# Verification report — OWR-17290-002 (Oberwolfach Report 47/2019, Problem 1)

Verification date: 2026-09-29.

**Verdict.** The ℓ-question is answered yes, in the complete setting that the report assumes. For every integer
ℓ ≥ 0, the almost-Riemannian structures X1 = ∂x, X2 = c(z)x(x^{2ℓ}+ρ(z))∂z (c > 0, ρ ≥ 0, c and cρ bounded, on
R × R or R × circle), all of which are complete, have an essentially self-adjoint Laplace–Beltrami operator on the
regular region. This family includes a structure that coincides with the model on a strip around the origin; the
same holds for Prandi–Rizzi–Seri's Examples 7.7 and 7.8 for all ℓ and n. No localisation theorem is proved: a
complete structure that agrees with the model only near the origin is covered by the paper's results only when it
belongs to this family or to the class of Corollary 7.1. The case ℓ = 1 (and ℓ ≤ n/2) had been settled by
Prandi–Rizzi–Seri in the complete (torus) setting of their Example 7.7. Taken literally on all of R², the model is incomplete, and its operator is not
essentially self-adjoint (even for ℓ = 1), for a reason unrelated to the singular set. The more general real-analytic
no-tangency conjecture of the report remains open; it is proved only for structures with compact singular set
satisfying x∂x log|f| ≥ 1 − κ|x| in normal coordinates (the report does not assume that the singular set is
compact). The note is unrefereed; it was revised after a second internal referee report (see below).

## Statement checked
- **Primary source.** Oberwolfach Reports 16 (2019), no. 4, 2911–2950, Report 47/2019, *Mini-Workshop: Self-adjoint
  Extensions in New Settings*, doi:10.4171/OWR/2019/47. The problem is in the abstract of L. Rizzi's talk (joint work
  with V. Franceschi, D. Prandi, M. Seri), pp. 2917–2921, §3.2 "Problem 1", p. 2920.
  - §3.1 of the abstract assumes that (M,d) is complete and asks whether Δ_g with domain C_c^∞(R) is essentially
    self-adjoint in L²(R, μ_g).
  - Problem 1 takes X1 = ∂x, X2 = x(x^{2ℓ}+z²)∂z on R², notes that ℓ = 1 is covered by [PRS, Example 7.7], and says
    the authors expect the same for all ℓ ≥ 1 but cannot prove it.
  - It then proposes that the Laplace–Beltrami operator of every real-analytic 2D ARS without tangency points is
    essentially self-adjoint.
  - The independent verifier and the second referee re-fetched the report anonymously; both downloads are
    byte-identical to the local copy (md5 86690afe26f542078c6b23baa52cf051).
- **Origin.** D. Prandi, L. Rizzi, M. Seri, J. Spectr. Theory 8 (2018) 1221–1280, doi:10.4171/JST/226, Example 7.7:
  the family on R × T^{n−1} with z² replaced by f ≥ 0; the authors say they do not know the answer for ℓ > n/2, even for
  n = 2, ℓ = 2 (checked in arXiv:1609.01724v3). All theorem and equation numbers of PRS and FPR cited in the paper
  refer to arXiv 1609.01724v3 and 1708.09626v3, as the paper now states.
- **Corpus record.** ulamai/UnsolvedMath v1.6.0, OWR-17290-002 (literature status: partially solved). The statement
  asks about the ℓ-family and, "more generally", the real-analytic no-tangency conjecture.

## Readings
| Reading | Answer | Where |
|---|---|---|
| ℓ-family inside a complete ARS, as §3.1 of the report assumes (intended reading) | yes, for every ℓ ≥ 0 | Thm 1.1, Cor. 4.2 |
| the n-dimensional versions, PRS Examples 7.7 and 7.8 (all ℓ, n, smooth f ≥ 0 on the torus) | yes | Cor. 4.3 |
| model (1) literally on all of R² (incomplete: z = ±∞ at finite distance) | no, because of the ends z → ±∞, not the origin | Prop. 6.1 |
| "more generally": real-analytic 2D ARS without tangency points | open; proved when the singular set S is compact and x∂x log\|f\| ≥ 1 − κ\|x\| in normal coordinates near S (the report does not assume S compact) | Cor. 7.1; Prop. 7.2 shows the limit of the method |

## Results in the paper
- **Lemma 2.1.** For m > 0 and a := −t m′/m, ∫m|w′|² = ∫m|w′ − bw/t|² + ∫(b(1+a−b) − tb′) m|w|²/t². The choice
  b ≡ 1 gives ∫m|w′|² ≥ ∫ a m|w|²/t²; b = (1+a)/2 reproduces the effective-potential route.
- **Theorem 3.2 (warped products).** On (R∖{0}) × N with g = dx² + h_x and ω = m dx dμ, suppose (A1) −x∂x log m ≥ 1
  and (A2) N admits cut-offs with small gradient on strips (automatic if N is compact). Then Δ_ω is essentially
  self-adjoint. The proof is the Hardy inequality with constant 1 plus an Agmon-type cut-off argument with profile
  G(r) = r/r0. It uses dominated convergence throughout.
- **Theorem 1.1 / Corollary 4.2.** For 2D structures X1 = ∂x, X2 = f∂z: the conditions (A1′) x∂x log|f| ≥ 1
  (|f|/|x| nondecreasing in |x|) and (A2′) N1 compact or f bounded on strips give essential self-adjointness. For
  f = c x(x^{2ℓ}+ρ) one has x∂x log|f| = 1 + 2ℓx^{2ℓ}/(x^{2ℓ}+ρ) ≥ 1. Lemma 4.1 gives δ = |x| and completeness.
- **Corollary 4.3.** PRS Examples 7.7 and 7.8 for all ℓ, n: a = (n−1)(1 + 2ℓt^{2ℓ}/(t^{2ℓ}+f)) ≥ 1.
- **Theorem 5.1.** In the criteria of PRS (Thm 3.1, Thm 7.9) and FPR (Thm 1.1), the hypothesis
  V_eff ≥ 3/(4δ²) − κ/δ can be replaced by −Δ_ωδ ≥ 1/δ − κ. Only Step 1 of the weak Hardy inequality changes.
  - Remark 5.2: this is an alternative, not a strengthening. For dω = t^α dt dμ it holds iff α ≤ −1, while the V_eff
    condition holds iff α ≤ −1 or α ≥ 3.
  - Both are special cases of one b-criterion.
  - Example 5.3: PRS Example 4.1, case 2, for every ℓ.
- **Proposition 6.1.** The literal model on R² is incomplete and not essentially self-adjoint. For
  U = φ(x)χ(z) and W = φ(x)χ(z)/z in the maximal domain, the boundary form equals ∫xφ² ≠ 0.
- **Corollary 7.1.** A complete 2D ARS M with compact singular set S without tangency points, and
  x∂x log|f| ≥ 1 − κ|x| in normal coordinates near S, is essentially self-adjoint. This covers Grushin points, regular structures, f = x^k h with |h|e^{κ|x|}
  nondecreasing, and the circle versions of Thm 1.1.
- **Proposition 7.2 (ridge example).** For f = x(σ(z−x)+x⁴) near the origin, with c1y² ≤ σ ≤ y², the weak Hardy
  inequality ∫|∇w|² ≥ a∫_{δ<η}(δ^{−2} − κδ^{−1})|w|² + C‖w‖² fails for every a > 0, κ, η, C.
  - This includes the complete real-analytic structure f = x(sin²(z−x)+x⁴) on R × R/πZ.
  - The proof has explicit constants for 0 < z0 < 1/8: D > z0^{−4}/45, A ≤ 21c1^{−1}z0^{−3}, B ≤ 12096z0³,
    N ≤ 16z0²D (the strict bound z0 < 1/8 makes supp w compact in the open square).
  - Consequently neither the V_eff criterion nor Thm 5.1 (nor any b) decides this structure. No counterexample to the
    conjecture is claimed.

## Computations (scripts and outputs in reproducibility/)
- **Finder** (`claimant/`).
  - `s1_exact_identities.py`: exact rational arithmetic (own polynomial library, no CAS), 114 PASS lines, 0 FAIL.
    It checks the formula for Δ_g, a = 1+2ℓu, the V_eff formula (3) against PRS (241)–(242) for n = 2..5, the
    Lemma 2.1 identity, min x²V_eff = 3/4 − (ℓ−1)²/3, the literal-model formulas, the family weights, PRS Ex. 4.1,
    the ridge values and the b-family.
  - `s2`–`s4` are numerical (TESTED): fibre Hardy constants > 1 for ℓ ≤ 6 (FD and FEM); the literal boundary form to
    twelve digits for ℓ = 1, 2, 3; ridge quotients (A+B)/D ≈ 13.3 z0.
  - All four outputs were reproduced byte for byte on 2026-09-29.
  - In the paper these are described as 114 exact checks (not all are identities: a minimisation, finiteness and
    distance checks, and a scaling check are included).
- **Referee** (`referee/ref_checks.py`, written from the paper's text, independent of the finder's library).
  - Exact Fraction-jet re-derivation of all identities, including Lemma 2.1 for random polynomial m, b, w.
  - Exact check of every constant in the proof of Prop. 7.2.
  - Numerical A, B, D, N for σ = y² and sin² at z0 from 1/8 to 1/1000, all within the proved bounds.
  - FEM fibre constants ≥ 1 for ℓ ∈ {1,2,3,5}, ρ ∈ {1, 1e−3, 1e−6, 0}, reproducing (ℓ+1)² for ρ = 0; control weight
    0.58 < 1.
  - The frame determinant of PRS Ex. 7.8 (n = 2, 3, 4).
  - The literal boundary form with a different φ and χ (agreement to 1e−12).
  - All PASS.
- **Second referee** (`referee2/`, written from the paper's text before reading any finder or referee-1 script).
  - `ref2_exact.py`: standard library only, own polynomial and rational-function arithmetic over Q; 311/311 exact
    checks PASS, including four negative controls that are correctly rejected. It covers a = 1+2ℓu, x²V_eff, PRS
    (241)–(242) for n = 2..6 and ℓ = 1..5, the minimum and sign thresholds, Lemma 2.1 (real and complex w),
    Remark 2.2, the Thm 1.1 family, the frame determinants of PRS Ex. 7.7 and 7.8 (n = 2..5), PRS Ex. 4.1,
    Remark 5.2, the identities of Prop. 6.1, the ridge values, every constant of Prop. 7.2, the Nenciu–Nenciu
    comparison and the examples after Cor. 7.1.
  - `ref2_numeric.py` (numpy, scipy): 48/48 numerical checks PASS: the Prop. 6.1 boundary form by 2-D quadrature,
    FEM fibre Hardy constants for ℓ = 1..6 (all ≥ 1, tending to 1), the integrals A, B, D, N of Prop. 7.2 for
    σ = y² and sin² at z0 = 2^{−3}, …, 2^{−11}, and Hörmander's condition for the ridge example.
  - The second referee also re-ran the claimant scripts s1–s4 and `referee/ref_checks.py` from an unzipped copy of
    `source.zip`: all five outputs are byte-identical to the recorded ones.
  - After the revision (2026-09-29), all seven recorded outputs (claimant s1–s4, `referee/ref_checks.py`,
    `referee2/ref2_exact.py`, `referee2/ref2_numeric.py`) were reproduced byte for byte from an unzipped copy of the
    revised source archive.

## Independent adversarial audit
Verdict of the independent verifier (2026-09-29): PAPER_CANDIDATE; correct: yes; answers the question as intended:
yes. The verifier checked every step of the main theorem by hand. It re-ran all scripts (byte-identical outputs)
and made independent FEM and quadrature checks. It confirmed Theorem 5.1 against the texts of FPR Prop. 4.6 and PRS
Prop. 3.3.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (report re-fetched; md5 matches) |
| Main proof (Thm 1.1 / 3.2) | CONFIRMED |
| First-order criterion (Thm 5.1) | CONFIRMED, with the scope fix below |
| Literal-model negative (Prop. 6.1) | CONFIRMED (independent quadrature) |
| Computations | CONFIRMED |
| Novelty | CONFIRMED as far as can be checked, after adding Nenciu–Nenciu 2022 |
| Presentation | CONFIRMED_WITH_FIXES |

All five required fixes were applied:
1. Nenciu–Nenciu, arXiv:2205.10494, is now credited, with the reason it does not cover (1). Its Assumption A_ρ needs
   (|∇ln r| + |∇γ|)δ^s bounded with s < 1, while ∂z ln ρ = −x^{−ℓ} at z = x^ℓ. The multiplier / ground-state Hardy
   literature (Barbatis–Filippas–Tertikas; Brusentsev; Robinson) is also cited. Novelty is framed as the choice
   b ≡ 1 (trial function δ).
2. Theorem 5.1 is stated as an alternative sufficient criterion, not a strengthening (Remark 5.2).
3. The record update separates the ℓ-family (yes in the complete setting), the literal R² operator (not
   essentially self-adjoint, because of z = ±∞) and the general conjecture (open outside the proved class; overall
   label partially_solved).
4. The ridge example: the failure of the weak Hardy inequality is now proved in full with explicit constants
   (Prop. 7.2), checked exactly and numerically. The earlier phrase "every δ-based argument provably fails" is
   replaced by the precise statement.
5. Dominated (not monotone) convergence is used for the limits in the cut-off argument.

Additional corrections made while preparing the paper:
- The s1 output has 114 PASS lines (an earlier summary said 115).
- For PRS Example 7.8 the frame determinant gives the exponent n−1, while PRS say n−1 is replaced by 2(n−1). The
  conclusion does not depend on this (footnote in Cor. 4.3).
- Hörmander's condition is stated as a sufficient condition (∂x^j f(0,z) ≠ 0 for some j), not an equivalence.

### Second referee report (2026-09-29)
Verdict of the second independent adversarial referee: **correct, with minor fixes; not fatal**. The referee checked
every proof line by line and found no mathematical error. The main result answers the ℓ-question of OWR 47/2019,
Problem 1, for every ℓ, for the complete versions of the model; Corollary 4.3, Theorem 5.1, Proposition 6.1,
Corollary 7.1 and Proposition 7.2 also hold. No prior publication of the result was found.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (with a wording qualification, fixes 3–4) |
| Proofs | CONFIRMED (one trivial boundary-case slip, fix 2) |
| Computations | CONFIRMED (own code: 311/311 exact, incl. 4 negative controls, and 48/48 numerical checks; the finder's and referee 1's outputs reproduce byte for byte) |
| Novelty | CONFIRMED as far as can be checked (no prior resolution found) |
| Presentation / house style | CONFIRMED_WITH_MINOR_FIXES |
| Fatal | No |

All required fixes of the second referee report were applied:
1. Bibliography: Robinson (2021) is J. Funct. Anal. 281, no. 4 (not no. 8), 109067 (checked against Crossref).
2. Proposition 7.2: 0 < z0 < 1/8 (strict), and 0 < z0 < min{1/8, η/8} in the conclusion, so that supp w is compact
   in the open square {0 < x < 1/2, 0 < z < 1/2} (for z0 = 1/8 it contained the point (1/2, 1/4)).
3. Abstract and introduction: the family of Theorem 1.1 (all of whose members are complete) is said to have an
   essentially self-adjoint Laplace–Beltrami operator, with an explicit statement that no localisation theorem is
   proved for arbitrary complete structures agreeing with (1) near the origin. The case ℓ = 1 is attributed to PRS
   "in the complete (torus) setting of PRS Example 7.7", noting that even ℓ = 1 fails on the literal R²
   (Prop. 6.1). "Answers the question as intended" is softened to "answers what we take to be the intended
   question".
4. Scope of the general conjecture: the introduction, Scope and priority, the paragraph after Prop. 7.2 and this
   report now say that Corollary 7.1 needs a compact singular set, which the report's conjecture does not assume.
5. Citation pointers: in Remark 3.3, G is the function of the proof of FPR Prop. 4.7, eq. (80) (equivalently FPR
   Remark 4.2 with a = 1); the introduction states that all PRS/FPR theorem and equation numbers refer to arXiv
   1609.01724v3 and 1708.09626v3 (also noted in the bibliography).
6. Notation: M is the manifold (Cor. 4.3, Cor. 7.1; Theorem 5.1 now uses Ω for the Riemannian region), N only the
   fibre of Setting 3.1, X1, …, XK the frame in the introduction, S (script) the singular set throughout (Z no longer
   used for it), (A1′)/(A2′) the hypotheses of Cor. 4.2 (formerly (H)/(B)), and Λ_R the bound on |f| in Lemma 4.1
   (formerly M_R).
7. Verification item 1: "114 polynomial identities" became "114 exact checks", noting that some are not identities.

Optional fixes also applied:
8. The second referee's scripts and outputs are included in `reproducibility/referee2/` and described in its
   README; Verification item 3 of the paper and this report mention them; `source.zip`, the Zenodo copies and the
   hashes and sizes in `ZENODO_METADATA.md` were regenerated.
9. The OWR entry now says "published 2020" (vol. 16 (2019), no. 4; Crossref issued 2020-11-18). E. Pozzoli,
   "Quantum confinement in α-Grushin planes" (Springer INdAM Series, 2020, doi:10.1007/978-3-030-60453-0_11,
   checked against Crossref) is cited alongside Beschastnyi–Quan, and A. D. Ward's two papers (2016, 2017; DOIs
   checked against Crossref), which the referee also named, are credited in Scope and priority. "Settles the cases
   left open in PRS Examples 7.7 and 7.8" became "covers all ℓ and n in PRS Examples 7.7 and 7.8, including those where the effective-potential
   criterion does not apply".

After the fixes the paper was rebuilt with tectonic (no errors, no overfull or underfull boxes, no undefined
references; 11 pages), and every page was re-rendered and inspected.

## Relation to the literature, novelty and scope
- **Prior work.**
  - ℓ = 1, and ℓ ≤ n/2 in PRS Example 7.7, are due to Prandi, Rizzi and Seri (effective-potential criterion).
  - The Agmon-type argument is from Nenciu–Nenciu (2009), PRS and FPR.
  - Hardy inequalities via multipliers or ground states are classical.
  - Nenciu–Nenciu (arXiv:2205.10494) apply weak anisotropic Hardy inequalities to 2D ARS, but under a density
    assumption that fails for (1).
  - Boscain–Laurent (2013), Gallone–Michelangeli–Pozzoli (2019), Pozzoli (2020), Beschastnyi (2023),
    Beschastnyi–Boscain–Pozzoli (2023) and Beschastnyi–Quan (preprint) treat two-step, Grushin-type, generic,
    curvature-perturbed or α-Grushin structures, not this family.
  - A. D. Ward's work on essential self-adjointness on domains and on the associated Hardy constant (Manuscripta
    Math. 2016; J. Aust. Math. Soc. 2017), named by the second referee, is now cited; it does not cover the family.
- **Searches (September 2026; all anonymous and logged in `queries.log`).**
  - arXiv API: almost-Riemannian with self-adjoint; quantum confinement; Laplace–Beltrami; sub-Laplacian; tangency
    points; authors Beschastnyi, Milatovic; Nenciu–Nenciu 2205.10494; weighted Hardy with self-adjointness.
  - OpenAlex: all works citing PRS and FPR.
  - zbMATH; Crossref (metadata and DOIs).
  - Two web searches (one by the finder, one during paper preparation), plus one by the verifier and one by the
    second referee.
  - The second referee ran its own arXiv, OpenAlex (all 22 works citing PRS and all 20 citing FPR, through 2026),
    zbMATH and Crossref searches; the revision added five anonymous Crossref DOI look-ups (logged in `queries.log`).
  - No resolution of Problem 1 or of PRS Example 7.7(2) was found.
- **Novelty.** The only new input is the choice b ≡ 1 (the multiplier δ) in place of the effective potential. We
  cannot exclude that criterion −Δ_ωδ ≥ 1/δ − κ appears elsewhere in some form. This negative search is not a proof
  of priority.
- **Scope.**
  - The positive results require completeness, which is the report's standing assumption. On R² literally, the
    answer is no (even for ℓ = 1) for a trivial reason.
  - Theorem 1.1 is a statement about the family (2); no localisation theorem is proved.
  - The general real-analytic conjecture is proved only under x∂x log|f| ≥ 1 − κ|x| and for compact singular set,
    which the report's conjecture does not assume. The ridge structure x(sin²(z−x)+x⁴) remains undecided.

## 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/