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, $\mathbf{\text{virtual}}\mathbf{\text{logical}}\mathbf{\text{depth}}$ (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) $\mathit{\text{Knowledge}}\mathit{\text{capacity}}\mathit{\text{vs.}}\mathit{\text{parameters}}$: at fixed parameter count, VLD leaves knowledge capacity nearly unchanged, while across models capacity still scales with parameters. (2) $\mathit{\text{Reasoning}}\mathit{\text{vs.}}\mathit{\text{reuse}}$: properly implemented VLD substantially improves reasoning ability \emph{without} more parameters, decoupling reasoning from size. This suggests a new scaling path beyond token-wise test-time methods. (3) $\mathit{\text{Robustness}}\mathit{\text{and}}\mathit{\text{generality}}$: 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/virtual_logical_depth-8024/.