To improve performance in contemporary deep learning, one is interested in scaling up the neural network in terms of both the number and the size of the layers. When ramping up the width of a single layer, graceful scaling of training has been linked to the need to normalize the weights and their updates in the "natural norm" particular to that layer. In this paper, we significantly generalize this idea by defining the modular norm, which is the natural norm on the full weight space of any neural network architecture. The modular norm is defined recursively in tandem with the network architecture itself. We show that the modular norm has several promising applications.

Large Language Models (LLMs) excel in reasoning but remain constrained by their Chain-of-Thought (CoT) approach, which struggles with complex tasks requiring more nuanced topological reasoning. We introduce SOLAR, Scalable Optimization of Large-scale Architecture for Reasoning, a framework that dynamically optimizes various reasoning topologies to enhance accuracy and efficiency. Our Topological Annotation Generation (TAG) system automates topological dataset creation and segmentation, improving post-training and evaluation. Additionally, we propose Topological-Scaling, a reward-driven framework that aligns training and inference scaling, equipping LLMs with adaptive, task-aware reasoning. SOLAR achieves substantial gains on MATH and GSM8K: +5% accuracy with Topological Tuning, +9% with Topological Reward, and +10.02% with Hybrid Scaling. It also reduces response length by over 5% for complex problems, lowering inference latency. To foster the reward system, we train a multi-task Topological Reward Model (M-TRM), which autonomously selects the best reasoning topology and answer in a single pass, eliminating the need for training and inference on multiple single-task TRMs (S-TRMs), thus reducing both training cost and inference latency. In addition, in terms of performance, M-TRM surpasses all S-TRMs, improving accuracy by +10% and rank correlation by +9%. To the best of our knowledge, SOLAR sets a new benchmark for scalable, high-precision LLM reasoning while introducing an automated annotation process and a dynamic reasoning topology competition mechanism.