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

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 kₙ₊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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-16163-011 (M. Gerhold, Bimonotone quantum Lévy processes, Oberwolfach Rep. 15 (2018), Report No. 22/2018, pp. 1354–1355).