Orientation Over Capacity: A Stabiliser-Theoretic Analysis of Representational Vulnerability in Neural Networks

Abstract Neural networks with identical architecture and capacity can exhibit different sensitivity to the same family of input perturbations. We formalise this observation through the symmetry-matching condition Stab(G,F) ∩ E = {e}, originally developed for check-digit codes over finite even-sized alphabets. In that setting the condition characterises deterministic error detection; here we extend it to continuous representations and define two measurable quantities for neural networks: a representation-level stabiliser Srep (the set of perturbations invisible to the encoding) and a scalar representational vulnerability V ∈ [0,1] (the fraction of a perturbation family that lies in the stabiliser). We prove that the containment Srep ⊆ Sout holds for any layered architecture, transfer the stabiliser monotonicity theorem to the neural setting under an injectivity condition, and establish an entropy–vulnerability separation theorem showing that V depends on the weight-matrix orientation at fixed capacity. On an Elhage-style toy model of superposition (20 features, d ∈ {2,…,20}, sparsity 0.05) we find that (i) V decreases with hidden dimension (the capacity effect), (ii) at fixed d = 5, V ranges from 0.047 to 0.148 across 15 learned weight matrices (the orientation effect), and (iii) the best-oriented d = 5 network (V = 0.047) outperforms the average d = 10 network (V = 0.085). On two pretrained transformer language models (Qwen 2.5-0.5B, SmolLM2-360M) we further show that (iv) mean per-layer sensitivity decreases monotonically with depth while VGini increases, indicating growing selectivity rather than growing blindness, and (v) six structured perturbation families (semantic substitution, typo, negation, synonym, number swap, word-order swap) form a stable sensitivity hierarchy across both models, with numeric-value perturbations exhibiting near-synonym-level insensitivity — a potential blind spot for safety-critical applications.