Beyond Parameters: Exploring Virtual Logic Depth for Scaling Laws

Scaling large language models typically involves three dimensions: depth, width, and parameter count. In this work, we explore a fourth dimension, virtual logical depth (VLD), which increases effective algorithmic depth without changing parameter count by reusing weights. While parameter reuse is not new, its role in scaling has been underexplored. Unlike recent test-time methods that scale token-wise, VLD alters the internal computation graph during training and inference. Through controlled experiments, we obtain three key insights. (1) textitKnowledge capacity vs. parameters: at fixed parameter count, VLD leaves knowledge capacity nearly unchanged, while across models capacity still scales with parameters. (2) textitReasoning vs. reuse: properly implemented VLD substantially improves reasoning ability without more parameters, decoupling reasoning from size. This suggests a new scaling path beyond token-wise test-time methods. (3) textitRobustness and generality: reasoning gains persist across architectures and reuse schedules, showing VLD captures a general scaling behavior. These results provide insight into future scaling strategies and raise a deeper question: does superintelligence require ever-larger models, or can it be achieved by reusing parameters and increasing logical depth? We argue many unknown dynamics in scaling remain to be explored. Code is available at https://anonymous.4open.science/r/virtuallogicaldepth-8024/.