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

Kingsuk Maitra, Shagun Sood Morteza Hosseini, Suman Gunnala, Vikram Gupta

We circumvent the Sanford-Hsu-Telgarsky (SHT) single-layer induction obstruction within a linear, one-step, causal, bilinear, symplectically consistent design class on the post-RoPE substrate, by lifting attention onto a symplectic phase space, mirroring Hairer's lift of Stormer-Verlet. The lift exits the premise of the SHT counting argument rather than the bound itself. The reframing exhibits the obstruction as a filter-order gap: a one-layer bilinear score realises a $z$-transform of joint order $(0,0)$, whereas the induction discriminator requires key-side order $\geq 1$. Applying the symplectic upper shear $M_γ:(q,p)\mapsto(q+γp,p)$ to the post-RoPE query and key streams closes it. We prove this lift is unique within the factorised subclass, exactly symplectic at operator level, and requires post-RoPE placement; and in an explicit $T_4$-only Gaussian reduction we derive a closed-form two-branch induction phase transition, held out at $r=0.9876$ with zero fitted parameters. That law is an analytically solvable limit, not a robust prediction: restoring the $T_3$ channel moves $γ_c$ at $d_k=64$ from 1.030 to 0.569 and removes the crossover. Deployability follows by exact…