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Structural Instability of Feature Composition

Sparse Autoencoders (SAEs) have emerged as a powerful paradigm for disentangling feature superposition in transformer-based architectures, enabling precise control via activation steering. However, the theoretical foundations of compositional steering -- the simultaneous activation of distinct semantic latents -- remain under-explored. The prevailing Linear Representation Hypothesis often abstracts away non-linear interference effects that arise in overcomplete dictionaries. We present a geometric framework for analyzing the instability of feature unions. Modeling the activation space as a high-dimensional sparse cone manifold, we derive an asymptotic compositional-collapse threshold under a spherical dictionary model, characterized by the Gaussian mean width (statistical dimension) of the signal cone. We further show that, in the high-bias regime, ReLU rectification converts microscopic correlation-induced variance fluctuations into a systematic drift that accumulates under composition, yielding interference growth consistent with a ratchet effect. We validate the predicted scaling trends on structured semantic features extracted from CLEVR, where hierarchical correlations accelerate the transition relative to random baselines. Together, our results highlight geometric constraints on the scalability of union-based steering and motivate composition mechanisms that explicitly manage interference beyond naive linear superposition.

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