# The Topological Dual of a Dataset: A Logic-to-Topology Encoding for AlphaGeometry-Style Data

arXiv: https://arxiv.org/abs/2604.18050 · HTML: https://mlsanity.com/paper/2604.18050/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 8 of its 16 references that are in the map.
- First posted on arXiv 20 Apr 2026.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 8 of 8)

- [DeepSeek-Prover-V2: Advancing Formal Mathematical Reasoning via Reinforcement Learning for Subgoal Decomposition](https://mlsanity.com/paper/2504.21801/) (2025)
- [PutnamBench: Evaluating Neural Theorem-Provers on the Putnam Mathematical Competition](https://mlsanity.com/paper/2407.11214/) (2024)
- [Goedel-Prover-V2: Scaling Formal Theorem Proving with Scaffolded Data Synthesis and Self-Correction](https://mlsanity.com/paper/2508.03613/) (2025)
- [Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2](https://mlsanity.com/paper/2502.03544/) (2025)
- [Hilbert: Recursively Building Formal Proofs with Informal Reasoning](https://mlsanity.com/paper/2509.22819/) (2025)
- [Seed-Prover 1.5: Mastering Undergraduate-Level Theorem Proving via Learning from Experience](https://mlsanity.com/paper/2512.17260/) (2025)
- [Ax-Prover: A Deep Reasoning Agentic Framework for Theorem Proving in Mathematics and Quantum Physics](https://mlsanity.com/paper/2510.12787/) (2025)
- [Bourbaki: Self-Generated and Goal-Conditioned MDPs for Theorem Proving](https://mlsanity.com/paper/2507.02726/) (2025)

## Abstract (arXiv)

AlphaGeometry represents a milestone in neuro-symbolic reasoning, yet its architecture faces a log-linear scaling bottleneck within its symbolic deduction engine that limits its efficiency as problem complexity increases. Recent technical reports suggest that current domain-specific languages may be isomorphic as input representations to natural language, interchanging them acts as a performance-invariant transformation, implying that current neural guidance relies on superficial encodings rather than structural understanding. This paper addresses this representation bottleneck by proposing a logic-to-topology encoding designed to reveal the structural invariants of a model's latent space under a transformation of its input space. By leveraging the Logic of Observation, we utilize the duality between provability in observable theories and topologies to propose a logic-to-topology encoder for the input space. We introduce the concept of the "topological dual of a dataset", a transformation that bridges formal logic, topology, and neural processing. This framework serves as a Rosetta Stone for neuro-symbolic AI, providing a principled pathway for the mechanistic interpretability of how models navigate complex discovery paths.