{
  "abstract": "Let $W$ be a representation of the Weyl relations on a separable Hilbert space and write $\\mathcal R_W(K)=\\{W(f):f\\in K\\}''$. If the symplectic pairing of $K_1$ and $K_2$ is nonzero, there is a self-adjoint unitary $B\\in\\mathcal R_W(K_2)$ such that, for every normal state $ρ$, one can choose a self-adjoint unitary $A_ρ\\in\\mathcal R_W(K_1)$ with $\\operatorname{Var}_ρ(-i[A_ρ,B])=4$. Hence the optimized mean-input-energy-constrained coefficient equals $4$ at every admissible energy threshold. For the fixed trigonometric Weyl witness $h$, an exact identity for $4-h^2$ reduces the deficit to a phase-fixed problem for commuting squared Weyl translations. For the one-mode quadratic energy $G_M=\\frac12 R^TMR-\\frac12\\sqrt{\\det M}$, with $M>0$ and $u^TΩv=π$, we prove $4-γ_{G_M,E}(h)=\\frac{u^TMu+v^TMv+2π\\sqrt{\\det M}}{4E}+O(E^{-2})$. The lower bound holds over all normal states satisfying the mean-energy constraint, and a localized Zak construction attains the same coefficient. For $G=dΓ(H_1)$ and witness directions $u,v\\in\\operatorname{dom}H_1^{1/2}$, the fixed-witness deficit is $Θ(E^{-1})$ whenever at least one direction lies outside $\\ker H_1$; if both directions are zero modes, the constrained supremum equals the endpoint at every positive threshold.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19163",
  "author_total": 1,
  "authors": [
    {
      "name": "Hassan Nasreddine"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6evq9bkdex6jd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19163",
  "fulltext_url": null,
  "id": "6evq9bkdex6jd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19163",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7213601557",
  "pdf_url": "https://arxiv.org/pdf/2609.19163",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6evq9bkdex6jd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:19Z",
      "native_id": "2609.19163",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19163"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Energy-Constrained Commutator Variance for Weyl Pairs",
  "title_text": "Energy-Constrained Commutator Variance for Weyl Pairs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/6evq9bkdex6jd",
  "venue": "arXiv",
  "versions": []
}