theory of modular forms. As one application of these ideas, we define and study an overconvergent Eichler-Shimura map in the context of compact Shimura curves over $\mathbb{Q}$, proving stronger analogues of results of Andreatta-Iovita-Stevens.

math.NT↗
arXiv2014

Strong local-global compatibility in the p-adic Langlands program for U(2)

We prove that certain Galois-isotypic parts of the completed cohomology group for U(2) can be written as a completed tensor product of a representation coming from the p-adic Langlands correspondence for $GL_2(\mathbb{Q}_p)$ and a representation arising via the local Langlands correspondence in famillies of Emerton and Helm.

math.NT↗
arXiv2014

Ordinary representations and companion points for U(3) in the indecomposable case

We prove that certain p-adic Banach representations, associated to local ordinary Galois representations, constructed by Breuil and Herzig appears in the completed cohomology of a definite unitary group in three variables. This confirms part of their conjecture. Our main technique is making use of p-adic automorphic forms for definite unitary groups and the eigenvarieties which parameterize them.

math.NT↗
arXiv2014

On non-abelian Lubin-Tate theory and analytic cohomology

We prove that the p-adic local Langlands correspondence for GL_2(Q_p) appears in the etale cohomology of the Lubin-Tate tower at infinity. We use global methods using recent results of Emerton on the local-global compatibility and hence our proof applies to local Galois representations which come via a restriction from global pro-modular Galois representations. We also discuss a folklore conjecture which states that the p-adic local Langlands correspondence appears in the de Rham cohomology of the Lubin-Tate tower (Drinfeld tower). We show that a study of the de Rham cohomology for perfectoid spaces reduces to a study of the analytic cohomology and we state a natural conjecture related to it.

math.NT↗
arXiv2013

Weak local-global compatibility in the p-adic Langlands program for U(2)

Inspired by Emerton's work for GL(2), we study the completed cohomology of the tower of finite sets associated with a definite unitary group in two variables. When p splits (and other technical assumptions are fulfilled), we show that the p-adic local Langlands correspondence for GL(2) (over Q_p) occurs in the cohomology. We give an application to the Fontaine-Mazur conjecture over CM fields.

math.NT↗
arXiv2013

On mod p non-abelian Lubin-Tate theory for GL_2(Q_p)

We analyse the mod p étale cohomology of the Lubin-Tate tower both with compact support and without support. We prove that there are no supersingular representations in the H^1_c of the Lubin-Tate tower. On the other hand, we show that in H^1 of the Lubin-Tate tower appears the mod p local Langlands correspondence and the mod p local Jacquet-Langlands correspondence, which we define in the text. We discuss the local-global compatibility part of the Buzzard-Diamond-Jarvis conjecture which appears naturally in this context.

math.NT↗