# When a Conceptor Complement Becomes Gain: Operator Geometry and Behavioral Attribution in Language Models

arXiv: https://arxiv.org/abs/2610.16325 · HTML: https://mlsanity.com/paper/2610.16325/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 186 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Builds on (most cited first, 20 of 20)

- [Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena](https://mlsanity.com/paper/2306.05685/) (2023)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [Null It Out: Guarding Protected Attributes by Iterative Nullspace Projection](https://mlsanity.com/paper/2004.07667/) (2020)
- [Persona Vectors: Monitoring and Controlling Character Traits in Language Models](https://mlsanity.com/paper/2507.21509/) (2025)
- [Function Vectors in Large Language Models](https://mlsanity.com/paper/2310.15213/) (2023)
- [LEACE: Perfect linear concept erasure in closed form](https://mlsanity.com/paper/2306.03819/) (2023)
- [AxBench: Steering LLMs? Even Simple Baselines Outperform Sparse Autoencoders](https://mlsanity.com/paper/2501.17148/) (2025)
- [Analyzing the Generalization and Reliability of Steering Vectors](https://mlsanity.com/paper/2407.12404/) (2024)
- [Improving Activation Steering in Language Models with Mean-Centring](https://mlsanity.com/paper/2312.03813/) (2023)
- [Controlling Language and Diffusion Models by Transporting Activations](https://mlsanity.com/paper/2410.23054/) (2024)
- [Spectral Editing of Activations for Large Language Model Alignment](https://mlsanity.com/paper/2405.09719/) (2024)
- [Angular Steering: Behavior Control via Rotation in Activation Space](https://mlsanity.com/paper/2510.26243/) (2025)
- [The Hidden Dimensions of LLM Alignment: A Multi-Dimensional Analysis of Orthogonal Safety Directions](https://mlsanity.com/paper/2502.09674/) (2025)
- [Steering Large Language Models using Conceptors: Improving Addition-Based Activation Engineering](https://mlsanity.com/paper/2410.16314/) (2024)
- [Towards Reliable Evaluation of Behavior Steering Interventions in LLMs](https://mlsanity.com/paper/2410.17245/) (2024)
- [Steering off Course: Reliability Challenges in Steering Language Models](https://mlsanity.com/paper/2504.04635/) (2025)
- [LinEAS: End-to-end Learning of Activation Steering with a Distributional Loss](https://mlsanity.com/paper/2503.10679/) (2025)
- [Conceptor-Aided Debiasing of Large Language Models](https://mlsanity.com/paper/2211.11087/) (2022)
- [Conceptors: an easy introduction](https://mlsanity.com/paper/1406.2671/) (2014)
- [Conceptors for Semantic Steering](https://mlsanity.com/paper/2605.04980/) (2026)

## Abstract (arXiv)

When a steering intervention changes a language model's behavior, which part of the intervention deserves the credit? We test this question for a target-centered, scaled conceptor complement by removing its fitted covariance filter while preserving the same target-specific center and gain. At the original aperture α=.05 and gain β=2, all 21 fitted maps across three models and seven targets expand every direction at the generation intervention layers. The matrix correction is 0.093%--7.815% of the matched-control displacement on held-out extraction activations. In 480 paired comparisons within a 1,920-response study, retaining the matrix changes many individual texts but does not establish a consistent additional target-suppression benefit on the four-target panel. A geometric aperture sweep shows that the operating regime changes with the filter setting. Separately, an invertible soft filter preserves what an unrestricted affine readout can recover, whereas dedicated erasure methods reduce prediction by newly fitted probes. Together, the results show why a whole-intervention comparison cannot by itself attribute generated behavior to the fitted matrix: analyze the complete transformation, hold shared components fixed, and test its output consequences directly.