# Constructive Circuit Amplification: Improving Math Reasoning in LLMs via Targeted Sub-Network Updates

arXiv: https://arxiv.org/abs/2512.16914 · HTML: https://mlsanity.com/paper/2512.16914/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 31 of its 42 references that are in the map.
- First posted on arXiv 18 Dec 2025.

## Placed in

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

## Builds on (most cited first, 30 of 31)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [Program of Thoughts Prompting: Disentangling Computation from Reasoning for Numerical Reasoning Tasks](https://mlsanity.com/paper/2211.12588/) (2022)
- [Gemma: Open Models Based on Gemini Research and Technology](https://mlsanity.com/paper/2403.08295/) (2024)
- [An Empirical Study of Catastrophic Forgetting in Large Language Models During Continual Fine-tuning](https://mlsanity.com/paper/2308.08747/) (2023)
- [OLMo: Accelerating the Science of Language Models](https://mlsanity.com/paper/2402.00838/) (2024)
- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [Sparse Feature Circuits: Discovering and Editing Interpretable Causal Graphs in Language Models](https://mlsanity.com/paper/2403.19647/) (2024)
- [The Illusion of Thinking: Understanding the Strengths and Limitations of Reasoning Models via the Lens of Problem Complexity](https://mlsanity.com/paper/2506.06941/) (2025)
- [How does GPT-2 compute greater-than?: Interpreting mathematical abilities in a pre-trained language model](https://mlsanity.com/paper/2305.00586/) (2023)
- [Fine-Tuning Enhances Existing Mechanisms: A Case Study on Entity Tracking](https://mlsanity.com/paper/2402.14811/) (2024)
- [Mechanistically analyzing the effects of fine-tuning on procedurally defined tasks](https://mlsanity.com/paper/2311.12786/) (2023)
- [Circuit Component Reuse Across Tasks in Transformer Language Models](https://mlsanity.com/paper/2310.08744/) (2023)
- [Physics of Language Models: Part 2.1, Grade-School Math and the Hidden Reasoning Process](https://mlsanity.com/paper/2407.20311/) (2024)
- [DeepSeek-R1 Thoughtology: Let's think about LLM Reasoning](https://mlsanity.com/paper/2504.07128/) (2025)
- [A Survey on Large Language Models for Mathematical Reasoning](https://mlsanity.com/paper/2506.08446/) (2025)
- [Reinforcement Learning Finetunes Small Subnetworks in Large Language Models](https://mlsanity.com/paper/2505.11711/) (2025)
- [NNsight and NDIF: Democratizing Access to Open-Weight Foundation Model Internals](https://mlsanity.com/paper/2407.14561/) (2024)
- [The relationship between reasoning and performance in large language models--o3 (mini) thinks harder, not longer](https://mlsanity.com/paper/2502.15631/) (2025)
- [Language Models use Lookbacks to Track Beliefs](https://mlsanity.com/paper/2505.14685/) (2025)
- [Discovering Variable Binding Circuitry with Desiderata](https://mlsanity.com/paper/2307.03637/) (2023)
- [Mechanistic?](https://mlsanity.com/paper/2410.09087/) (2024)
- [The Quest for the Right Mediator: Surveying Mechanistic Interpretability Through the Lens of Causal Mediation Analysis](https://mlsanity.com/paper/2408.01416/) (2024)
- [Probing for Arithmetic Errors in Language Models](https://mlsanity.com/paper/2507.12379/) (2025)
- [ThinkEdit: Interpretable Weight Editing to Mitigate Overly Short Thinking in Reasoning Models](https://mlsanity.com/paper/2503.22048/) (2025)
- [LIFT the Veil for the Truth: Principal Weights Emerge after Rank Reduction for Reasoning-Focused Supervised Fine-Tuning](https://mlsanity.com/paper/2506.00772/) (2025)
- [Failure by Interference: Language Models Make Balanced Parentheses Errors When Faulty Mechanisms Overshadow Sound Ones](https://mlsanity.com/paper/2507.00322/) (2025)

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

- [Circuit Condensation: Post-Training that Concentrates a Behavior's Causal Circuit](https://mlsanity.com/paper/2608.27254/) (2026)

## Abstract (arXiv)

Prior studies investigating the internal workings of LLMs have uncovered sparse subnetworks, often referred to as circuits, that are responsible for performing specific tasks. Additionally, it has been shown that model performance improvement through fine-tuning often results from the strengthening of existing circuits in the model. Taken together, these findings suggest the possibility of intervening directly on such circuits to make precise, task-targeted updates. Motivated by these findings, we propose a novel method called Constructive Circuit Amplification which identifies pivotal tokens from model reasoning traces as well as model components responsible for the desired task, and updates only those components. Applied to mathematical reasoning, it improves accuracy by up to +11.4% across multiple models while modifying as little as 1.59% of model components, with minimal impact on other abilities as measured by MMLU, TriviaQA, and TruthfulQA. These results demonstrate that targeted capabilities can be reliably enhanced by selectively updating a sparse set of model components.