Phase Space Attention:A Hairer Lift Circumvents the Single-Layer Induction Obstruction
Abstract
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 ≥ 1. Applying the symplectic upper shear Mγ:(q,p)↦(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 T4-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 T3 channel moves γc at dk=64 from 1.030 to 0.569 and removes the crossover. Deployability follows by exact derivation: the KV cache is unchanged, prefix reuse and speculative decoding are preserved, overhead is 6d FLOPs per token per layer, INT8 headroom grows by at most log2(1+2γ) bits, fused kernels are unmodified, and no parameters are added. At 91.3M parameters a supercritical sweep locates an emergence band: induction forms 3/3 seeds at γ=0.80 in a mean of 717 steps, against 2/3 seeds and 2700 steps at γ=0. Adverse results are reported as directly: a key-only half-lift reaches 0.949 against 0.811 for the symmetric operator, so if induction accuracy is the objective, the half-lift is the better construction. Forty-one notebooks and result files ship as ancillary material.
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BibTeX
@misc{maitra2026phase,
title = {{Phase Space Attention:A Hairer Lift Circumvents the Single-Layer Induction Obstruction}},
author = {Kingsuk Maitra and Shagun Sood Morteza Hosseini and Suman Gunnala and Vikram Gupta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32319},
eprint = {2609.32319},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32319},
note = {CC0},
}
RIS
TY - UNPB
TI - Phase Space Attention:A Hairer Lift Circumvents the Single-Layer Induction Obstruction
AU - Kingsuk Maitra
AU - Shagun Sood Morteza Hosseini
AU - Suman Gunnala
AU - Vikram Gupta
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32319
UR - https://arxiv.org/abs/2609.32319
AB - 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 ≥ 1. Applying the symplectic upper shear M_γ:(q,p)↦(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 derivation: the KV cache is unchanged, prefix reuse and speculative decoding are preserved, overhead is 6d FLOPs per token per layer, INT8 headroom grows by at most log_2(1+2γ) bits, fused kernels are unmodified, and no parameters are added. At 91.3M parameters a supercritical sweep locates an emergence band: induction forms 3/3 seeds at γ=0.80 in a mean of 717 steps, against 2/3 seeds and 2700 steps at γ=0. Adverse results are reported as directly: a key-only half-lift reaches 0.949 against 0.811 for the symmetric operator, so if induction accuracy is the objective, the half-lift is the better construction. Forty-one notebooks and result files ship as ancillary material.
ER -