{
  "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 $\\geq 1$. 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 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32319",
  "author_total": 4,
  "authors": [
    {
      "name": "Kingsuk Maitra"
    },
    {
      "name": "Shagun Sood Morteza Hosseini"
    },
    {
      "name": "Suman Gunnala"
    },
    {
      "name": "Vikram Gupta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a0fjam1r8tdbk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32319",
  "fulltext_url": null,
  "id": "a0fjam1r8tdbk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32319",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.mtrl-sci"
    ],
    [
      "arxiv",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "openalex",
      "1702|Topic Modeling"
    ]
  ],
  "openalex": "W7214722154",
  "pdf_url": "https://arxiv.org/pdf/2609.32319",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a0fjam1r8tdbk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2609.32319",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32319"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mtrl-sci",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Phase Space Attention:A Hairer Lift Circumvents the Single-Layer Induction Obstruction",
  "title_text": "Phase Space Attention:A Hairer Lift Circumvents the Single-Layer Induction Obstruction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:12Z",
  "url": "https://research.draconitic.com/paper/a0fjam1r8tdbk",
  "venue": "arXiv",
  "versions": []
}