Phase Space Attention: A Hairer Lift Resolves the Single-Layer Induction Obstruction

We resolve the Sanford-Hsu-Telgarsky (SHT) single-layer induction obstruction within the linear, one-step, causal, bilinear, symplectically consistent design class CPSA on the post-RoPE substrate, by lifting standard transformer attention onto a symplectic phase space in direct correspondence with Hairer’s lift of Stormer-Verlet onto a one-step symplectic integrator. The reframing recasts SHT as a filter-order gap: a standard one-layer bilinear score realises a z-transform of joint order (0,0), whereas the induction discriminator requires (1,1). We close the gap by applying the symplectic upper shear Mgamma : (q,p) -> (q + gamma p, p) to the post-RoPE query and key streams. We prove that this lift is the unique solution within CPSA (Theorem 4); is operator-level symplectic with zero secular drift (Theorem 6); requires post-RoPE placement (Corollary 7); and, under explicit Assumption Set A, induces a closed-form induction phase transition at gammac = (d_k log(T-1))^(1/4) (Theorem 8). The SHT lower bound binds where total parameter budget approaches the bit budget of the two-layer induction circuit it forbids: negligible at frontier scale (>= 1B), but decisive in the sub-100M regime that ships in the billions on consumer edge form factors – phones, wearables, microcontrollers, embedded controllers – where it governs whether on-device in-context learning is feasible. We work at 4-92M by design. We report 74.2% single-layer induction at gamma = 2.0, Pearson r >= 0.998 for the closed-form transfer function, r = -0.679 for the embedding-axis low-pass induction filter; and the post-softmax map Frechet-linearises to Differential Transformer (Ye et al., 2024) at first order under the small-signal regime (SS).