Reasoning: From Reflection to Solution

TL;DR

This paper defines reasoning as iterative operator application in state spaces, proposes a theoretical framework, and introduces a new architecture that significantly outperforms current large language models on reasoning tasks.

Contribution

It offers a formal definition of reasoning, develops a theoretical model, and presents a novel architecture achieving high accuracy on reasoning benchmarks.

Findings

OpenXOR puzzle solved with 76% accuracy

Theoretical framework explains reasoning failures

New architecture outperforms state-of-the-art models

Abstract

What is reasoning? This question has driven centuries of philosophical inquiry, from Aristotle's syllogisms to modern computational complexity theory. In the age of large language models achieving superhuman performance on benchmarks like GSM8K (95\% accuracy) and HumanEval (90\% pass@1), we must ask: have these systems learned to \emph{reason}, or have they learned to \emph{pattern-match over reasoning traces}? This paper argues for a specific answer: \textbf{reasoning is iterative operator application in state spaces, converging to fixed points}. This definition is not merely philosophical -- it has concrete architectural implications that explain both the failures of current systems and the path to genuine reasoning capabilities. Our investigation begins with a puzzle (OpenXOR), progresses through theory (OpenOperator), and culminates in a working solution (OpenLM) that achieves 76\% accuracy where state-of-the-art LLMs achieve 0\%. This is not about criticizing existing systems, but about \emph{understanding what reasoning requires} and \emph{building architectures that provide it}.