Numbers Already Carry Their Own Embeddings

SuHyun Bae, Donghun Lee

Keywords: numerical representation, algebraic structure, embeddings

TL;DR: We present Adelic Operation-preserved Embedding (AOE), a training-free numerical representation that combines real and $p$-adic components to preserve additive and multiplicative structure, dropping seamlessly into standard Transformers.

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 drop 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.

Submission Number: 136