AlphaGeometry: Automated Geometric Reasoning

• AlphaGeometry is a multifaceted computational framework integrating neuro-symbolic theorem proving, positive geometry constructions, and generalized metric analysis for advanced geometric problem solving.
• It features AG1 and AG2 systems, with AG2 achieving 84% IMO problem solving and near-cubic time algorithms via a shared ensemble search paradigm.
• It extends into physics with positive geometry in scattering amplitudes and into machine learning with robust metric measures, enabling innovative interdisciplinary applications.

AlphaGeometry is a multifaceted research program and suite of computational frameworks advancing geometric problem-solving across symbolic AI, mathematical physics, and geometric data analysis. It chiefly denotes two domains: (1) neuro-symbolic theorem proving for Olympiad-level geometry, and (2) new positive geometry constructions in the context of scattering amplitudes and generalized metric spaces. This article details both aspects, focusing on formal architectures, language design, algorithmic innovations, empirical performance, and connections to broader research areas.

1. Neuro-Symbolic Automated Geometry (AlphaGeometry1, AlphaGeometry2)

AlphaGeometry was introduced as a neuro-symbolic system for synthetic Euclidean geometry reasoning at International Mathematical Olympiad (IMO) levels, integrating domain-specific formal language, symbolic engines, statistical proof search, and deep learning components (Sinha et al., 9 Apr 2024, Chervonyi et al., 5 Feb 2025). The original implementation, AlphaGeometry1 (AG1), formalized geometry problems using nine predicates, processed by a transformer-based LLM and the DDAR1 symbolic engine. AG1 was limited by narrow language expressivity and computational bottlenecks—its coverage of IMO geometry problems (2000–2024) was 66%, with a 54% overall solving rate.

AlphaGeometry2 (AG2) represents a major advancement. The domain language now covers locus-type statements (movement predicates), linear angle/ratio/distance equations, non-constructive “find x” problem types, and explicit diagram predicates, increasing coverage to 88%. AG2 employs the DDAR2 symbolic engine, which supports double point constructions via auxiliary intersection points, near-cubic time algorithms for triangle/cyclic figure detection, and C++ Gaussian elimination for 300× speedups compared to prior versions. Synthetic data generation is enhanced to produce diverse, larger-dimension problems and balanced premise types.

AG2’s proof search is orchestrated by the SKEST algorithm: a Shared Knowledge Ensemble of Search Trees, where multiple search trees independently explore solution paths, sharing a centralized fact database to improve search efficiency. The LLM is Gemini-based, trained on over 300M synthetic proofs, with multimodal and math-pretrained variants. The LM is conditioned on premise-derived, goal-inferring, and diagram-verified fact strings.

Empirically, AG2 solves 84% of all IMO geometry problems from 2000–2024—the highest among all published systems—and outperforms top IMO gold medalists on the IMO-AG-50 benchmark (42/50 problems). AG2 contributed to the silver-medalist system at IMO 2024; solutions were often rated as “superhumanly creative.”

2. Symbolic Engines, Algebraic Methods, and Hybridization

AlphaGeometry’s symbolic backbone originates in Deductive Database (DD) and Angle/Ratio/Distance (AR) reasoning. Geometric facts are incrementally deduced from a base of ≈70 human-crafted inference rules, leveraging forward chaining in DD and Gaussian elimination in AR modules for linear relationships. Problems unsolved by pure DD+AR (14/30 on AG-30) are typically those requiring sophisticated construction or algebraic elimination.

Wu’s method, a classical algebraic approach based on polynomial elimination and Ritt–Wu characteristic set computation, provides complementary strength. While not human-readable, its algebraic proofs often handle degenerate or symmetric cases inaccessible to synthetic pattern matching (Sinha et al., 9 Apr 2024). Hybrid systems combining AG with Wu’s method can solve the maximum known number of AG-30 IMO problems (27/30), exceeding gold-medalist human performance. This suggests that symbolically and algebraically complementary systems define the current best practice; future research may further integrate algebraic engines and synthetic symbolic reasoning beyond planar geometry.

3. Language, Domain Extensions, and Formalization

The AlphaGeometry language, especially in AG2, is intentionally crafted for high expressivity. Movements (“locus-type”), explicit equality and inequality of distances/angles/ratios (distmeq, distseq, angeq), and point-construction relaxation cater for the full spectrum of IMO-level geometric problems. Non-constructive goals (acompute/rcompute) and diagram predicates (sameclock, cyclic_with_center, noverlap) handle orientation, non-degeneracy, and double-point reasoning.

Automatic translation of problem statements into formal language representations is achieved via few-shot prompting with Gemini, enabling auto-formalization of English text into AG2 format. Automated diagram generation for non-constructive problems involves a two-stage optimization (ADAM plus Gauss–Newton–Levenberg) over coordinates to satisfy explicit geometric and non-degeneracy constraints, scaling to all formalizable IMO problems on practical hardware.

4. Comparison, Benchmarks, and Quantitative Performance

AlphaGeometry defines multiple performance benchmarks:

System	IMO Coverage	AG-30 Problems Solved	IMO-AG-50 Problems Solved	AG-SL-30 (Hard Shortlist)
AG1 (2024)	66%	25/30	27/50	n/a
DDAR2 pure	n/a	16/50	n/a	n/a
AG2 (2025)	88%	n/a	42/50 (84%)	20/30
Wu’s Method	n/a	15/30	n/a	n/a
DD+AR+Wu (symbolic ensemble)	n/a	21/30	n/a	n/a
AG+Wu (hybrid)	n/a	27/30	n/a	n/a
TongGeometry (recent system)	n/a	≤30/50	n/a	n/a

These results demonstrate AG2 and its hybrid variants are state-of-the-art in automated synthetic geometry, with the symbolically enriched engines (DDAR2, SKEST) providing robust solutions on classical and non-constructive problems (Chervonyi et al., 5 Feb 2025, Sinha et al., 9 Apr 2024).

5. Automation, End-to-End Pipelines, and Extensions

Toward fully automated geometric reasoning, AlphaGeometry2 deploys:

• Auto-formalization via Gemini-based LM
• Numeric optimization-based diagram generation for metric and locus problems
• Integration with reinforcement learning and subproblem decomposition, with ongoing work targeting inequalities, non-linear constraints, and adaptive variable-size configurations
• Reduction of formalization errors by supervised fine-tuning with larger datasets

This pipeline forms the basis for next-generation end-to-end systems capable of solving geometric problems directly from natural-language input, enabling future applications in educational technology, interactive mathematics, and scientific computing.

6. Positive Geometry: Stringy Amplitudes and the Associahedral Grid

In theoretical physics, “AlphaGeometry” also denotes the associahedral grid, a positive geometry construction realizing the inverse string-theory KLT kernel, generalizing rational amplitudes to stringy, trigonometric domains (Bartsch et al., 27 Aug 2025). For n-point scattering, the ABHY associahedron in kinematic space is uplifted as an infinite lattice of translates:

$$\mathcal{A}_n^{\alpha'} = \mathcal{A}_n + \frac{1}{\alpha'} \mathbb{Z}^{n-3}$$

The canonical form on this grid encodes:

• Poles at $X_{ij} = k/\alpha'$ representing resonance towers
• Factorization properties (pinching boundaries yields products of lower-dimensional grids)
• Kinematic δ-shifts, geometricizing BAS–NLSM relations
• Trigonometric amplitude expressions with string-theoretic $\alpha'$-dependence

In the field-theory limit $\alpha' \to 0$, the grid collapses to the classical associahedron. Thus, AlphaGeometry provides a unifying geometric object for string amplitudes, pion amplitudes, and their field-theory relations.

7. Generalized Metric Spaces: Alpha-Procrustes Geometry

A related thread extends AlphaGeometry to robust metric geometry of SPD matrices in machine learning and functional data analysis (Goomanee et al., 12 Nov 2025). The Alpha-Procrustes family generalizes classical Bures–Wasserstein, Log-Euclidean, and Wasserstein distances using unitized Hilbert–Schmidt operators and learnable Mahalanobis kernels with regularization:

$$d^\alpha_{ProM_\infty}[X+\mu I, Y+\mu I] = \min_{Q \in U(H)} \| (X+\mu I)^\alpha - (Y+\mu I)^\alpha Q \|_{M^{-1}_\infty}$$

Special cases include GBW, generalized Log-Hilbert–Schmidt, and Wasserstein metrics. Regularization parameter $\rho$ ensures spectral stability in infinite dimensions. Empirical findings demonstrate that properly regularized generalized metrics improve performance in discriminating functional shapes, covariance kernels, and Gaussian process datasets. This suggests AlphaGeometry-inspired metrics serve as robust building blocks for high-dimensional functional analysis and geometric machine learning.

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AlphaGeometry thus encompasses a set of powerful frameworks for automated geometric problem solving, positive geometry in scattering amplitudes, and generalized metric geometry. Each domain leverages domain-specific languages, algorithmic innovation, and data-driven approaches to expand the boundaries of automated reasoning and computational geometry (Chervonyi et al., 5 Feb 2025, Sinha et al., 9 Apr 2024, Bartsch et al., 27 Aug 2025, Goomanee et al., 12 Nov 2025).