{
  "schema_version": 1,
  "problem_number": "OWR-16163-011",
  "title": "The Law of the Sum of the Two Faces of Bi-Monotone Brownian Motion Has Atoms: A Partial Answer to a Question of M. Gerhold",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "M. Gerhold asked for a concrete description of the vacuum distributions μ_t of the operators λ_t* + λ_t + ρ_t* + ρ_t on the monotone Fock space, that is, of the sum of the two faces of bi-monotone Brownian motion (Oberwolfach Rep. 15 (2018)); the same question closes his paper on bi-monotone Brownian motion, where the even moments of μ_1 are identified as #PP(2n)/n! with the numbers 1, 4, 48, 928, 24448, … of bi-monotone pair partitions, for which no explicit or recursive formula was known. We give a partial answer: we do not find a closed formula, and the question remains open; we prove structural facts. (1) The measures μ_t are the dilations by √t of one measure μ = μ_1; this scaling is implicit in the source. (2) The compression of the operator to the orthogonal complement of the vacuum has norm at most √6, while the operator itself has norm c > √6. Hence the edge c of the support of μ is an isolated atom: μ = w(δ_c + δ_{−c}) + μ_0 with μ_0 carried by [−√6, √6]. This is proved without computation. Enclosures in exact rational arithmetic (computer-assisted) give c = 2.585826606048204448… and w = 0.251331654638419314…, so the two atoms carry more than half of the mass. (3) The Boolean cumulants of μ are k_{n+2} = 4⟨B^n 1⟩ for an explicit linear operator B on polynomials in three variables and an explicit linear functional. This gives all moments exactly in polynomial time; the table of the source (eleven values) is extended to m_306. (4) Computer-assisted, exact: μ has mass in (2.01148, 2.01149) and in (2.001, 2.003), so μ_0 is not carried by [−2, 2]. Only observed numerically, and not proved: further spectral points accumulating at ±2 that look like atoms, a nearly flat density of total mass about 0.49 on [−2, 2], Jacobi parameters tending slowly to 1, and the absence of a linear recurrence with polynomial coefficients for the moments within the sizes searched. An explicit formula for μ, the nature of its part in [−√6, √6], and a formula for the numbers #PP(2n) remain open. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.OA",
    "math.PR",
    "math.FA",
    "math.CO"
  ],
  "keywords": [
    "bi-monotone Brownian motion",
    "bi-monotone independence",
    "monotone Fock space",
    "vacuum distribution",
    "atoms",
    "Boolean cumulants",
    "Schur complement",
    "Schur test",
    "bi-monotone pair partitions",
    "moment problem",
    "computer-assisted proof",
    "partial answer",
    "Oberwolfach Reports open problems",
    "UnsolvedMath",
    "OWR-16163-011",
    "math.OA",
    "math.PR",
    "math.FA",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-10",
  "publication_date": "2026-10-10",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-10",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-16163-011/",
  "pdf_url": "https://eulersolve.org/papers/owr-16163-011/paper.pdf?v=b3b001107ea0",
  "doi": "10.5281/zenodo.23277381",
  "zenodo_record_url": "https://zenodo.org/records/23277381",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Partial answer to the question of M. Gerhold (Oberwolfach Reports 15 (2018), Report 22/2018; arXiv:1708.03510): the note proves structural facts about the law of the sum of the two faces of bi-monotone Brownian motion (one measure up to dilation; the edge of the support is an isolated atom; the two atoms carry more than half of the mass; an exact polynomial model for the moments; mass just above 2) and does not give a closed formula. The explicit description asked for stays open. Parts of the results are computer-assisted, in exact rational arithmetic; what is only observed numerically is labelled as such in the note.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}