# Beyond Parameters: Exploring Virtual Logic Depth for Scaling Laws

Authors: Ruike Zhu, Hanwen Zhang, Kevin Li, Tianyu Shi, Yiqun Duan, Chi Wang, Tianyi Zhou, Arindam Banerjee, Zengyi Qin.

Repeats transformer layers with shared parameters in several reuse patterns. Compares how added computation affects memorization and reasoning at a fixed parameter count, using synthetic tasks and language-model benchmarks.

First public: 2025-06-23.
ArXiv initial submission: 2025-06-23.

[Original paper](https://arxiv.org/abs/2506.18233) · [PDF](https://arxiv.org/pdf/2506.18233) · [HTML](https://arxiv.org/html/2506.18233v3) · [Reading-list entry](https://latent-implicit-thinking.github.io/papers/2506.18233/)
[Preprint BibTeX](https://latent-implicit-thinking.github.io/papers/2506.18233/citation.bib) · [Preprint CSL-JSON](https://latent-implicit-thinking.github.io/papers/2506.18233/citation.csl.json) · [JSON record](https://latent-implicit-thinking.github.io/papers/2506.18233/record.json)

## Approaches

- [Recurrent reasoning](https://latent-implicit-thinking.github.io/topics/recurrent-reasoning/): Repeat parts of the model or keep intermediate information in memory while working through a problem. Some methods also learn how many passes to run.
- [Analysis and evaluation](https://latent-implicit-thinking.github.io/topics/implicit-reasoning-evaluation/): Measure how well these methods work, investigate what their internal computation does, and identify where they fail. This includes experiments, studies of model internals, and mathematical results.

Reading-list version 0.15.2; literature checked through 2026-11-03.