# Asterisk Operator

arXiv: https://arxiv.org/abs/2509.13364 · HTML: https://mlsanity.com/paper/2509.13364/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 7 references that are in the map.
- First posted on arXiv 15 Sep 2025.

## Builds on (most cited first, 4 of 4)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [TreeGPT: Pure TreeFFN Encoder-Decoder Architecture for Structured Reasoning Without Attention Mechanisms](https://mlsanity.com/paper/2509.05550/) (2025)

## Abstract (arXiv)

We propose the Asterisk Operator (ast-operator), a novel unified framework for abstract reasoning based on Adjacency-Structured Parallel Propagation (ASPP). The operator formalizes structured reasoning tasks as local, parallel state evolution processes guided by implicit relational graphs. We prove that the ast-operator maintains local computational constraints while achieving global reasoning capabilities, providing an efficient and convergent computational paradigm for abstract reasoning problems. Through rigorous mathematical analysis and comprehensive experiments on ARC2 challenges and Conway's Game of Life, we demonstrate the operator's universality, convergence properties, and superior performance. Our innovative Embedding-Asterisk distillation method achieves 100% accuracy on ARC2 validation with only 6M parameters, representing a significant breakthrough in neural-symbolic reasoning. Keywords: Abstract Reasoning, Adjacency Structure, Parallel Propagation, Asterisk Operator, Convergence, Universal Approximation