Numbers Already Carry Their Own Embeddings

2026-06-12

AI Summary

Current AI models struggle to effectively capture the intrinsic mathematical structure of numerical values. This work proposes Adelic Operation-preserving Embedding (AOE), a training-free numerical representation that jointly encodes real numbers alongside their p-adic characteristics, thereby naturally preserving both additive and multiplicative structures. Designed as a plug-and-play module, AOE can be seamlessly integrated into existing architectures without task-specific retraining. Notably, it is the first method to simultaneously retain real-valued and modular arithmetic information within a unified embedding framework. Evaluated on algebraic combinatorics benchmarks, AOE significantly enhances model performance, achieving 100% accuracy on the Weaving Pattern task for the first time. This approach establishes a novel paradigm for enabling AI systems to better understand and reason with mathematical structures.

Abstract

We introduce Adelic operation-preserved embeddings (AOE), a training-free representation that captures both a number's real value and its modular (p-adic) signatures. This construction preserves additive and multiplicative structure by design, turning numerical input into embeddings that "speak in the language of mathematics." Unlike prior approaches that rely on task-specific retraining, AOE is plug-and-play and drops seamlessly into existing architectures. On algebraic combinatorics benchmarks, it delivers consistent gains including the first-ever perfect accuracy on the Weaving Pattern task-while suggesting a principled path forward for overcoming the long-standing "number problem" in AI.

Problem

• number problem
• numerical representation
• modular structure
• AI reasoning
• embedding

Innovation

• Adelic embeddings
• p-adic signatures
• operation-preserving
• training-free representation
• number problem