Mechanistic studies of sequence models often treat layerwise state encodings as architectural traits: recurrent models concentrate readable state, attention-based models distribute it. We find that the same architecture reverses this profile when the task changes. Across Transformers, Mamba, Mamba-2, LSTMs, and GRUs, Parity is concentrated late in Mamba and the recurrent baselines and built gradually by Transformer; on bounded-depth Dyck-k the pattern flips. The same flip appears in fine-tuned Mamba-130M and Pythia-160M, and the Pythia Dyck bottleneck persists at 410M. Two explanations are conflated in the literature: algebraic structure (commutativity) versus computational structure (prefix update vs. stack). To separate them we add a third task: non-commutative S_3 permutation composition. S_3 groups with Parity, not Dyck, on layerwise probing across all five architectures and on Mamba-specific Conv1D attribution, so the grouping tracks computational structure rather than commutativity. Causal interventions show that, in the 4-layer formal models, linearly readable directions are often functionally necessary and can remain important at out-of-distribution lengths on Parity and Dyck. At pretrained scale the picture splits. Fine-tuned Pythia Dyck has a strong middle-layer bottleneck (L6-L7 ablation drops accuracy by roughly 81% at 160M; broader L4-L18 plateau at 410M), far weaker at the best-probe layer. Pretrained Mamba shows the complementary failure mode: its final layer is highly readable, no single probe direction breaks the task on Parity, Dyck, or S_3, yet mid-position activation patching there recovers about 97-98% of the clean-corrupted logit gap. Probing localizes where state is linearly available, not always where the computation is bottlenecked. Mechanistic signatures are properties of architecture and task together.