Representations + Lie Algebra = Explanations

Graph Neural Networks (GNNs) are powerful yet remain `black box' models, and their use in high-stakes domains requires explanation methods that are both mathematically principled and semantically faithful. Existing factual explainers identify influential subgraphs via discrete combinatorial search, while counterfactual methods perturb the latent embedding without geometric constraints, frequently producing out-of-distribution (OOD) artifacts that are infeasible or structurally invalid. We address both limitations by framing GNN reasoning as symmetry transformation on a learned latent manifold. To this end, we introduce GLIDE: Graph LIe Decomposition Explainer. We model the GNN latent space as a smooth manifold acted upon by a continuous Lie group $G$, and we learn a basis $\{A_k\}$ for the associated Lie algebra $\mathfrak{g}$. Factual explanations correspond to the label-stabilizer orbit (group actions generated by the subalgebra $\mathfrak{h}\subset \mathfrak{g}$ that leave the predicted class invariant). Counterfactual explanations correspond to transverse flows generated by the complement $\mathfrak{m}$, which traverse the decision boundary along the shortest algebraic path. We prove that this decomposition is reductive, guarantee that all generated trajectories remain within a bounded neighborhood of the data manifold, establish that orthogonal generator bases produce locally independent counterfactual directions, and show that winner-take-all sparsity recovers the unique minimal-norm counterfactual generator. Empirical evaluation on synthetic (BA-2Motifs) and biochemical (MUTAG, PROTEINS, NCI1, AIDS) benchmarks demonstrates that our framework achieves high-quality factual and counterfactual explanations with strict manifold proximity, outperforming all baseline methods.