# Verification report — OWR-16163-011 (M. Gerhold: the law of λ_t* + λ_t + ρ_t* + ρ_t, the sum of the two faces of bi-monotone Brownian motion)

Verification date: 2026-10-10.

**Verdict.** The note gives a **partial answer**. The source asks for a concrete description (an explicit
formula) of the vacuum distributions μ_t; the note does not give one, and the question stays open. Every statement
the note makes as a theorem is proved, with the following status.
- **Proved without computation.** μ_t is the dilation by √t of one measure μ = μ_1 (Theorem 1.1). The compression
  b_Q of the operator to the orthogonal complement of the vacuum has norm at most √6 (Theorem 1.2(a)). There are
  c > √6 and w ∈ (0, 1/3) with μ = w(δ_c + δ_{−c}) + μ_0 and supp μ_0 ⊂ [−√6, √6]; c = max supp μ = ‖b‖ is a simple
  eigenvalue, and the only point of the spectrum above √6 (Theorem 1.2(b), (c), (e)). The Boolean cumulants of μ
  are k_{n+2} = 4⟨B^n 1⟩ for an explicit operator B on Q[x_1, x_3, s] and an explicit functional (Theorem 1.3).
  ‖b‖ = max supp μ and ‖b_Q‖ = max supp ν = lim (k_{2n+2}/k_{2n})^{1/2} (Proposition 7.1, Corollary 7.2(a), (b)).
- **Proved, computer-assisted (exact rational arithmetic).** The enclosures of c and w in Theorem 1.2(d):
  c = 2.585826606048204448631515236…, w = 0.2513316546384193145305056…; in particular 2w > 1/2. The bound
  ‖b_Q‖ > 2.1019907 (Corollary 7.2(c)). Positive μ-measure of the windows (2.01148, 2.01149), (2.001, 2.003),
  (2.011485, 2.011486), (2.0017, 2.0018) (Theorem 1.4).
- **Not new.** The scaling of Theorem 1.1 is implicit in the source (proof of its Theorem 7.1). The mechanism (Schur
  complement at the vacuum, Boolean cumulants, an eigenvalue outside the spectrum of the compression) is classical;
  new are the proof that it applies here and the numbers.
- **Only observed numerically, not proved.** Further spectral points above 2 that look like atoms and seem to
  accumulate at ±2; a nearly flat density of total mass about 0.49 on [−2, 2]; Jacobi parameters tending slowly to 1;
  no linear recurrence with polynomial coefficients of the sizes searched; no small algebraic or integer relation
  for c and w (Section 11).
- **Open.** A closed formula for μ or its Cauchy transform; the nature of μ_0; a formula or recursion for the
  numbers of bi-monotone pair partitions.

The note is unrefereed.

## Statement checked
- **Primary source.** M. Gerhold, "Bimonotone quantum Lévy processes", abstract in Oberwolfach Report
  No. 22/2018, Oberwolfach Rep. 15 (2018), no. 2, pp. 1354–1355 (report: pp. 1293–1379), DOI 10.4171/OWR/2018/22.
  - Read in the file of the report on the publisher's site, fetched on 2026-10-10 for the first results, by all
    three verification runs and at the writing (865,042 bytes each time; sha256 of the copies fetched at the
    writing and by the third run: `efe1c7946a168a6fcad4e6128ad74c761b40085f3f6280a19b36e9674939316d`).
  - The question is the last sentence of the abstract (p. 1355): it is called there "an open problem to give a
    concrete description of the corresponding probability measures". The measures are the vacuum distributions of
    the self-adjoint operators λ_t* + λ_t + ρ_t* + ρ_t on the monotone Fock space.
  - It is not item (3) of the "Open Problems" session of the same report (p. 1373).
- **The paper behind it.** M. Gerhold, "Bimonotone Brownian motion", arXiv:1708.03510 (one version, 11 August
  2017), read in full in the TeX source. Definitions 3 and 4 (monotone Fock space; the four operators), Definition 8
  (bi-monotone pair partitions), Theorems 4.1, 4.2, 6.5, 7.1 and Corollary 7.2 were compared with the statement of
  the note by all three verification runs (the third on the final text). The paper ends with the wish for an explicit formula of the measure for t = 1
  and states that no explicit or recursive formula for the numbers 1, 4, 48, 928, 24448, … is known to its author.
  - The published version (QP–PQ: Quantum Probability and White Noise Analysis 32 (2023) 59–76,
    DOI 10.1142/9789811275999_0005) **could not be consulted** (automated access refused; one further anonymous
    request by the third run was answered with HTTP 403 and was not worked around).
- **Normalisations.** f = 1_[0,t]; vacuum state; variance of μ_t is 4t; the case n = 0 of the two annihilation
  operators is the integral over the whole line, as forced by adjointness (Lemma 2.1). μ_t only sees the invariant
  subspace of functions supported in [0, t]; statements about the norm and the spectrum refer to it.
- **Corpus record.** ulamai/UnsolvedMath, OWR-16163-011 (dataset version 1.6.0; status open). Its statement is a
  faithful paraphrase of the sentence of the report.

## Readings
| Reading | Answer | Where |
|---|---|---|
| "Concrete description" = an explicit formula for μ_t (density and atoms, or Cauchy transform) | not found; open | Sections 1.4, 11 |
| The family μ_t, t > 0 | all are dilations of μ = μ_1: μ_t(A) = μ(A/√t) | Theorem 1.1 |
| Is μ absolutely continuous, like the arcsine law of one face? | no: atoms at ±c, c = 2.5858…, of mass w = 0.2513… each | Theorem 1.2 |
| Support and norm | supp μ ⊂ [−√6, √6] ∪ {±c}; ‖b‖ = c; no mass in (√6, c) | Theorem 1.2(b), Corollary 7.2 |
| Is the rest of μ carried by [−2, 2]? | no: positive mass in windows just above 2 | Theorem 1.4 |
| Moments (numbers of bi-monotone pair partitions) | exact polynomial-time algorithm; values to m_306 (second run: m_560); no closed formula or recursion | Theorem 1.3, Table 1 |
| Other intervals of length t | same law (stationarity of the increments, Theorem 4.2 of the source) | Section 1.1 |
| Difference of the two faces, or one face | δ_0, respectively the arcsine law; not the question | Remarks 3.1, 5.4(2) |
| "Bi-monotonic independence" of Gu–Hasebe–Skoufranis | their type II is the notion of the source; their central limit theorem is for type I, a different object (for full correlation the sum is 2 × arcsine) | Section 1.3 |

## Results in the paper
- **Theorem 1.1.** μ_t(A) = μ(A/√t); μ symmetric, variance 4, carried by [−4, 4]. Proof: the unitary
  (U_t g)(s) = t^{n/2} g(ts) conjugates b_t into √t·b.
- **Lemma 4.1.** Schur complement at the vacuum: G(z) = 1/(z − K(z)), K(z) = ⟨ξ, (z − b_Q)^{-1} ξ⟩ = 4∫dν/(z − x),
  ξ = bΩ; k_{n+2} = ⟨ξ, b_Q^n ξ⟩.
- **Lemmas 5.1, 5.2, Proposition 5.3.** A positive supersolution h_n(t) = Σ_j φ(t_j) Π_{i≠j} (γ + 2|t_i − t_j|)^{−1/2}
  with (Th)_n ≤ 2√(1+γ) h_n under a one-dimensional condition on φ, and a Schur test. Certificate A: γ = 2/3, φ = 1,
  bound 2√(5/3). Certificate B: γ = 1/2, φ(s) = 1 + (3/5)(2s − 1)², bound √6; the condition reduces to E(Y) ≥ 0 on
  [1, √5] for the quintic E(Y) = −(2/25)Y⁵ + (3/10)Y⁴ + (2/5)Y³ − (8/5)Y² − (8/5)Y + 209/50, with
  E′(Y) = −(2/5)(Y − 2)(Y³ − Y² − 5Y − 2) and minimum E(2) = 1/50.
- **Theorem 1.2(b), (c), (e)** (Section 6). F(z) = z − K(z) is increasing on (√6, ∞); max supp μ ≥ (m_6/m_4)^{1/2} =
  (58/9)^{1/2} > √6 gives its zero c; the eigenvector Ω + (c − b_Q)^{-1} ξ gives the mass w = 1/F′(c).
- **Proposition 7.1, Corollary 7.2.** Positivity: the norms of b and b_Q are the largest points of the supports
  of μ and ν; 382/95 ≤ ‖b_Q‖², 2.1019907 < ‖b_Q‖ ≤ √6.
- **Theorem 1.3** (Section 8). Product formula for the powers of one face (Lemma 8.1), the lift Π from polynomials
  to Fock vectors and the relation bΠ(f) = Π(Bf) + 2⟨f⟩Ω (Lemma 8.2); cumulants; complexity (O(N⁵) operations on
  rationals with O(N²) binary digits; the proof of the size bound was completed after the third run).
- **Theorem 1.2(d)** (Section 9). Two-sided remainder bounds (Lemma 9.1) evaluated at rational points.
- **Theorem 1.4** (Section 10). Window certificates: one rational number I(q) = ∫(x² − a²)(x² − b²) q(x)² dμ < 0.
- **Tables.** Table 1: exact moments and cumulants; Table 2: enclosures; Table 3: window certificates; Table 4:
  numerical data (not proved).

## Computations (programs and outputs in reproducibility/)
- **What is computer-assisted.** Theorem 1.2(d), the second lower bound in Corollary 7.2(c), and Theorem 1.4. Each
  is the exact evaluation of finitely many rational numbers from exact moments. The moments m_0, …, m_44 are
  computed directly from the definition of the operators; higher moments rest on Theorem 1.3.
  - Theorem 1.2(d), first line: needs Theorem 1.2(a) and m_0, …, m_44 from the definition. It gives w > 0.2500111,
    so 2w > 1/2 is certified without Theorem 1.3, with the small margin 1.1·10⁻⁵ and only with the constant √6.
  - Theorem 1.2(d), second line: needs in addition Theorem 1.3 and the cumulants k_2, …, k_306.
  - Theorem 1.4(a): m_0, …, m_304 (Theorem 1.3). Theorem 1.4(b): m_0, …, m_556, computed by verification run B.
- **Programs of the note** (`original/`): `moments_fock.py` (from the definition), `moments_sphere.py`
  (Theorem 1.3), `enclose_atom.py`, `window_certificate.py`, `certificate_B.py` (Sturm), and programs for the
  numerical section. The run to m_306 was stopped at a time limit after 28 minutes; it had printed k_2, …, k_306,
  from which the moment file was rebuilt (`cumulants_to_moments.py`); the file agrees with the complete run to
  m_200 and with the independent values of run B.
- **Verification run A** (`verification_run_A/`): moments m_0, …, m_44 from the definition (identical); count of
  bi-monotone pair partitions from the definition of the source for n ≤ 6; own implementation of Theorem 1.3
  (cumulants equal to the direct ones up to k_44 and to those of the note up to k_160); symbolic test of Lemma 8.1
  and of the intertwining relation on 39 monomials, with a mutation test; Galerkin truncations of dimension up to
  10⁷ (largest eigenvalue 2.58548, extrapolated 2.58581; vacuum weight 0.25171, extrapolated 0.25135); own
  enclosures 2.58579474 < c < 2.58603695, 0.25001113 < w < 0.25312134 (from m_0, …, m_44) and
  2.585826606048203 < c < 2.585826606048213, 0.25133165463828 < w < 0.25133165463861 (from its own m_0, …, m_160).
- **Verification run B** (`verification_run_B/`): three routes to the moments (counting, n ≤ 10; the definition,
  to m_48; Theorem 1.3 with two implementations, to k_46 and k_560), all equal to each other, to the table of the
  source and to all moment files of the note (154 moments, 153 cumulants); the intertwining relation on 455
  monomials; all enclosures (from its own k_2, …, k_306 the intervals of Theorem 1.2(d); from k_2, …, k_560 an
  enclosure of width 9.0·10⁻⁵¹); Certificate B (quintic recomputed; own Sturm sequence; interval subdivision;
  minimum 1/50); all window certificates of the note with identical values, and those with K = 278; Jacobi
  parameters to k = 280, Gauss nodes; searches for recurrences (order ≤ 24 on 281 terms) and for closed forms.
- **Third verification run** (`independent_run_2/`; all programs written before the other programs of the package
  were opened): moments m_0, …, m_44 from the definition (identical); numbers of bi-monotone pair partitions and of
  the irreducible ones from the definition of the source for n ≤ 8 (equal to n!·m_2n and n!·k_2n); two
  implementations of Theorem 1.3 (to k_122 and to k_562), equal to the cumulants from the definition up to k_44 and
  to all 153 cumulants and 154 moments of the note and all 280 cumulants of run B; the relation
  bΠ(f) = Π(Bf) + 2⟨f⟩Ω on 28 monomials with the operators taken from the definition; Table 1; Corollary 7.2(c);
  Theorem 1.2(d) (first line from its own direct moments: 2.585794744033 < c < 2.586036943331,
  0.250011133605 < w < 0.253121331264; second line from its own cumulants) and all rows of Table 2; all six rows of
  Table 3 with its own construction and evaluation of q, and the two attempts without conclusion; Table 4 and the
  numbers of Section 11 except the truncations of Observation 5 and the last search of Observation 6; the quintic
  and its factorisation; a numerical test of Lemma 5.1; the size statement of Theorem 1.3 for n ≤ 60.
  - By-product, not used in the note: for the two windows without conclusion a second construction of q gives
    I(q) < 0 exactly ((2.0011, 2.0024) from m_0, …, m_304 and (2.0004, 2.0009) from m_0, …, m_556); the polynomials are
    in `independent_run_2/out/certs/` and can be re-evaluated with the standard library.
- **Program written with the note** (`writing_stage/check_paper.py`, standard library, about half a minute;
  TOTAL failures: 0): recomputes from the recorded files the numbers printed in the note, among them the
  enclosures of Theorem 1.2(d) and Table 2 by a bisection of its own.
- **Re-runs at the writing (2026-10-10).** The programs with a running time below ten minutes were run again, one
  process at a time; see `reproducibility/README.md` and `reproducibility/RERUN_LOG.txt`. All new outputs are
  identical to the recorded ones, or identical up to running times and lines appended by the shell of the
  original run. Among them: the fast implementation of Theorem 1.3 by run B to k_320 and to k_560, the comparison
  of its cumulants with the moment file of the note (equal on all 153 common indices), all enclosures of Table 2,
  and all window certificates of Table 3, including those with K = 278. Not re-run: the 28-minute run of the note
  to m_306, the 23-minute run of run A to k_160, its large Galerkin runs, and the counts of run B for 2n = 18, 20.
- **Re-run by the third verification run (2026-10-10).** `run_quick.sh` from an extracted copy of the archive:
  950 seconds, 55 results, none different. After the last corrections of the text `check_paper.py` was run again
  with the final source of the note (TOTAL failures: 0).

## Independent verification runs
Three independent verification runs, all AI-assisted, were made on 2026-10-10. Each wrote its own programs. Runs A
and B examined the first written version of the results, and neither saw the report of the other: run A examined
the statement, the proofs of Theorems 1.1–1.3 and the logic of Theorem 1.2(d) and Theorem 1.4; run B recomputed
all numbers and examined the computer-assisted statements, the labelling of tested statements and the
literature. The third run (the second round of verification; programs in `independent_run_2/`) examined the final
text of the note: the statement against the two sources, every proof line by line, everything that had been added
when the note was written, all numbers, the references and the literature.

| Item | Run A | Run B | Third run (final text) |
|---|---|---|---|
| Statement, definitions, normalisation against the two sources | CONFIRMED | CONFIRMED (sources read again) | CONFIRMED (sources fetched and read again; numbering and quotations checked) |
| Theorem 1.1, Lemmas 2.1, 2.2, 4.1 | CONFIRMED_WITH_FIXES (two editorial corrections) | Lemma 2.1 re-derived for its programs; Lemma 4.1 read, no error | CONFIRMED (re-derived) |
| Theorem 1.2(a): Lemmas 5.1, 5.2, Certificates A and B | CONFIRMED (re-derived; own proof of Certificate B) | Certificate B CONFIRMED (recomputed, certified twice); Lemmas 5.1, 5.2 read, no error; Lemma 5.1 tested numerically | CONFIRMED (all six steps and the case n = 1 of Lemma 5.1; the bookkeeping of Lemma 5.2; the quintic and the factorisation of E′ derived again) |
| Theorem 1.2(b), (c), (e) | CONFIRMED (re-derived in nine steps) | not its part | CONFIRMED (Steps 1–6; one sentence after the theorem corrected) |
| Proposition 7.1, Corollary 7.2 | contributed by this run | not seen (it computed the bound of Corollary 7.2(c)) | CONFIRMED (the inequality for the modulus of Aψ, the density hypothesis, both applications) |
| Theorem 1.3 (proof) | CONFIRMED (re-derived; own implementation; symbolic tests) | proof read, no error; tested on 455 monomials; two implementations | CONFIRMED (re-derived; two implementations; exact test on 28 monomials) |
| Theorem 1.3, last sentence (complexity, size of the numbers) | not in the version examined | not in the version examined | statement CORRECT; its proof was incomplete and is now written out |
| Moments and cumulants | m_0, …, m_44 and k_2, …, k_160 reproduced | CONFIRMED: all 154 moments and 153 cumulants of the note equal to its own | CONFIRMED: m_0, …, m_44 from the definition, k_2, …, k_562 from the model; all files equal |
| Theorem 1.2(d) | logic CONFIRMED; first line reproduced; second line to 14 digits (c) and 12 digits (w) | CONFIRMED: all intervals reproduced | CONFIRMED: both lines and all rows of Table 2 reproduced |
| Theorem 1.4 | logic CONFIRMED | CONFIRMED: all values reproduced; part (b) added | CONFIRMED: all six rows of Table 3 reproduced with its own q |
| Section 11, Table 4 | — | reproduced and extended to k = 280 | reproduced; the range in Observation 3 corrected |
| Labelling of what is only tested | one recommendation | CONFIRMED_WITH_FIXES (three wording points) | CONFIRMED |
| Additions made at the writing | — | — | CONFIRMED (see below) |
| Novelty and credit | one web search; nothing found | CONFIRMED_WITH_FIXES: nothing found for Theorems 1.2–1.4; credits and two disclosures required | nothing found; all references verified; one related paper added |

No run found a wrong statement of a theorem or a gap that could not be closed.

**Corrections required by the runs**, all applied in the note (details in the Verification paragraph of the note):
1. (Run A) Lemma 2.1: the identity for ‖λ*g‖² with the vacuum term.
2. (Run A) Lemma 4.1(c): the reason for k_{2n} > 0.
3. (Run B) The value 2.10199076… is presented as a numerical value for max supp ν; Proposition 7.1 (from run A)
   proves max supp ν = ‖b_Q‖; proved bounds are in Corollary 7.2(c).
4. (Run B) The OEIS was queried directly: no entry for 1, 4, 48, 928, 24448, 811776, for four subsequences, and for
   the cumulant sequence 4, 16, 160, 2432, 48896.
5. (Run B) Varšo's thesis, which the first two runs could not obtain, was to be listed among the sources not
   consulted. It has since been obtained and read; the note cites it with what it contains (see below).
6. (Run B) Credits: Gerhold; Muraki and Lu; Gu–Hasebe–Skoufranis (type II is the notion of the source, found
   independently; their central limit theorem is for type I); Speicher–Woroudi; the classical mechanism
   (Cohen–Trenholme; Bożejko–Wysoczański).
7. (Run B) No largest ratio of the numerical tests of Lemma 5.1 is quoted as a margin: the ratio tends to the
   constant of the lemma (Remark 5.4(1)).
8. (Run B) The limit of k(β_k − 1) is described as not determined by the data.
9. (Third run) Theorem 1.3: the proof of the statement on the size of the numbers is written out (common
   denominator L_1 ⋯ L_n with L_m = 2^m lcm(1, …, m+1); growth of the numerators by 2^{O(n)} per step; for the last
   step, non-negative terms and n!·m_2n ∈ ℤ).
10. (Third run) Observation 3: the total weight of the Gauss nodes in [−2, 2] is between 0.4902 and 0.4907 for all K
    from 100 to 280 (the range 0.4904–0.4907 had been given as found for the four values of K of Table 4); it
    drops when a node passes the point 2 (K = 154, K = 277).
11. (Third run) The sentence after Theorem 1.2: part (b) uses m_2 = 4 besides m_4 and m_6.
12. (Third run) Credit: V. Crismale and Y. G. Lu, J. Operator Theory 83 (2020) 495–515 (for sums of position
    operators on the monotone Fock space over ℓ²(ℕ) the norm is the right endpoint of the support of the vacuum
    distribution), cited in Section 1.3 as a statement of the same kind as Corollary 7.2(a).

Also changed after the third run, without being required: display (6) is broken into two lines; "≤ A|ψ|" is
written in the proof of Proposition 7.1; Remark 8.6(1) mentions the enumeration for n ≤ 8.

**Taken over from the runs.** From run A: the proof of Proposition 5.3(b) through the factorisation of E′; the use
of m_6/m_4 > 6 in Section 6; Proposition 7.1. From run B: Theorem 1.4(b) and rows 4–6 of Table 3; row 5 of Table 2
and the 49 digits of c; the faster formulation of the model (Remark 8.6(2)); the Jacobi parameters beyond k = 153.

**Added when the note was written, after runs A and B; examined by the third run.**
- The formulas for the measures with Jacobi parameters (θ, 1, 1, …) in Section 1.3 and the computation of
  Remark 8.3: re-derived (zero of z − θG_sc(z) at θ/√(θ−1), residue (θ−2)/(2(θ−1)); density 1/(2π) on the disc of
  radius √2) and confirmed in exact arithmetic. CONFIRMED.
- The statement on the size of the numbers in Theorem 1.3: correct; proof completed (correction 9); tested for
  n ≤ 60.
- Remark 8.6(1): the remark at the end of Section 5 of the source states the decomposition formula with the words
  "it is not difficult to see that" and gives no proof; summing it over the patterns gives the Boolean
  moment–cumulant relation, so n!·k_2n is the number of irreducible bi-monotone pair partitions. Enumeration from
  the definition: 4, 16, 160, 2432, 48896, 1215744, 35909120, 1226442240 for n = 1, …, 8. CONFIRMED.
- What is said about the thesis of Varšo, about Gu–Hasebe–Skoufranis and about Bożejko–Wysoczański: see the next
  section. CONFIRMED.

## Relation to the literature, novelty and scope
- **Searches (10 October 2026).** arXiv (bi-monotone, bimonotone, bi-monotonic, monotone Fock space, also with
  eigenvalue, atom, norm; two-faced central limit theorems; the papers of the author of the question), zbMATH (the
  same terms; 24 entries for "monotone Fock space"), Crossref, the lists of works citing the source and citing
  Gu–Hasebe–Skoufranis in OpenAlex (seven and nine records; titles, available abstracts, keyword searches in the
  arXiv sources of several of them), eight web searches, and direct queries of the OEIS. The third run repeated
  the queries to arXiv (17 records for the three spellings, 10 for "monotone Fock space"), zbMATH (43 and 24),
  Crossref, OpenAlex (the same seven and nine records) and the OEIS (no entry), and made one web search.
  - Nothing was found on the law of the sum, its support, its norm, atoms, a Cauchy transform, a recursion or a
    generating function.
- **What was read.** The arXiv version of the source in full; the abstract in the Oberwolfach report; of
  Gu–Hasebe–Skoufranis (arXiv version) the passages on the two types in the introduction, the subsection on
  type II and the central limit example; of Gerhold's paper on the Schoenberg correspondence (arXiv version) the
  outlook section; of Bożejko–Wysoczański (open copy on Numdam) the definition of the t-transformation,
  1/G_{μ_t} = t/G_μ + (1 − t)z, and its effect on the Boolean cumulants, K_{μ_t} = t·K_μ (the description in
  Section 1.3 is accurate; their example with atoms at ±1/√(1−t) for t < 1/2 agrees with the formula of
  Section 1.3 after a dilation); of Crismale–Lu the abstract. The other references are cited for standard facts
  or context and were not read for the note. The bibliographic data of all references were checked with Crossref,
  zbMATH or arXiv (again by the third run: no correction needed).
- **Gu–Hasebe–Skoufranis, checked by the third run** (arXiv:1708.05334v5): the introduction says that the second
  type "was recently and independently discovered and studied" in Gerhold's paper and that the authors prefer the
  first; the central limit theorem is stated for type I. For α = β = γ = t = 1 the formula of their example equals
  (g(w) − g(z))/(z − w) with g(z) = (z² − 2)^{−1/2}, the two-variable Cauchy transform of (X, X) with X arcsine of
  variance 1; the sum has the moments 4, 24, 160 (m_6 = 464/3 for μ). The sentences of Section 1.3 are accurate.
- **The thesis of P. Varšo** ("Studies on Positive and Symmetric Two-Faced Universal Products", University of
  Greifswald, 2021, urn:nbn:de:gbv:9-opus-75524), to which the outlook of Gerhold's Schoenberg paper points, was
  obtained and read after the first two verification runs: Sections 6.4 and 6.5 in full, the rest by a full-text search
  for the words bimonotone, Brownian, Hankel, central limit and Gerhold. Its Section 6.4 tests symmetric two-faced
  universal products (partition-induced products and q-deformed families) for positivity and reports computations,
  for these products, of the moments up to order 10 and of the Hankel determinants Δ_0, …, Δ_5 of the sum of the
  two faces under the standard Gaussian functional. The bi-monotone product is not symmetric and is not among them; it occurs in the thesis
  only as an example of a two-faced product and as the model for a product of pointed representations. The law
  studied in the note, its moments, support or atoms do not occur there.
  - The third verification run examined this in a copy of the thesis provided by the author of the note:
    Section 6.4 (pp. 257–262) and Section 6.5 (pp. 262–264) were read in full, and the whole text was searched for
    bimonotone, Brownian, Hankel, central limit, Gerhold, arcsine and atom. Every clause of what the note says is
    accurate: the section gives the moments M_0, …, M_10 and the Hankel determinants Δ_0, …, Δ_5 of the sum of the
    two faces under the "standard Gaussian" functional for the partition-induced products, and for the two
    q-deformed families as polynomials in the parameter with numbers for one parameter value; "bimonotone" occurs
    three times in the text (as an example of a two-faced independence in the introduction; as the model for the
    product of pointed representations in the introduction and in Chapter 6) and otherwise only in the
    bibliography; "Brownian" occurs only in two titles of the bibliography.
- **Caveats.** One source could not be consulted because automated access was refused: the published version of
  the source (World Scientific, QP–PQ 32 (2023) 59–76). No novelty statement of the note depends on it. MathSciNet
  was not used. A search that finds nothing is not a proof of novelty.
- **Scope.** A partial answer: structural facts, an algorithm and rigorous numbers, no closed formula. No priority
  is claimed for Theorem 1.1 or for the mechanism; for the other results the note says only that they were not
  found in the literature that could be read.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/