{
  "abstract": "For $s>0$, and $z\\in\\mathbb H=\\{z=x+iy:y>0\\}$, define $θ(s;z)=\\sum_{m,n\\in\\mathbb Z}e^{-πs\\big(m^2y+\\frac{(mx-n)^2}{y}\\big)}$ and $ J(z;a,b)=\\sum_{m,n\\in\\mathbb Z}e^{-π\\big(m^2y+\\frac{(mx-n)^2}{y}\\big)} \\cos\\bigl(2π(ma+nb)\\bigr)$ for the classical and shifted theta functions (Gaussian lattice sums), respectively. We prove that, up to the modular group, there exist three thresholds $0<α_a<α_b<α_c<1$ such that \\begin{equation}\\aligned\\nonumber \\Minima_{z\\in\\mathbb H,\\,(a,b)\\in\\mathbb R^2}\\Big(θ(1;z)+αJ(z;a,b)\\Big)= \\begin{cases} \\;\\big(e^{iπ/{3}};1/3,1/3\\big), &\\hbox{if}\\;\\; α\\in[0,α_a),\\\\ \\;\\big(e^{iπ/{3}};1/3,1/3\\big)\\;\\hbox{or}\\;\\big(e^{i{φ_{α_a}}};1/2,1/2\\big), &\\hbox{if}\\;\\; α=α_a,\\\\ \\;\\big(e^{i{φ_α}};1/2,1/2\\big),\\;φ_α\\in(φ_{α_a},\\fracπ{2}),\\;\\; &\\hbox{if}\\;\\; α\\in(α_a,α_b),\\\\ \\;\\big(i;1/2,1/2\\big),\\; &\\hbox{if}\\;\\; α\\in[α_b,α_c],\\\\ \\;\\big(iy_α;1/2,1/2\\big),\\; y_α\\in(1,\\sqrt3],\\;\\;y_1=\\sqrt3,\\;\\;&\\hbox{if}\\;\\; α\\in(α_c,1], \\end{cases} \\endaligned\\end{equation} where the complex variable $z$ and vector $(a,b)$, represent the total vortex shapes and relative positions of the two-component Bose gas, respectively. Consequently, this completely proves the conjecture of Mueller and Ho~\\cite{Mue2002} (2002) arising in Bose--Einstein condensates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12203",
  "author_total": 2,
  "authors": [
    {
      "name": "Senping Luo"
    },
    {
      "name": "Juncheng Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1dxra4hp4b5wz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12203",
  "fulltext_url": null,
  "id": "1dxra4hp4b5wz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12203",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.12203",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1dxra4hp4b5wz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:54Z",
      "native_id": "2610.12203",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12203"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "A Complete Proof of Mueller--Ho Conjecture",
  "title_text": "A Complete Proof of Mueller--Ho Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/1dxra4hp4b5wz",
  "venue": "arXiv",
  "versions": []
}