# Verification report — OWR-14298808-012 (Bühler–Hug: high-dimensional behaviour of Y*_{d,k})

Verification date: 2026-09-29.

**Verdict.** The question is answered completely. Let (d_j, k_j) be admissible (1 ≤ k ≤ d − 1, 2k > d + 1) with d_j →
∞, and put m = 2k − d − 1, m_c = (2πe(k−1))^{1/2} and y = (m − m_c)/m_c^{1/2}. Then the Lévy distance between
Y*_{d_j,k_j} and N(0, Φ(−y_j/√2)) tends to 0, where Φ is the standard normal distribution function. Hence Y*
converges in distribution if and only if y_j converges in [−∞, ∞]; every limit is a centred, possibly degenerate,
Gaussian N(0, τ²) with τ² = Φ(−y/√2) ∈ [0, 1] (N(0,0) = δ₀); and every τ² ∈ [0, 1] occurs. At the critical rate left
open by Bühler–Hug–Thäle (ECP 31 (2026), Remark 1.2), k = d/2 + ½(eπd)^{1/2} + O(1), the limit is N(0, 1/2). The
N(0,1)/δ₀ dichotomy away from the critical scale is published (Bühler–Hug–Thäle, Theorem 1.1) and is recovered, not
claimed. Two independent referees found the proofs correct (the second one found one false remark, now corrected; see
below). The note has not been peer reviewed.

## Statement checked
- **Primary source.** T. Bühler (joint with D. Hug), "Intersection processes of k-flats in hyperbolic space",
  Oberwolfach Reports 21 (2024), no. 4, Report 58/2024, pp. 3357–3359, doi:10.4171/OWR/2024/58. The question is on
  p. 3359.
  - For admissible sequences with d_j → ∞ it asks for the asymptotic behaviour of the standardized limit law
    Y*_{d_j,k_j}.
  - It asks whether conditions such as k_j/d_j → 1/2 ensure convergence in distribution, and whether the limit is
    then Gaussian.
  - Motivation given there: plotted densities looked close to N(0,1) for k ≈ d/2, while every cumulant of order
    at least 3 diverges.
- **Same question in the paper.** T. Bühler, D. Hug, Stoch. Proc. Appl. 185 (2025) 104613,
  doi:10.1016/j.spa.2025.104613 (arXiv:2410.10413), end of Section 6, with Z*_{d,k} in place of Y*_{d,k}.
- **Corpus record.** ulamai/UnsolvedMath, OWR-14298808-012 (status `open`).

## Readings
| Reading | Status | Comment |
|---|---|---|
| Y*_{d,k} = standardized limit law of the report | same as Z*_{d,k} = Z_{d,k}/σ_{d,k} | Y_{d,k} is a positive constant multiple of the centred variable Z_{d,k} (Bühler–Hug 2025, Sections 5–6), so standardization removes the constant |
| Lévy measure of Z_{d,k} in the s-form of Bühler–Hug §6 or the x-form (1.2) of Bühler–Hug–Thäle | equivalent | substitution x = cosh^{1−k} s; also checked numerically (max deviation 8·10⁻¹¹) |
| "asymptotic behaviour" = behaviour along every sequence, not only along convergent ones | answered | Theorem 1.3(a) gives a Gaussian approximation in the Lévy metric along every sequence |

## Results in the paper
- **Lemma 2.1 (Beta representation; the identities of Bühler–Hug–Thäle (4.1)–(4.3)).** σ² = π^a Γ(b)/Γ(N) with
  a = (d−k)/2, b = (2k−d−1)/2, N = (k−1)/2. The normalized Kolmogorov measure x²ν*(dx) of Z* is the law of
  X = W^N/σ with W ~ Beta(b, a). Consequences: cum_n(Z*) = E X^{n−2}, and p = P(X ≤ 1) = I_{σ^{1/N}}(b, a).
- **Lemma 3.1 (log-Beta variables).** L = −log W is infinitely divisible with an explicit Lévy measure (from
  Gauss's integral for ψ). It satisfies:
  - a/(bN) ≤ κ₂ ≤ (a/(bN))(1 + 2/b);
  - κ₄/κ₂² ≤ 6/a + 2/b + 24/b² + 24/(ab);
  - a CLT, uniform in x, when a, b → ∞ (asymptotic normality of log W is classical; the paper gives a self-contained
    proof with explicit cumulant bounds and does not claim it as new);
  - log-concavity of the density of log W for a ≥ 1.
- **Lemma 4.1 (reduction).** If the Kolmogorov measures K_j put vanishing mass on every [ε, M], then the
  characteristic function is uniformly close to exp(−p_j t²/2) on compacts, with p_j = K_j((0,1]). This extends
  Bühler–Hug–Thäle, Lemma 2.1 (the cases p_j → 1 and p_j → 0).
- **Lemma 4.2 (log-concave densities).** sup f ≤ 2e²/s, with a self-contained proof. The sharp bound sup f ≤ 1/s
  (Bobkov–Chistyakov 2015, Prop. 2.1) is cited.
- **Lemma 4.3 (anti-concentration).** K_j([ε, M]) → 0 for every admissible sequence with d_j → ∞; if d − k stays
  bounded then p_j → 0.
- **Lemma 4.4 (Markov bound).** For N ≥ π, 1 − p ≤ (eρ)^{1/2}, where ρ = m²/m_c².
- **Lemma 5.1 (threshold).** p = P(ξ ≥ A/F) exactly. Moreover A = (a/2) log ρ + (b/2 + ¼) log θ + E with
  |E| ≤ N/b.
- **Lemma 5.2 (asymptotics of A/F).**
  - If limsup ρ < 1 then A/F → −∞.
  - If liminf ρ > 1 then A/F → +∞; the positivity of φ(θ_*) is stated explicitly.
  - If ρ → 1 then A/F = (y/√2)(1 + o(1)) + o(1).
- **Theorem 1.3.**
  - (a) The Lévy distance to N(0, Φ(−y_j/√2)) tends to 0; the compactness argument is written out in Step 3.
  - (b) Y* converges if and only if y_j converges.
  - (c) All subsequential limits are Gaussian, and every τ² ∈ [0,1] occurs.
- **Corollary 1.4.**
  - (a) The regimes ρ ≷ 1, which recover Bühler–Hug–Thäle, Theorem 1.1.
  - (b) The equivalent d-form y' = (m − (eπd)^{1/2})/(eπd)^{1/4}.
  - (c) The shift k = d/2 + ½(eπd)^{1/2} + c d^{1/4} + O(1) gives N(0, Φ(−√2 c/(eπ)^{1/4})). The value c = 0 is the
    critical rate, with limit N(0, 1/2).
- **Remark 1.5 (cumulants).** The divergent cumulants come from the part of the Kolmogorov measure on {X > 1}. That
  part escapes to infinity and does not affect the limit.
- **Remark 7.2 (bounded codimension; corrected after referee 2).** If c = d − k stays bounded, p → 0 slowly, like a
  power of d. Numerically (a computation), p ≈ Q(c/2, (c/4) log(N/π)) with Q the regularized upper incomplete Gamma
  function (relative error below 3·10⁻³ for 1 ≤ c ≤ 4 and 10⁴ ≤ d ≤ 10¹²), which is of order d^{−c/4}(log d)^{c/2−1}.
  The previous version said "only logarithmically", which is false.

## Computations (checks only; scripts and outputs in reproducibility/)
- **Finder** (`claimant/`, NumPy/SciPy).
  - Lemma 2.1 against direct quadrature of the original s-form: 11 pairs with d ≤ 60, max deviation 8·10⁻¹¹; the
    cumulants agree with (6.2) of Bühler–Hug.
  - All explicit inequalities (pre-audit constants) on 8336 pairs with d ≤ 10⁶: 0 violations.
  - p by incomplete Beta and by a Gamma-mixture quadrature, agreeing to 10⁻⁷. At the critical rate
    p = 0.929, 0.886, 0.736, 0.611, 0.546, 0.518, 0.507, 0.502 for d = 10³, 10⁴, 10⁶, 10⁸, 10¹⁰, 10¹², 10¹⁴, 10¹⁶,
    tending to 1/2. For y = ±1, ±2 the values tend to Φ(∓1/√2) and Φ(∓√2).
  - The characteristic function is within ~10⁻⁹ of exp(−pt²/2) at k = 10¹² (t ∈ [0.25, 5]), while its distance to
    the N(0,1) characteristic function grows to 0.24 at y ≈ 0.
- **Release audit** (`referee/audit_check.py`, standard library only).
  - p by Simpson quadrature of the log W density (no incomplete-Beta routine) reproduces the finder's values to six
    digits. The c·d^{1/4} shifts c = ±1 approach the predicted 0.796 and 0.204.
  - The inequalities with the paper's constants (including |E| ≤ (17/24)N/b, the κ₄ bound with 2/b, and the sharp
    log-concave bound) were checked on 8593 pairs with d ≤ 6.3·10⁶, using the audit's own digamma/polygamma code:
    0 violations.
- **Second referee** (`referee2/`, NumPy/SciPy; code written before reading the finder's scripts).
  - Lemma 2.1 against (1.2) of Bühler–Hug–Thäle, the cosh-parametrization and (6.2) of Bühler–Hug, 13 pairs with
    d ≤ 50: σ² within 5·10⁻¹⁴, cumulants n = 3, 4, 6 within 7·10⁻¹³, characteristic functions within 3.3·10⁻¹¹.
  - The inequalities with the paper's constants on 42,234 admissible pairs with d ≤ 6·10⁶: 0 violations. The largest
    value of |E|/(N/b) is 0.64; the sharp bound 1/F is attained up to rounding when a = 1.
  - p by `betainc` and by Simpson quadrature of the log W density: agreement within 1.2·10⁻⁷ up to d = 10¹⁶; every
    printed entry of Table 1 and the values 0.1117, 0.0549, 0.0040 for d − k = 1 reproduce.
  - Verification item 4 (y ≈ 0): sup over t ∈ [0.25, 5] of |cf − e^{−pt²/2}| is 2.4·10⁻³ at k = 10³ and 1.3·10⁻⁹
    at k = 10¹²; the distance to e^{−t²/2} grows from 0.043 to 0.240.
  - Bounded codimension: p ≈ Q(c/2, (c/4) log(N/π)) (see Remark 7.2 above), so the decay is polynomial in d.
  - Re-run from the release copy: all six recorded outputs reproduce byte for byte.
- **Convergence is slow**, of order N^{−1/4} log N in the window. A heuristic second-order term (not proved) explains
  the numbers, and why moderate-d plots look standard normal.

## Independent adversarial audit
Verdict of the independent verifier (2026-09-29): **PAPER_CANDIDATE**; correct: yes; answers the question as
intended: yes.
- The verifier read the source (OWR 58/2024, pp. 3357–3359) and checked the substitution v = tanh² s by hand.
- The verifier re-derived every lemma. It found |E| ≲ 1.33 N/b, within the constant then claimed; the audit
  sharpened the constant to N/b.
- The verifier re-ran the finder's scripts, with identical outputs.
- The verifier made two independent computations: p by quadrature of the log-Beta density (agreement with
  `betainc` within 10⁻⁹ up to d = 10¹⁴), and the characteristic function from the x-form Lévy measure (1.2) of
  Bühler–Hug–Thäle.

All six required fixes of the first verifier were applied:
1. The compactness argument from characteristic functions to Lévy distance (Theorem 1.3(a), Step 3).
2. The sharp log-concave bound (Bobkov–Chistyakov) is cited; the self-contained weaker lemma is kept.
3. φ(θ_*) > 0 is stated explicitly before the chord bound (Lemma 5.2(ii)).
4. The novelty caveat was re-checked; the results are listed below.
5. The note is framed as completing Bühler–Hug–Thäle, Theorem 1.1 and Remark 1.2. The alternating k_j/d_j → 1/2
   example is credited to their Theorem 1.1(a),(b).
6. HF status: the recommendation is **partially_solved** until the note is posted publicly.

**Second independent referee (2026-09-29).** Overall verdict: the mathematics is correct and the paper answers the
Oberwolfach question completely; no error in any proof; every numerical claim reproduces; no prior or concurrent
publication of the result found; **not fatal**. Verdicts by item:
- Statement fidelity: **pass**. The referee fetched OWR 58/2024 (pp. 3357–3359) and arXiv:2410.10413v1 and confirmed
  Y*_{d,k} = Z*_{d,k}, the Lévy measure (2) and the statement of Bühler–Hug–Thäle, Theorem 1.1.
- Proofs: **all correct**, checked line by line (Lemma 2.1, (G1)–(G3), Lemma 3.1(a)–(d), Lemmas 4.1–4.4, (5),
  Lemmas 5.1–5.2, Steps 1–4 of Section 6, Corollary 1.4 and the alternating example).
- Computations: **all reproduced** with independent code (see Computations above); one remark (7.2) was
  quantitatively misdescribed ("logarithmic" decay; the decay is polynomial).
- Novelty: **no prior or concurrent resolution** of the critical case found (arXiv, OpenAlex, Crossref, zbMATH, one
  web search, and a re-read of OWR 57/2025). The novelty wording was judged appropriately modest.
- Presentation, references and house style: good; all DOIs verified against Crossref; the build has no overfull or
  underfull boxes.

All six required fixes of the second referee were applied:
1. Remark 7.2: "p tends to 0 only logarithmically" replaced by the correct statement (p tends to 0 slowly, like a
   power of d; numerically p ≈ Q(c/2, (c/4) log(N/π)), of order d^{−c/4}(log d)^{c/2−1}; labelled as a computation).
   The values 0.11, 0.055, 0.004 are kept. RESULT.md was corrected in the same way.
2. Verification item 2 now says that the SciPy check on 8336 pairs used the earlier, weaker constants
   (|A − T| ≤ 2N/b, and 6/b in the κ₄ bound), and that the constants as stated were checked by the standard-library
   code (8593 pairs, d ≤ 6.3·10⁶) and by the second referee with SciPy (42,234 pairs, d ≤ 6·10⁶).
3. Abstract and Zenodo description: Φ is defined as the standard normal distribution function, and the limit laws are
   called centred, possibly degenerate, Gaussians, with N(0,0) = δ₀.
4. "Relation to [BHT26]": the log-Beta CLT is no longer listed as a new ingredient; it is described as a
   self-contained CLT with explicit cumulant bounds, and the asymptotic normality is called classical.
5. Verification item 5 and "Scope and priority" record the second referee's check (proofs re-checked; Lemma 2.1,
   Table 1, Remark 7.2 and item 4 recomputed with independent code) and the fourth web search, with no new result.
6. Release package: `reproducibility/audit/` renamed to `reproducibility/referee/`; the second referee's scripts and
   outputs added in `reproducibility/referee2/` (not the literature downloads or the fetch helper); README updated;
   paper rebuilt (11 pages, no overfull boxes, every page inspected), main.tex re-copied, source.zip rebuilt, the three
   Zenodo files re-copied and their sha256, size and md5 recomputed.

Optional suggestions of the second referee that were also applied: Lemma 2.1(iv) is stated as an identity with p
defined afterwards; the factor (a − 1) ≥ 0 is explicit in Lemma 3.1(d); Lemma 5.2(iii) notes θ ≥ 1/N once b ≥ 1;
Remark 7.1 says which terms of order N^{−1/4} the heuristic neglects; Kabluchko–Rosen–Thäle is cited for the
hyperplane case. The pre-revision source is kept as `paper/main_v1_prereferee_2026-09-29.tex` in the working folder.

## Relation to the literature, novelty and scope
- **Prior work.**
  - Bühler–Hug–Thäle, "High-dimensional limits arising from hyperbolic Poisson k-plane processes", Electron.
    Commun. Probab. 31 (2026), paper 44, doi:10.1214/26-ECP794 (arXiv:2511.20519). It proves the N(0,1)/δ₀
    dichotomy (Theorem 1.1) and uses the same Beta and incomplete-Beta identities.
  - Its Remark 1.2 states that the critical case 2γ = (eπ)^{1/2} remains open.
  - Its Theorem 1.3 concerns a different rescaling and is not treated here.
  - Kabluchko–Rosen–Thäle, "Fluctuations of λ-geodesic Poisson hyperplanes in hyperbolic space", Israel J. Math.
    269 (2025), 545–586, doi:10.1007/s11856-025-2766-6 (arXiv:2205.12820), determined the non-Gaussian limit law
    in the hyperplane case k = d − 1; it is now cited in the introduction, as the Oberwolfach report does.
- **What is new here.**
  - The mixed-limit reduction and the anti-concentration lemma.
  - The fact that every limit is Gaussian.
  - The criterion "Y* converges if and only if y_j converges", with limit N(0, Φ(−y/√2)).
  - The d^{1/4}-wide window.
  - The value N(0, 1/2) at the open critical rate.
- **Searches (September 2026).** Sources searched: arXiv (API: authors Bühler, Hug and Thäle, and keywords), Crossref,
  OpenAlex, zbMATH, Semantic Scholar and OpenCitations, plus four web searches (finder, verifier, audit, second referee). No
  treatment of the critical case was found. The second referee repeated the arXiv, OpenAlex, Crossref and zbMATH
  searches independently (all requests anonymous and logged) and found no new result.
  - Crossref and Scopus (via KITopen) each report one citation of the ECP paper, which could not be identified.
  - The Oberwolfach report 57/2025 (D. Hug, pp. 3048–3049, doi:10.4171/OWR/2025/57) cites arXiv:2511.20519 in a
    survey paragraph and has no result on the critical case.
  - arXiv shows only v1 of 2511.20519.
  - The Project Euclid page was not accessible (bot challenge; not bypassed), and the authors' personal pages were not
    located. This negative search is not a proof of priority.
- **Scope.** The note does not cover rates of convergence (only a heuristic is given) or joint limits in R and d of
  the prelimit variables. It also does not cover the rescaled Lévy measures of Bühler–Hug–Thäle, Theorem 1.3.
- **Recommended corpus status.** partially_solved. The dichotomy is published by Bühler–Hug–Thäle, and the critical
  window is settled in this unrefereed note. Change to solved once the note is public.

## 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/