Causal Intervention in Large Language Models via Circular Spectrum Observation

This paper investigates the controllability of logical representations in the hidden space of large language models (LLMs). By applying Logical Differential Vector (LDV) injection to the Qwen3-4B model, we observe a nonlinear phase transition from ambiguous semantics to rigorous logical deduction. To monitor this process in real-time, we propose an innovative "Circular Spectrum Observation" method that maps 2560-dimensional hidden states onto a polar coordinate system. Our experiments reveal that the model exhibits high intervention sensitivity at layer 23, while layer 29 demonstrates a "geometric singularity" with exceptional resistance to intervention. This work provides an intuitive visualization diagnostic tool and theoretical foundation for logic alignment without fine-tuning.