Recent work has shown that contiguous vision transformer (ViT) blocks (a) can be replaced by a linear map and (b) organize into recurrent phases of computation. We ask whether these observations coincide: does ViT depth implement approximately \textit{autonomous linear} dynamics, admitting a single operator
K applied recurrently across a contiguous span? We test this using Dynamic Mode Decomposition (DMD), which fits
K from selected, consecutive hidden-state pairs and predicts
p steps ahead via
Kp. On four pretrained DINO ViTs, we study the regularization, rank, and calibration budget required for stable fitting. For short spans (
p≤4),
Kp tracks an unconstrained endpoint map to within
0.02 cosine similarity on DINOv3-H/16+, while also recovering intermediate activations at each skipped block. At early cut starts, the fitted operators compress to rank
≪d with minimal calibration data, and across tokens, \texttt{cls} is most amenable to linearization; both properties decay monotonically with depth. Yet this local fidelity does not transfer downstream. At the final hidden state, after propagating through the remaining blocks, an identity baseline becomes competitive.