# Quantifying the Necessity of Chain of Thought through Opaque Serial Depth

arXiv: https://arxiv.org/abs/2603.09786 · HTML: https://mlsanity.com/paper/2603.09786/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Prompting and chain-of-thought reasoning, one of 17 directions in LLM reasoning.
- Builds on 14 of its 29 references that are in the map.
- First posted on arXiv 10 Mar 2026.

## Placed in

- LLM reasoning / [Prompting and chain-of-thought reasoning](https://mlsanity.com/lr/prompting-and-chain-of-thought-reasoning/)

## Builds on (most cited first, 14 of 14)

- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [Gemma 3 Technical Report](https://mlsanity.com/paper/2503.19786/) (2025)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [Training Large Language Models to Reason in a Continuous Latent Space](https://mlsanity.com/paper/2412.06769/) (2024)
- [Chain of Thought Monitorability: A New and Fragile Opportunity for AI Safety](https://mlsanity.com/paper/2507.11473/) (2025)
- [Soft Thinking: Unlocking the Reasoning Potential of LLMs in Continuous Concept Space](https://mlsanity.com/paper/2505.15778/) (2025)
- [When Chain of Thought is Necessary, Language Models Struggle to Evade Monitors](https://mlsanity.com/paper/2507.05246/) (2025)
- [Monitoring Monitorability](https://mlsanity.com/paper/2512.18311/) (2025)
- [LLMs are Single-threaded Reasoners: Demystifying the Working Mechanism of Soft Thinking](https://mlsanity.com/paper/2508.03440/) (2025)
- [Text Generation Beyond Discrete Token Sampling](https://mlsanity.com/paper/2505.14827/) (2025)
- [MONA: Myopic Optimization with Non-myopic Approval Can Mitigate Multi-step Reward Hacking](https://mlsanity.com/paper/2501.13011/) (2025)
- [A Pragmatic Way to Measure Chain-of-Thought Monitorability](https://mlsanity.com/paper/2510.23966/) (2025)
- [State over Tokens: Characterizing the Role of Reasoning Tokens](https://mlsanity.com/paper/2512.12777/) (2025)

## Built on by (most cited first, 8 of 8)

- [Think Fast: Estimating No-CoT Task-Completion Time Horizons of Frontier AI Models](https://mlsanity.com/paper/2606.07157/) (2026)
- [Aligned, Orthogonal or In-conflict: When can we safely optimize Chain-of-Thought?](https://mlsanity.com/paper/2603.30036/) (2026)
- [Full-bandwidth transformer](https://mlsanity.com/paper/2608.08888/) (2026)
- [How Transparent is DiffusionGemma?](https://mlsanity.com/paper/2606.20560/) (2026)
- [Chain-of-Thought Faithfulness of Reasoning Models Varies with Where and How Preference Cues Are Delivered](https://mlsanity.com/paper/2608.29464/) (2026)
- [How Many Thoughts Can a Vector Hold? The Capacity of Reasoning by Superposition](https://mlsanity.com/paper/2609.13747/) (2026)
- [Hidden Reasoning Must Leak, but Need Not Be Readable: Fundamental Opportunities and Limits for Chain-of-Thought Monitoring](https://mlsanity.com/paper/2609.37312/) (2026)
- [Correct Answers, Invalid Traces: What Verifiable Grade-School Math Reveals About Chain-of-Thought Traces](https://mlsanity.com/paper/2609.38107/) (2026)

## Abstract (arXiv)

Large language models (LLMs) tend to externalize their reasoning in their chain of thought, making the chain of thought a good target for monitoring. This is partially an inherent feature of the Transformer architecture: sufficiently long serial cognition must pass through the chain of thought (Korbak et al., 2025). We formalize this argument through the notion of opaque serial depth, given by the length of the longest computation that can be done without the use of interpretable intermediate steps like chain of thought. Given this formalization, we compute numeric upper bounds on the opaque serial depth of Gemma 3 models, as well as asymptotic results for additional architectures beyond standard LLMs. We also open-source an automated method that can calculate upper bounds on the opaque serial depth of arbitrary neural networks, and use it to demonstrate that Mixture-of-Experts models likely have lower depth than dense models. Overall, our results suggest that opaque serial depth is a useful tool for understanding the potential for models to do significant reasoning that is not externalized.