Exact Null-space Inverse Operator

The EEG inverse problem is fundamentally ill-posed due to the nontrivial null space of the forward operator mapping cortical activity to sensor measurements. Classical inverse methods address this ambiguity through regularization and prior assumptions, trading reconstruction accuracy for stability. In this work, we present an inverse model that operates directly in the null space of a fixed forward model. For a given forward operator, the proposed method reconstructs arbitrary source-space signals with error bounded by numerical precision. Reconstruction accuracy is invariant to additive sensor-space noise and requires negligible computational cost at inference time, enabling real-time application. We empirically validate exact reconstruction across diverse source priors and analyze failure modes under forward-model mismatch and null-space rank degradation. The results demonstrate that null-space inversion, when explicitly parameterized, admits exact and noise-robust reconstruction for fixed forward models. This work presents a self-contained demonstration of exact null-space inversion for fixed forward models.