# Verification report — OWR-15208-008 (conformal parametrization of nonnegatively curved surfaces at integer smoothness)

Verification date: 2026-09-30 (revised the same day after a second independent verification run).

**Verdict.** The answer is no. Let M_0 = ℂ and M_1 = S², with metrics g_κ of constant curvature κ ∈ {0, 1}. For
every integer k ≥ 0 and κ ∈ {0, 1}, the map Π_κ(u, φ) = φ^*(e^{-2u} g_κ) is a continuous bijection from
O^k_{≥0}(M_κ) × D^{k+1}(M_κ) onto R^k_{≥0}(M_κ), but it is not a homeomorphism: its inverse is discontinuous at every
point. The same holds on the positively curved metrics R_{>0}, and for the map (u, φ) ↦ φ_*(e^{-2u} g_κ) in which the
Oberwolfach report states the question. For k ≥ 1 the conformal-factor component alone is discontinuous. The note is
unrefereed.

## Statement checked
- **Primary source.** I. Belegradek, "Spaces of nonnegatively curved surfaces" (joint work with J. Hu and T. Banakh),
  in: Mini-Workshop "Spaces and Moduli Spaces of Riemannian Metrics", Oberwolfach Report 3/2017, pp. 148–150,
  doi:10.4171/OWR/2017/3. The question is on p. 149.
  - Setting: M_0 = ℂ, M_1 = S²; D(M_κ) = orientation-preserving diffeomorphisms fixing 0, 1 (and ∞ for κ = 1).
  - The report writes the parametrization with the pushforward φ_*: by uniformization (and Blanc–Fiala for κ = 0)
    every g ∈ R_{≥0}(M_κ) is uniquely φ_*(e^{-2u} g_κ), and the map is (u, φ) ↦ φ_*(e^{-2u} g_κ).
  - Earle–Schatz gives a homeomorphism when g, u and φ_* vary in C^{r,α}, 0 < α < 1. The report asks whether the
    map is a homeomorphism and says the authors "expect this to fail when α = 0".
- **Pullback and pushforward.** BH15, BB18 and the paper use the pullback φ^*. The two formulations are equivalent.
  Since φ_* = (φ^{-1})^*, the report's map is Π_κ ∘ (id × inv) with inv(φ) = φ^{-1}, and inv is a homeomorphism of
  D^j(M_κ) for every integer j ≥ 1, because these are topological groups (BB18, Sections 8 and 9). For the paper's
  sequences, which differ from a fixed diffeomorphism only on a fixed compact subset of ℂ, this is elementary.
  Hence Theorem 1.1 holds verbatim for the report's map, with φ^{-1} and φ_l^{-1} in place of φ and φ_l and with the
  same u and u_l. Since φ_*(e^{-2u} g_κ) involves the differential of φ^{-1}, letting φ_* vary in C^{r,α}
  corresponds to φ ∈ C^{r+1,α}, i.e. to the topology of D^{k+1}(M_κ) for α = 0 and r = k. The paper explains this in
  the paragraph after Theorem 1.1 and in "Scope and priority".
- **Precise setting.** T. Banakh, I. Belegradek, J. Math. Soc. Japan 70 (2018) 733–756, doi:10.2969/jmsj/07027344
  (read in arXiv:1510.07269v2, 1 Mar 2017): (8.3), (9.2) and Theorem 8.4. Here φ ∈ D^{γ+1}, u ∈ O^γ, g ∈ R^γ, all with
  uniform C^γ convergence on compact sets; the map is a homeomorphism for γ ∉ ℤ and γ = ∞, and the authors expect
  failure for integer γ. For ℂ also I. Belegradek, J. Hu, Math. Ann. 362 (2015), Theorem 4.1 (read in arXiv v3,
  which states it for C^{k+γ} with 0 < γ < 1; by BB18, proof of Theorem 8.4, the published statement assumed integer
  exponents). The erratum (Math. Ann. 364 (2016) 711–712) was not read; the paper relies on the account of it in
  [BB18, proof of Theorem 8.4].
- **Corpus record.** ulamai/UnsolvedMath, record OWR-15208-008, as found in two local copies of the dataset (the live
  Hugging Face page was not read for this report):
  - release v1.6.0 (commit c6e7f41): `statement.clean` is null; the upstream statement asks whether the
    parametrization (u, φ) ↦ φ^{*}e^{-2u}g_κ is a homeomorphism "in the specified metric topologies"; category
    "Algebraic Geometry"; `upstream_status` partially_solved.
  - a later copy (commit 372682f, saved 2026-09-29; the record was updated on 2026-09-16): status partially_solved,
    category "topology", and a clean statement (statement status corrected_verified) asking whether the
    parametrization remains a homeomorphism "in the borderline C^{r,0} metric, function, and diffeomorphism
    topologies".
  - Both are answered by the case k = r of Theorem 1.1, with φ in C^{r+1}. Both versions write the map with the
    pullback, as the paper does.

## Readings
| Reading | Answer | Where |
|---|---|---|
| BB18 setting: φ ∈ D^{k+1}, u ∈ O^k, g ∈ R^k, compact-open, any integer k ≥ 0, ℂ or S² | not a homeomorphism; inverse discontinuous at every point | Theorem 1.1 |
| the same for positively curved metrics (O_{>0}, R_{>0}) | same | Theorem 1.1 |
| the report's formulation (u, φ) ↦ φ_*(e^{-2u} g_κ), same topologies | same answer (φ_l^{-1} in place of φ_l) | paragraph after Theorem 1.1 |
| any topology on the diffeomorphism group, k ≥ 1 | u-component of Π^{-1} discontinuous at every point | Theorem 1.1(b), Remark 7.2 |
| any topology on the diffeomorphism group, k = 0 | u-component discontinuous in C^0 at every flat metric on ℂ; other points not treated | Proposition 7.3 |
| D topologized by C^0 convergence of φ^*g_0 (weaker than C^1), k = 0 | Π_0^{-1} discontinuous at flat metrics | Proposition 7.3 |
| all smooth metrics on S² (map (8.1) of BB18), integer γ | inverse discontinuous at every point | Remark 7.4(i) |
| RP², or R_{≥λ} with λ ≠ 0 | not treated | Remark 7.4(ii) |

## Results in the paper
- **Theorem 1.1.** For every g = Π_κ(u, φ) there are g_l = Π_κ(u_l, φ_l) ∈ R_{≥0} with g_l → g in C^k and:
  - k = 0: g_l isometric to g, u_l = u, φ_l does not converge in C^1 (spirals, Proposition 3.1);
  - k ≥ 1: φ_l → φ in C^k, φ_l does not converge in C^{k+1}, u_l does not converge in C^k.
- **Section 4 (log-twisted maps).** ψ_ε = ε z^{k+1} H(|z|²) with H(s) = −∫_s^∞ χ(t) dt/(t+δ²), δ = e^{-1/ε}.
  - Lemma 4.3 (i)–(vi) proves the needed estimates in full: symbol classes S^m, the bound ε log(1/ρ) ≤ 1, Faà di Bruno
    for μ_ε = ∂̄ψ/(1+∂ψ), and why D^kμ_ε has no logarithmic term.
  - Lemma 4.5: ∂^{k+1}(φ_ε − id)(0) = −(k+1)!(2 + ε c_δ) and ∂^k log|∂φ_ε|(0) = ½ ∂^{k+1}(φ_ε − id)(0).
- **Section 5 (curvature).** For k ≥ 2, K ≥ 0 follows from C² convergence. For k = 1 a radial correction w_ε with
  Δw_ε = 16ε(|z|²+2δ²)(|z|²+δ²)^{-3/2} near 0 is added; Lemma 5.3 rests on the identity
  (σ+2)²(σ+1)³ − σ(σ²+3σ+3)² = σ⁴+4σ³+7σ²+7σ+4. In Proposition 5.1(c) the constant C_U depends, for k = 1, also on the
  auxiliary data of Section 5.2 (through w_ε).
- **Section 6.** Proof of the theorem: non-flat metrics via a chart around a point of positive curvature; flat metrics on
  ℂ via h_t = e^{-2(c+tv)}g_0 and a diagonal choice with ε_l ≤ 1/l.
- **Remark 7.1 (k ≥ 1 only).** Π^{-1} is continuous into O^{k−1+β} × D^{k+β} for β ∈ (0,1), so only the endpoint is lost.
- **Proposition 7.3.** Smoothed cones at the flat plane: g_ε → g_0 uniformly on ℂ, u_ε(0) = −1/(1−ε).

## Computations (sanity checks; scripts and outputs in reproducibility/)
- **Finder** (`claimant/`, exact Taylor-jet arithmetic, no finite differences). Parameters a = 0.2, b = 0.9, a_w = 0.1,
  b_w = 0.3, [a_β, b_β] = [0.35, 0.9], A = 16; cut-offs are steps and a bump in |z| (stated in Section 8 of the
  paper); k = 1, 2, 3; κ = 0, 1; ε = 0.04 … 0.01; min K is taken over a grid on |z| ≤ 0.95.
  - The C^k distances ‖φ_ε − id‖, ‖μ_ε‖, ‖g_ε − h‖ decrease approximately linearly in ε (for k = 1 the metric
    distances contain O(ε²) terms from e^{-2w_ε}).
  - ∂^{k+1}(φ_ε − id)(0) equals −(k+1)!(2 + εc_δ) to all printed digits.
  - min K(g_ε) > 0 on the grid for the tested ε = 0.02, 0.015, 0.01. For k = 1, κ = 0 and ε = 0.04, 0.03 a negative
    value occurs in the outer annulus (ε above the smallness threshold of Proposition 5.1 for these cut-offs).
  - Without the correction w_ε (k = 1), min K ≈ −2·10^{42} at ε = 0.01.
  - Re-run on 2026-09-30 from the release copy: all outputs byte-identical.
- **Verifier** (`verifier/indep_check.py`, written from the paper for this release; no jets, no Brioschi formula;
  cut-offs chosen independently of the finder's).
  - Lemma 2.2 checked with K from the Riemann tensor (finite differences): (K_γ − 4 Re ∂²μ)/t² stays between 7.9 and 8.4 for t = 10^{-1}, 10^{-2}, 10^{-3}.
  - Formulas (3), (4) for ∂̄ψ, ∂ψ against finite differences (170-digit arithmetic near 0); Lemma 4.5 values to 10
    digits, extracted as Fourier coefficients on |z| = 10^{-6}δ.
  - K(g_ε) computed as e^{2F}ΔF for the conformal metric e^{-2F}|dw|², F = V∘φ_ε^{-1} (Newton inversion, 5-point
    Laplacian, 120–238 digits near 0 down to |z| = δ/100), for k = 1, 2, 3, κ = 0, 1, ε = 0.02, 0.0125, 0.008, 0.005 and
    two cut-off sets (48 configurations). K > 0 at every grid point in all configurations with ε ≤ 0.0125 and for
    ε = 0.02 except k = 1, κ = 0, where the outer-annulus minimum is −0.008 and −0.14 (ε not yet small enough for
    these cut-offs; the proof requires ε ≤ ε_1).
  - The correction is needed: for k = 1, ε = 0.02, at z = −2δ, K ≈ 3.6·10^{20} with w_ε and ≈ −4.1·10^{20} without.
  - Spirals and smoothed cones by finite differences: sup|g_ε − g_0| between 2.0ε and 2.7ε on 10^{-3}η ≤ |z| ≤ 10^6 for the cones,
    u_ε(0) = −1/(1−ε).
- **First independent verification run** (AI-assisted, 2026-09-30). It checked every step of the proofs, re-ran the
  finder's scripts (byte-identical outputs), computed K(g_ε) = e^{2V}Δ_{φ^*|dz|²}V by finite differences in 36
  configurations (k = 1, 2, 3; κ = 0, 1; ε = 0.02, 0.0125, 0.008; the finder's cut-offs and another set; all min K > 0),
  and made a 60-digit spot check at the scale δ. Its scratch programs were not retained.
- **Second independent verification run** (AI-assisted, 2026-09-30; code and outputs in
  `reproducibility/independent_run_2/`; Python standard library only, `decimal` with 56–480 digits). Its programs were
  written from the paper's definitions before the finder's scripts were read.
  - Construction (`rj.py`, `rconstr.py`, `run_grid.py`): own truncated bivariate Taylor-jet arithmetic; the paper's
    radii and A = 16; cut-offs that are steps and a bump in s = |z|² built from e^{-1/x} ("family A"; they coincide
    with the verifier's set P1) and, for comparison, from e^{-1/x²} ("family B"); H, c_δ, c_w, Q and W by tanh-sinh
    quadrature.
  - K(g_ε) by two formulas, neither of which is Lemma 2.2: the Riemann tensor from Christoffel symbols, and
    e^{2V}Δ_{φ^*|dw|²}V with V = U + w + log|∂φ|; relative agreement below 2·10^{-51} at every grid point. A third code
    path (`fd_check.py`: Gauss–Legendre quadrature, fourth-order finite differences and Brioschi's formula, 200–270
    digits) agrees to 1.5·10^{-20} at 224 points (ε = 0.02, 0.01; four models with k = 1, 2, 3; |z| = δ/3, δ, 5δ and
    every cut-off annulus).
  - Confirmed: Lemma 4.5 to all printed digits for k = 1, 2, 3; ‖φ_ε − id‖_{C^k}/ε ≈ 0.94, 3.3 and 30.9 for k = 1, 2, 3
    and ‖log|∂φ_ε|‖_{C^k} ≥ (k+1)!/2; for random cubic μ = t·m (t = 10^{-1}, 10^{-2}, 10^{-3}) the ratios in Lemma 2.2
    stay at most 2.7 (R_1) and 4.9 (R_2); Lemma 5.3 is never violated (sharp as δ/|z| → 0); the spirals give
    sup|h_ε − h| = 0.479, 0.220, 0.105, 0.051 at ε = 0.2, 0.1, 0.05, 0.025, with Jacobian 1 to 2·10^{-16} and
    ∂φ_ε(0) = e^i; the cones give sup|log λ|/ε from 1.178 (ε = 0.2) to 1.004 (ε = 0.005), sup|μ|/ε ≤ 0.556 and
    u_ε(0) = −1/(1−ε). The run's report also states agreement of (3), (4) to 10^{-69}–10^{-88}; that scratch test is
    not among the retained programs.
  - min K(g_ε), family A, centred models of Table 1: for k = 1, κ = 0 it is −1.324, −0.672, −0.0088, 0.327, 0.665 and
    0.954 at ε = 0.04, 0.03, 0.02, 0.015, 0.01, 0.005; all other (k, κ) are positive at every tested ε except k = 2,
    κ = 0 at ε = 0.04 (−0.052). Without w_ε (k = 1): min K ≈ −8.3·10^{20} (ε = 0.02) and −2.16·10^{42} (ε = 0.01).
  - Proof constants for k = 1. On |z| ≤ a_w the lower bound of the proof holds with a small constant:
    e^{-2(U+w_ε)}K − ΔU − 8ε(|z|²+2δ²)(|z|²+δ²)^{-3/2} is ≥ 1.7ε for the centred models and ≥ −1.4ε in the S² chart
    described below. On the outer annulus min (e^{-2(U+w_ε)}K − ΔU)/ε lies between −48.2 and −45.1 for all tested ε
    between 0.001 and 0.04 and all tested U (family A), so C''' ≈ 48, uniformly in ε. At the minimum (r ≈ 0.69, k = 1, κ = 0), Δw_ε ≈ −26.8ε (the β-term) and
    4 Re ∂²μ_ε ≈ −21ε (the χ-transition); the remainder R_1 + R_2 is about +1.5ε to +2ε.
  - The observed sign changes match the threshold c_0/C''': centred κ = 0 model (ΔU ≈ 0.92 at the minimum, predicted
    ≈ 0.0195): negative at ε = 0.02, positive at 0.015; flat family U_t with t = 0.1 (predicted ≈ 0.002; grid
    |z| ≥ 0.1): −0.153, −0.0039, +0.045 at ε = 0.005, 0.002, 0.001; the S² chart of Section 6.2 at p = 1/2 with
    r_0 = 0.3 (ΔU ≈ 0.16, predicted ≈ 0.0034): −4.83, −1.94, −0.48 and +0.258 at ε = 0.02, 0.01, 0.005, 0.0025 (the
    last on |z| ≥ 0.1 only). For k = 2 in the same chart min K = −1.34 and −0.17 at ε = 0.02 and 0.01. With family B,
    C''' ≈ 240 and min K < 0 at ε = 0.02 and 0.01 for k = 1 and for k = 2, κ = 0. None of this contradicts
    Proposition 5.1, which asserts K > 0 only for ε ≤ ε_1.
  - Rerun of the release from the previous `source.zip` (sha256 890b633e…a03): the finder's `run_all.sh` reproduced all
    five recorded outputs byte for byte (16 s), and `verifier/indep_check.py` reproduced its output except for
    wall-clock timing fields. The directory entries of that zip had mode 0600 and could not be entered after
    `unzip` without `chmod`; the zip has been rebuilt (directories 0755, files 0644).

## Independent verification
Verdicts of the first independent verification run (2026-09-30):

| Item | Verdict |
|---|---|
| Source fidelity | OK (OWR 2017/3 re-downloaded; identical to the cached copy; BB18 v2 is the latest arXiv version) |
| Correctness | no error found; every step checked |
| Answers the question as intended | yes |
| Novelty | no prior treatment of the integer case found (OpenAlex citations of BB18, BH15, BH16; arXiv; one web search) |
| Significance | modest: the analytic mechanism is classical; new are the curvature-preserving construction for k ≥ 1 (especially the k = 1 correction) and the discontinuity at every point on both surfaces |
| Classification | paper candidate |

All eight required fixes were applied:
1. Two editorial remarks in the working document that did not concern the mathematics were removed; the paper does
   not contain them.
2. HF status vocabulary: the suggested status is `solved`, with the note "answered in the negative; own unrefereed
   proof; cite the Zenodo DOI". It is not presented as a literature result.
3. Category: `geometry` (id 6) is suggested in place of the local label "Algebraic Geometry"; the HF label `topology`
   is defensible, so changing it is optional.
4. Flat case (Section 6.3): ε_l ≤ 1/l is now explicit, and both conclusions (φ_l → id in C^k; u_l does not converge
   in C^k) are proved, using that ψ_ε and w_ε do not depend on the base metric.
5. The estimates labelled (E1)–(E6) in the working document are written out as Lemma 4.3 (i)–(vi) with full proofs
   (symbol classes, Faà di Bruno, the role of ε log(1/ρ) ≤ 1, and why D^kμ_ε has no logarithmic term).
6. Novelty is stated modestly: k = 0 is an elementary observation and the analytic mechanism is classical; the new
   part is the curvature-preserving construction for k ≥ 1, especially the k = 1 correction, and the discontinuity
   at every point.
7. The sharpness statement is restricted to k ≥ 1 (Remark 7.1), and "for every ε ≤ 0.02" now reads "for the tested
   ε = 0.02, 0.015, 0.01".
8. The erratum [BH16] is cited only through the account in [BB18, proof of Theorem 8.4].

Verdicts of the second independent verification run (2026-09-30):

| Item | Verdict |
|---|---|
| Statement fidelity | faithful; the report's pushforward notation had to be addressed (fix 1 below) |
| Correctness | no mathematical error and no unrepairable gap found; every step checked; one constant dependence to be stated (fix 4) |
| Answers the question as intended | yes; α = 0, every integer k ≥ 0, both surfaces, R_{≥0} and R_{>0}; discontinuity at every point |
| Computations | every numerical claim confirmed by independent code; the negative values at ε ≈ 0.02 are the smallness threshold of Proposition 5.1 |
| Novelty and credit | no prior treatment of the integer case found; credit appropriate |
| Fatal problems | none |

All seven required fixes were applied:
1. OWR notation: the paper (Section 1, a new paragraph after Theorem 1.1, and "Scope and priority") and this
   verification report now say that the Oberwolfach report writes the map with the pushforward φ_* and lets g, u and
   φ_* vary in C^{r,α}; that BH15/BB18 and the paper use the pullback; that the two are equivalent because
   φ_* = (φ^{-1})^* and inversion is a homeomorphism of D^{k+1}(M_κ); that Theorem 1.1 holds verbatim for the
   Oberwolfach map with φ_l^{-1} and the same u_l; and that φ_* in C^{r,α} corresponds to φ ∈ C^{r+1,α}. The earlier
   version of this verification report misquoted the Oberwolfach report as letting "φ^*" vary in C^{r,α}; this is
   corrected above.
2. The abstract and the Zenodo description now say "orientation-preserving diffeomorphism fixing 0 and 1" (without it
   the uniqueness claim fails because of complex conjugation). The body already defined D(M_κ) this way; the proof
   of Proposition 7.3 now also states that φ_ε preserves orientation.
3. The Verification paragraph of the paper and this report now describe the step-by-step checks only as
   AI-assisted verification runs; a wording that suggested manual checking was removed.
4. Proposition 5.1(c) now states that for k = 1 the constant C_U also depends on the auxiliary data of Section 5.2
   (χ_w, β, the radii a_w, b_w, a_β, b_β, and A) through w_ε. Theorem 1.1(a), (b) now say "outside a fixed compact
   subset of the chart ℂ ⊂ M_κ" (the old wording was vacuous on S²).
5. `source.zip` (and its Zenodo copy) was rebuilt with directory entries of mode 0755 and files of mode 0644; the
   checksums in `zenodo/ZENODO_METADATA.md` were updated.
6. The "Corpus record" bullet above now describes the record as found in the two local copies of the dataset.
7. Section 8 of the paper now states the cut-offs used for Table 1; explains that the ε below which the numerics show
   K ≥ 0 depends on the cut-off shapes and on U, since Proposition 5.1 gives a threshold of order c_0/C''' with C'''
   independent of ε; reports that two independent computations with the same radii and cut-offs in |z|² are already
   negative at ε = 0.02; and attributes the negative outer minimum to both the term −Aεc_wβ and the χ-transition
   term 4 Re ∂²∂̄ψ.

Optional suggestions a (Theorem 1.1 wording), b (Table 1 cut-offs, the grid of min K, "approximately linearly") and
d (BH15 arXiv v3 states Theorem 4.1 with 0 < γ < 1) were also applied; c, e, f and g were not.

## Relation to the literature, novelty and scope
- **Prior work credited.** Belegradek–Hu (Math. Ann. 2015, Theorem 1.1 and Theorem 4.1) and its erratum;
  Banakh–Belegradek (J. Math. Soc. Japan 2018); Belegradek's Oberwolfach abstract (2017); Earle–Schatz (1970);
  Blanc–Fiala (1941). The mechanism (the Beurling transform and Schauder estimates fail at integer exponents) is
  classical.
- **Searches (September 2026).**
  - arXiv: Belegradek's author listing (47 entries); his later papers 1705.01220, 1705.01223 and 1805.02239 were read.
  - OpenAlex: all works citing BB18 (5), BH15 (16) and BH16 (4); Tuschmann–Wiemeler (arXiv:1712.07052) and
    Coll–Whitt (arXiv:2001.03888) were read in full.
  - zbMATH; arXiv keyword queries; two web searches in earlier stages (a third attempt during the write-up failed and
    was not repeated).
  - Second independent verification run (2026-09-30): arXiv API (Belegradek's full listing and 10 keyword queries),
    zbMATH API (Belegradek's documents 2017–2026, a citation query and a title search), Crossref for all DOIs, and one
    web search; OpenAlex was rate-limited, so the saved citing-work lists were reused. Nothing treating the integer
    case was found among the items that could be read.
  - Nothing found treats the integer case. This negative search is not a proof of priority.
- **Scope.** ℂ and S², every integer k ≥ 0, R_{≥0} and R_{>0}, in the BB18 topologies (equivalently, in the report's
  pushforward formulation). For k = 0, the discontinuity of the u-component alone is shown only at flat metrics on ℂ.
  RP² and R_{≥λ} with λ ≠ 0 are not treated.

## 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/