# Knowledge Knows, Verbalization Tells: Disentangling Latent Directions for Mathematical Solvability in LLMs

arXiv: https://arxiv.org/abs/2607.05013 · HTML: https://mlsanity.com/paper/2607.05013/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 14 of its 29 references that are in the map.
- First posted on arXiv 6 Jul 2026.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

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

- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks](https://mlsanity.com/paper/1703.10593/) (2017)
- [Automatic diagnosis of the 12-lead ECG using a deep neural network](https://mlsanity.com/paper/1904.01949/) (2019)
- [Deep Learning for ECG Analysis: Benchmarks and Insights from PTB-XL](https://mlsanity.com/paper/2004.13701/) (2020)
- [HOLMES: Health OnLine Model Ensemble Serving for Deep Learning Models in Intensive Care Units](https://mlsanity.com/paper/2008.04063/) (2020)
- [Deep neural networks can predict mortality from 12-lead electrocardiogram voltage data](https://mlsanity.com/paper/1904.07032/) (2019)
- [ECG-Image-Kit: A Synthetic Image Generation Toolbox to Facilitate Deep Learning-Based Electrocardiogram Digitization](https://mlsanity.com/paper/2307.01946/) (2023)
- [AnyPPG: An ECG-Guided PPG Foundation Model Trained on Over 100,000 Hours of Recordings for Holistic Health Profiling](https://mlsanity.com/paper/2511.01747/) (2025)
- [Multi-Channel Masked Autoencoder and Comprehensive Evaluations for Reconstructing 12-Lead ECG from Arbitrary Single-Lead ECG](https://mlsanity.com/paper/2407.11481/) (2024)
- [ECG-R1: Protocol-Guided and Modality-Agnostic MLLM for Reliable ECG Interpretation](https://mlsanity.com/paper/2602.04279/) (2026)
- [PPGFlowECG: Latent Rectified Flow with Cross-Modal Encoding for PPG-Guided ECG Generation and Cardiovascular Disease Detection](https://mlsanity.com/paper/2509.19774/) (2025)
- [Self-Alignment Learning to Improve Myocardial Infarction Detection from Single-Lead ECG](https://mlsanity.com/paper/2509.19397/) (2025)
- [ECGrecover: a Deep Learning Approach for Electrocardiogram Signal Completion](https://mlsanity.com/paper/2406.16901/) (2024)
- [Digitizing Paper ECGs at Scale: An Open-Source Algorithm for Clinical Research](https://mlsanity.com/paper/2510.19590/) (2025)

## Built on by (most cited first, 2 of 2)

- [More Capable, Less Faithful: A Multilingual Analysis of Mathematical (Un)Solvability Detection in LLMs](https://mlsanity.com/paper/2608.30463/) (2026)
- [COMPASS: Finding Where Reasoning Lives in Language Models](https://mlsanity.com/paper/2610.07469/) (2026)

## Abstract (arXiv)

Although LLMs have made significant progress in mathematical reasoning, determining whether a mathematical problem is solvable remains a fundamental yet challenging capability. While recent studies have probed internal representations of model solvability beliefs, verbalization has primarily been studied behaviorally rather than as an internal representation, limiting its analysis and manipulation. We address this gap by separately probing representations of solvability knowledge and verbalization, allowing us to disentangle the two within model hidden states. Across multiple LLMs, we show that knowledge and verbalization are encoded as distinct, linearly decodable representations and that fabrication is primarily associated with changes in verbalization rather than the underlying knowledge. Prompting with unsolvability cues reduces fabrication primarily by shifting verbalization, while activation steering demonstrates that these representations can be echanistically manipulated to improve model abstention.