On the Intrinsic Limits of Transformer Image Embeddings in Non-Solvable Spatial Reasoning

Vision Transformers (ViTs) excel in semantic recognition but exhibit systematic failures in spatial reasoning tasks such as mental rotation. While often attributed to data scale, this work argues that the limitation arises from the intrinsic circuit complexity of the architecture. By formalizing spatial understanding as a Group Homomorphism Problem—where latent embeddings preserve the algebraic structure of physical transformations acting on images—we identify a fundamental computational bottleneck. Specifically, for non-solvable groups (e.g., $\mathrm{SO}(3)$), maintaining such structure-preserving embeddings is lower-bounded by the Word Problem, which is $\mathsf{NC^1}$-complete. In contrast, constant-depth ViTs with polynomial precision are strictly bounded by the complexity class $\mathsf{TC^0}$. Under the standard conjecture $\mathsf{TC^0} \subsetneq \mathsf{NC^1}$, a complexity boundary emerges: constant-depth architectures lack the logical depth required to capture non-solvable spatial structures in a single forward pass. To empirically validate this theoretical gap, we propose the Latent Space Algebra (LSA) benchmark, which reveals a significant degradation in ViT representations as the compositional depth of non-solvable tasks increases.