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

TL;DR

This work addresses the variability in post-RLVR reasoning across pre-trained LLMs by introducing the Soundness-Aware Level (SAL), a microscopic signature derived from internal representations. By framing reasoning as Horn clauses and extracting them from cross-layer sparse autoencoders, the authors quantify how well a model distinguishes sound knowledge from noise, using Jensen-Shannon Divergence across three soundness levels. SAL robustly predicts post-RLVR performance via an empirical law, ε = exp(-α · SAL^β), with high fit (R^2 ≈ 0.985 on observed models and R^2 ≈ 0.872 in leave-one-out validation) across multiple model families and scales, suggesting that pre-training implicitly tunes reasoning potential. The findings imply that scale and architecture influence a model’s internal ability to separate reliable knowledge, offering a practical, intrinsic metric to guide model selection and pre-training design toward stronger reasoning capabilities.

Abstract

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.

Figures (10)

• Figure 1: An illustration of our method for probing the internal logic of a pre-trained LLM. The process extracts and analyzes the reasoning rules that a model has already learned from its pre-training. Step 1: A cross-layer sparse autoencoder reads out the semantically meaningful features from the LLM's hidden activations. Step 2: By tracking feature co-occurrences, we extract the implicit logic rules the model has learned (e.g., $c_1\wedge c_3\rightarrow c_2$), estimating the conditional probabilities it has assigned for these entailments. Step 3: We then assess the quality of extracted rules. Each rule is labeled with a soundness level by an LLM judge, and we compute the Soundness-Aware Level (SAL) by measuring the JS-divergence between the distributions of different soundness levels. A larger SAL indicates the pre-trained model has more effectively learned to separate its sound knowledge from unsound ones, which in turn predicts its future reasoning potential.
• Figure 2: Probability distribution of extracted logic rules from a higher-potential (Qwen‑2.5‑7B) model and a lower-potential (Llama‑3.1‑8B) model. The higher-potential model shows significantly different distributions for No, Plausible, and Strict logic rules, whereas the lower-potential model collapses toward similar shapes, indicating a failure to recognize different soundness levels.
• Figure 3: Jensen-Shannon Divergence (JSD) quantifies distribution shifts between probabilities of soundness levels (S: "Strict", P: "Plausible", & N: "No").
• Figure 4: Left: Correlation between the SAL over extracted rules and models' post-RLVR performance. Right: The exponential power law $\epsilon=\exp(-\alpha\cdot s^{\,\beta})$ describes the correlation between SAL $s$ and the error rate $\epsilon$ of solving mathematical problems. The best fitted model is $\alpha = 4.246$ and $\beta = 1.090$ with $R^{2}=0.985$ for interpolation fitting observed models.
• Figure 5: Deconstructing the Soundness-Awareness Level (SAL). (Left) SAL increases with model scale within the Qwen-2.5 family. (Right) At a comparable 7B scale, SAL varies significantly across different model families, indicating that architecture and pre-training data are also key factors.