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Moritz Firsching

Publications and source records attributed to Moritz Firsching.

11 recordsLinked to original sources
arXiv2026

Advancing Mathematics Research with AI-Driven Formal Proof Search

Large language models (LLMs) increasingly excel at mathematical reasoning, but their unreliability limits their utility in mathematics research. A mitigation is using LLMs to generate formal proofs in languages like Lean. We perform the first large-scale evaluation of this method's ability to solve open problems. Our most capable agent autonomously resolved 9 of 353 open Erd\H{o}s problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures, and is being deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research. A basic agent alternating LLM-based generation with Lean-based verification replicated the Erd\H{o}s successes but proved costlier on the hardest problems. These findings demonstrate the power of AI-aided formal proof search and shed light on the agent designs that enable it.

cs.AI↗
arXiv2026

Formal Conjectures: An Open and Evolving Benchmark for Verified Discovery in Mathematics

As automated reasoning systems advance rapidly, there is a growing need for research-level formal mathematical problems to accurately evaluate their capabilities. To address this, we present Formal Conjectures, an evolving benchmark of currently 2615 mathematical problem statements formalized in Lean 4. Sourced from areas of active mathematical research, the dataset features 1029 open research conjectures providing a zero-contamination benchmark for mathematical proof discovery, and 836 solved problems for proof autoformalization. Notably, the repository provides a structured interface connecting mathematicians who formalize and clarify problems with the AI systems and humans attempting to solve them. Demonstrating its immediate utility, the benchmark has already been leveraged to make new mathematical discoveries, including the resolution of open research conjectures. We describe our approach to ensuring the correctness of these formalizations in a collaborative open-source project where contributions stem from an active community. In this framework, AI-generated proofs and disproofs serve as a valuable auditing mechanism to iteratively improve the fidelity of the benchmark. Finally, we provide a standardized evaluation setup and report baseline results on frozen evaluation subsets, demonstrating a climbable signal that measures the current frontier of automated reasoning on research-level mathematics.

cs.AI↗
arXiv2025

The JPEG XL Image Coding System: History, Features, Coding Tools, Design Rationale, and Future

JPEG XL is a new image coding system offering state-of-the-art compression performance, lossless JPEG recompression, and advanced features. It aims to replace JPEG, PNG, GIF, and other formats with a single universal codec. This article provides an overview of JPEG XL, including its history, design rationale, coding tools, and future potential. It can be used as a companion document to the standard (ISO/IEC 18181), or as a standalone article to better understand JPEG XL, either at a high level or in considerable technical detail.

cs.MM↗
arXiv2023

AutoNumerics-Zero: Automated Discovery of State-of-the-Art Mathematical Functions

Transcendental functions, such as the exponential, are central to scientific computing, yet they cannot be natively calculated by digital hardware. Instead, computers must approximate these functions by combining basic operations, such as $\{+, -, \times, \div\}$, using methods like Taylor series. These methods were developed over centuries by mathematicians, who focused on approaches that could attain arbitrary accuracy. However, computers can handle most applications by using only finite-precision types, like float32, where any accuracy beyond the type's precision is effectively discarded. We explore, therefore, whether forgoing arbitrary accuracy can lead to the discovery of more efficient approximations. The evolutionary method of symbolic regression is particularly suitable, as it can search for arbitrary operation combinations and can optimize non-differentiable objectives, such as the number of operations used. Our results show that evolution can discover computer programs that outperform established methods in this setting, despite having no prior mathematical knowledge beyond the calculation of the basic operations. Starting from empty code, symbolic regression constructs programs representing novel mathematical expressions. In particular, we discovered a 10-operation program that approximates the exponential function to 14 significant figures, exceeding the accuracy of previously known approximations of this size by more than 6 orders of magnitude.

cs.NE↗
arXiv2020

Intelligent Matrix Exponentiation

We present a novel machine learning architecture that uses the exponential of a single input-dependent matrix as its only nonlinearity. The mathematical simplicity of this architecture allows a detailed analysis of its behaviour, providing robustness guarantees via Lipschitz bounds. Despite its simplicity, a single matrix exponential layer already provides universal approximation properties and can learn fundamental functions of the input, such as periodic functions or multivariate polynomials. This architecture outperforms other general-purpose architectures on benchmark problems, including CIFAR-10, using substantially fewer parameters.

cs.LG↗
arXiv2019

Committee Draft of JPEG XL Image Coding System

JPEG XL is a practical approach focused on scalable web distribution and efficient compression of high-quality images. It provides various benefits compared to existing image formats: 60% size reduction at equivalent subjective quality; fast, parallelizable decoding and encoding configurations; features such as progressive, lossless, animation, and reversible transcoding of existing JPEG with 22% size reduction; support for high-quality applications including wide gamut, higher resolution/bit depth/dynamic range, and visually lossless coding. The JPEG XL architecture is traditional block-transform coding with upgrades to each component.

eess.IV↗
arXiv2019

SO(8) Supergravity and the Magic of Machine Learning

Using de Wit-Nicolai $D=4\;\mathcal{N}=8\;SO(8)$ supergravity as an example, we show how modern Machine Learning software libraries such as Google's TensorFlow can be employed to greatly simplify the analysis of high-dimensional scalar sectors of some M-Theory compactifications. We provide detailed information on the location, symmetries, and particle spectra and charges of 192 critical points on the scalar manifold of SO(8) supergravity, including one newly discovered $\mathcal{N}=1$ vacuum with $SO(3)$ residual symmetry, one new potentially stabilizable non-supersymmetric solution, and examples for "Galois conjugate pairs" of solutions, i.e. solution-pairs that share the same gauge group embedding into~$SO(8)$ and minimal polynomials for the cosmological constant. Where feasible, we give analytic expressions for solution coordinates and cosmological constants. As the authors' aspiration is to present the discussion in a form that is accessible to both the Machine Learning and String Theory communities and allows adopting our methods towards the study of other models, we provide an introductory overview over the relevant Physics as well as Machine Learning concepts. This includes short pedagogical code examples. In particular, we show how to formulate a requirement for residual Supersymmetry as a Machine Learning loss function and effectively guide the numerical search towards supersymmetric critical points. Numerical investigations suggest that there are no further supersymmetric vacua beyond this newly discovered fifth solution.

hep-th↗
arXiv2018

The complete enumeration of 4-polytopes and 3-spheres with nine vertices

We describe an algorithm to enumerate polytopes. This algorithm is then implemented to give a complete classification of combinatorial spheres of dimension 3 with 9 vertices and decide polytopality of those spheres. In particular, we completely enumerate all combinatorial types of 4-dimensional polytopes with 9 vertices. It is shown that all of those combinatorial types are rational: They can be realized with rational coordinates. We find 316014 combinatorial spheres on 9 vertices. Of those, 274148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41866 are non-polytopal.

math.MG↗
arXiv2015

Realizability and inscribability for simplicial polytopes via nonlinear optimization

We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. Thus we obtain a complete classification of neighborly polytopes of dimension $4$, $6$ and $7$ with $11$ vertices, of neighborly $5$-polytopes with $10$ vertices, as well as a complete classification of simplicial $3$-spheres with $10$ vertices into polytopal and non-polytopal spheres. Surprisingly many of the realizable polytopes are also inscribable.

math.MG↗
arXiv2014

Computing maximal copies of polytopes contained in a polytope

Kepler (1619) and Croft (1980) have considered largest homothetic copies of one regular polytope contained in another regular polytope. For arbitrary pairs of polytopes we propose to model this as a quadratically constrained optimization problem. These problems can then be solved numerically; in case the optimal solutions are algebraic, exact optima can be recovered by solving systems of equations to very high precision and then using integer relation algorithms. Based on this approach, we complete Croft's solution to the problem concerning maximal inclusions of regular three-dimensional polyhedra by describing inclusions for the six remaining cases.

math.MG↗