{
  "abstract": "We construct Komlós signing laws with discrepancy below $6.84$, independent Gaussian reference blocks and exponentially many balanced signings. One law preserves hard constraints and exact conditional means while its reference controls every joint convex cost. We resolve both Reis--Rothvoss Schatten conjectures. For $n$ symmetric $n\\times n$ matrices, one prescribed-mean signing law bounds all centered Schatten discrepancies with sharp powers of $n$ and a universal constant. One Gaussian event of measure $e^{-O(n)}$ controls every unitarily invariant norm at the square-function radius. We prove Reis's volume-ratio conjecture: every $r$-dimensional quotient of the $a\\times b$ trace-class unit ball has volume ratio $O(\\sqrt{1+\\min(a,b)/r})$. We resolve Nutz--Wang--Zhang's directional and supermartingale conjectures and the higher-dimensional Bass-flow conjecture. For square-integrable laws, full convex reference support is the universal finite-potential criterion in Tschiderer's $q$-Bass problem. Every driver, even atomic, admits one extended potential and shift for all maximal-covariance martingale optimizers of each irreducible pair with full terminal affine span. For Talagrand's problem, five copies of a compact balanced Gaussian half-mass set contain compact convex output of mass at least $3/4$, also in Wiener space. Cheeger's bound is asymptotically exact throughout subexponential Dirichlet spectral ranges of mixed weighted products. For standard Borel sources, a Blackwell-greatest perfectly private full-data release exists exactly when, after discarding conditional point masses, the rest coincide or share a two-point support. Under conditional atomlessness, existence is equivalent to independence. Covariance-matched analytic uniformly log-concave sources can have a greatest useful-data-only private release with infinite information, while no greatest full-data release exists.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38228",
  "author_total": 1,
  "authors": [
    {
      "name": "Eren Ercan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5efds7z5q6k4g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38228",
  "fulltext_url": null,
  "id": "5efds7z5q6k4g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38228",
  "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.GM"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7215110428",
  "pdf_url": "https://arxiv.org/pdf/2609.38228",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5efds7z5q6k4g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38228",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38228"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2610|Advanced Banach Space Theory",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Vector balancing in convex order",
  "title_text": "Vector balancing in convex order",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/5efds7z5q6k4g",
  "venue": "arXiv",
  "versions": []
}