# Soundness-Aware Level: A Microscopic Signature that Predicts LLM Reasoning Potential

arXiv: https://arxiv.org/abs/2510.15216 · HTML: https://mlsanity.com/paper/2510.15216/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Audio-language model reasoning, one of 17 directions in LLM reasoning.
- Builds on 24 of its 36 references that are in the map.
- First posted on arXiv 17 Oct 2025.

## Placed in

- LLM reasoning / [Audio-language model reasoning](https://mlsanity.com/lr/audio-language-model-reasoning/)
- Mechanistic interpretability / [Steering and interpreting LLM reasoning](https://mlsanity.com/mechinterp/steering-and-interpreting-llm-reasoning/)

## Builds on (most cited first, 24 of 24)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Transformer Feed-Forward Layers Are Key-Value Memories](https://mlsanity.com/paper/2012.14913/) (2020)
- [Qwen2.5-Coder Technical Report](https://mlsanity.com/paper/2409.12186/) (2024)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Open-Reasoner-Zero: An Open Source Approach to Scaling Up Reinforcement Learning on the Base Model](https://mlsanity.com/paper/2503.24290/) (2025)
- [Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2](https://mlsanity.com/paper/2408.05147/) (2024)
- [Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs](https://mlsanity.com/paper/2503.01307/) (2025)
- [Reinforcement Learning with Verifiable Rewards Implicitly Incentivizes Correct Reasoning in Base LLMs](https://mlsanity.com/paper/2506.14245/) (2025)
- [Reasoning with Exploration: An Entropy Perspective](https://mlsanity.com/paper/2506.14758/) (2025)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [On a generalization of the Jensen-Shannon divergence and the JS-symmetrization of distances relying on abstract means](https://mlsanity.com/paper/1904.04017/) (2019)
- [Topology of Reasoning: Understanding Large Reasoning Models through Reasoning Graph Properties](https://mlsanity.com/paper/2506.05744/) (2025)
- [How Much Backtracking is Enough? Exploring the Interplay of SFT and RL in Enhancing LLM Reasoning](https://mlsanity.com/paper/2505.24273/) (2025)
- [Understanding the Thinking Process of Reasoning Models: A Perspective from Schoenfeld's Episode Theory](https://mlsanity.com/paper/2509.14662/) (2025)
- [Learning Language Representations with Logical Inductive Bias](https://mlsanity.com/paper/2302.09458/) (2023)

## Built on by (most cited first, 1 of 1)

- [SFT Leaves a Reproducible Signature in Activation Geometry: Calibrating Post-Training Interpretability Against Run-to-Run Noise](https://mlsanity.com/paper/2610.17093/) (2026)

## Abstract (arXiv)

Reinforcement learning with verifiable rewards (RLVR) can elicit strong reasoning in large language models (LLMs), while their performance after RLVR varies dramatically across different base models. This raises a fundamental question: what microscopic property of pre-trained models leads to this variation? To investigate, we formalize reasoning as chains of Horn clauses ("if-then" rules) built from features extracted from the LLM's latent space via cross-layer sparse autoencoders (SAEs). We estimate the transition probabilities between its features, and further categorize each rule by its semantic soundness level (e.g., strict, plausible, noisy) with an LLM. Our key discovery is that high-potential models are inherently soundness-aware: their internal probability distributions systematically shift across rules' soundness levels, becoming highly distinct for "strict" versus "noisy" rules. In contrast, weaker models are soundness-agnostic, collapsing to one distribution regardless of soundness levels. To quantify this, we introduce the Soundness-Aware Level (SAL), a microscopic metric using the Jensen-Shannon Divergence to measure the separation between these distributions. We show that SAL's predictions of post-RLVR reasoning performance follow a precise empirical law (R^2=0.87) across diverse model families (Qwen, Mistral, Llama, DeepSeek) and scales (0.5B-14B). This reveals that a model's reasoning potential is tied to its intrinsic, pre-trained ability to distinguish sound knowledge from unsound ones. These findings underscore the critical role of model pre-training in shaping reasoning and offer a practical metric grounded in the model's internal mechanisms for selecting/designing stronger base models.