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Self-attention carries a quadratic cost in token count, which limits how far Vision Transformers can scale with input resolution. We introduce SSoG, a separable sum-of-Gaussians spatial operator that replaces dense self-attention with a learned mixture of axis-factorized Gaussian kernels, reducing the per-layer cost from $\mathcal{O}(N^2)$ to subquadratic in $N$. Unlike dot-product self-attention, whose Lipschitz constant is unbounded outside a compact input domain, SSoG with fixed kernels is globally Lipschitz continuous, which, via the Picard--Lindelöf theorem, guarantees that the continuous-depth variant has a unique solution on the whole integration interval, a global guarantee that dense-attention counterparts lack in general. On CIFAR-100 at $224\times224$, in a matched discrete-Transformer setting and in the continuous setting, SSoG improves Top-1 accuracy by 2.2--4.1 points across three model widths while using 8.8--16.6\% fewer parameters, with depth and training recipe held fixed. In a looped-Transformer setting, SSoG improves Top-1 accuracy by 5.4--8.1 points. We further probe inference-time scaling by evaluating on inputs up to $1024\times1024$ at matched batch size: in the continuous-depth (Neural ODE) architecture, SSoG reaches up to 154\% higher throughput and over 60\% lower peak memory than dense attention, with the gap widening as resolution grows, consistent with the difference in asymptotic cost. In the discrete and looped settings, SSoG also achieves over 30\% higher throughput than dense attention at this resolution. Together, these results establish SSoG as a practical, parameter-efficient, and theoretically grounded alternative to dense attention across discrete, weight-shared, and continuous-depth Transformer variants.