# Verification report — AMR-119-0003 (Tang: are conformal homeomorphisms of strongly pseudoconvex CR 3-manifolds CR?)

Verification date: 2026-10-20.

**Verdict.** For the Carnot–Carathéodory (CC) metric reading, the question is answered, modulo Theorem 1.1 of
Capogna–Citti–Le Donne–Ottazzi (J. Math. Pures Appl. 122 (2019)), a published theorem used without re-proof:
**no** as literally posed, **yes** under the Korányi–Reimann convention f*θ' = λθ with λ > 0 (coorientation
preserving). The analytic reading remains open. The answer is a corollary of the 2019 theorem.
- **Settled (Theorem 1.1 of the note).** A homeomorphism between strongly pseudoconvex CR 3-manifolds (C^∞; not
  necessarily compact, connected or embeddable) whose metric distortion H_f, for CC distances of smooth Hermitian
  metrics on the contact planes, is 1 at every point is a C^∞ diffeomorphism with df(H) = H' and horizontal
  differential a similarity; on each component it is CR or anti-CR; it is CR exactly where it preserves the
  coorientation of the contact planes (f*θ' = uθ with u > 0 for adapted forms). Conversely every C^1
  diffeomorphism that is CR or anti-CR on each component has H_f = 1 for all Hermitian metrics.
- **Literal answer no.** Complex conjugation σ(z, w) = (z̄, w̄) of S^3 and τ(x, y, t) = (x, −y, −t) of the
  Heisenberg group are CC isometries and anti-CR (exact identities; Section 5).
- **Not new mathematics.** The whole analytic content (1-quasiconformal maps between sub-Riemannian contact
  manifolds are smooth conformal maps) is CCLO 2019, Theorem 1.1; the rest is elementary linear algebra, which for
  the flat model is in Cowling–Li–Ottazzi–Wu (2020) (and, for Goursat groups, in Wu–Fu (2014)) and, almost
  everywhere on embedded hypersurfaces, in Wu–Wang (2012). No new technique is claimed; the novelty is low. The
  honest label is "answered by combining published results".
- **Open / outside the claim.** The analytic reading (Tang 1996, Definition 2.1: ACL, differentiable a.e.,
  horizontal differential a similarity a.e.) is covered only for maps that are in addition metrically
  quasiconformal (Remark 4.3). The higher-dimensional case and K-quasiconformal maps with K > 1 are not treated.

The note is unrefereed.

## Statement checked
- **Primary source.** G. Buzzard, J. E. Fornæss, E. A. Gavosto, S. G. Krantz, *The Several Complex Variables
  Problem List*, September 1995, arXiv:math/9509202 (four authors in the arXiv metadata; the LaTeX source has an
  empty author field and the date "September 1995"). The LaTeX source was read, and compiled to obtain the
  typeset numbering: Problem 2.0.3 in Section 2 "New problems in this edition", headed "Liouville type
  assertion on CR 3-manifolds", repeated as Problem 12.3.1 in Subsection 12.3 "Quasiconformal mappings"
  (the list's own cross-reference "(See Section 11.3)" counts the sections one lower; in the typeset list
  11.3 is "Complex Differential Geometry"); proposer Puqi Tang. The text defines a homeomorphism between
  two strongly pseudoconvex CR 3-manifolds to be conformal if it maps infinitesimal spheres with respect to a CC
  metric to infinitesimal spheres, and asks "Is a conformal homeomorphism CR?"
- **Companion problem** (the next item): quasiconformal diffeomorphisms of strongly pseudoconvex CR
  (2n+1)-manifolds distort the CR structures boundedly; with Hermitian metrics on the contact bundles the
  distortion is measured by how spheres are mapped to ellipsoids. This supports reading "CC metric" as the CC
  distance of a Hermitian metric, and the orientation convention (Beltrami coefficient |μ| < 1).
- **Corpus records.** ulamai/UnsolvedMath AMR-119-0003 ("Problem 3") and AMR-119-0078 (same text), status
  `open` (snapshot d371ee9, dataset version 1.8.0). The record text is a verbatim copy of the source.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Literal: CC distances of Hermitian metrics, H_f = 1 at every point, "CR" for the given J, J' | no (σ on S^3, τ on the Heisenberg group) | Corollary 1.2(a), Props. 5.1, 5.2 |
| Same, with the convention f*θ' = λθ, λ > 0 (coorientation preserving; Korányi–Reimann) | yes | Corollary 1.2(b), Theorem 1.1(3) |
| Sphere version H^=_f = 1 everywhere | same notion | Lemma 3.1 |
| lim instead of limsup | same notion (ratio ≥ 1 for small r) | Section 2 |
| H_f bounded and H_f = 1 a.e. | same conclusion (CCLO need only this) | after Corollary 1.2 |
| "CR or anti-CR" / "CR for J' or for −J'" | yes, on each component | Theorem 1.1(2), Corollary 1.2(c) |
| Disconnected M | componentwise sign | Theorem 1.1(2) |
| Non-Hermitian sub-Riemannian metric on H | false (explicit example) | Remark 4.4 |
| Analytic reading (Tang 1996, Def. 2.1) | open; covered only with metric quasiconformality in addition | Remark 4.3 |
| Higher dimension; K > 1 | not treated | Remark 4.5 |

## Results in the paper
- **Lemma 2.1 (adapted forms).** The sign of dθ(X, JX) is constant; L_θ is symmetric and J-invariant; adapted forms
  are unique up to positive factors, so H is always coorientable (a global adapted form exists).
- **Remark 2.3.** The CC distance with absolutely continuous horizontal curves equals CCLO's (smooth curves, bound
  on the speed); standard.
- **Lemma 3.1 (balls versus spheres).** For every homeomorphism and every point, ℓ_f = ℓ^=_f for small r and
  H_f = H^=_f. (CCLO note the equality of the infima in geodesic spaces in the proof of their Corollary 3.26 and
  prove H_f = H^=_f a.e. for quasiconformal maps.)
- **Lemma 3.2.** A C^1 diffeomorphism with horizontal differential a similarity has H_f = 1 everywhere.
- **Theorem 1.1 and Corollary 1.2** as in the verdict; proof in Section 4: Step 1 = CCLO 2019 Theorem 1.1
  (published numbering; arXiv v2 Theorem 1.19) via Definitions 3.1, 3.2, 3.13 and Theorem 3.3 with condition (3.4)
  (arXiv v2: 1.1, 1.2, 1.13, 1.3, (1.4)); Lemma 4.1 (a similarity between Euclidean planes with orthogonal complex
  structures commutes or anticommutes with them, not both; conversely); the clopen sign argument; the sign of u.
- **Remark 4.2.** Contact diffeomorphisms of 3-manifolds preserve θ∧dθ (f*(θ'∧dθ') = u²θ∧dθ), so the orientation
  of M does not detect anti-CR maps.
- **Remarks 4.3–4.5.** Analytic reading; Hermitian metrics are needed (F = A⁻¹∘R∘A with A = (2x, y/2, t));
  where each hypothesis is used.
- **Propositions 5.1, 5.2.** The anti-CR isometries σ and τ; the Heisenberg CR structure is the one induced from
  the Siegel domain Im w > |z|² (Z = ∂_z + i z̄ ∂_t, so JX = Y); the counterexample does not depend on this sign.

## Computations (sanity checks; programs and outputs in reproducibility/)
- `check_note.py` (standard library only, exact polynomial arithmetic): 33 checks, all PASS, 0.04 s — Section 5
  identities, the Siegel-domain CR structure, the sign rule on dilations/rotations and their compositions with τ,
  Lemma 4.1 for all 2400 nonzero integer 2×2 matrices with entries in [−3, 3], the matrices of Remark 4.4, and the
  non-example (z, w) ↦ (z̄, w) (value −24/25 at p = (3/5, 0, 0, 4/5)). A negative control (wrong maps) makes 7
  checks fail.
- `original/check_conjugation.py` (SymPy): the first check of the Section 5 identities; output re-produced.
- The programs of the two verification runs (SymPy; NumPy/SciPy for the numerical illustration) re-produced their
  recorded outputs byte for byte (`RERUN_LOG.txt`).
- `independent_run_2/run2_certificate.py` (SymPy, own routes): 36 checks, all PASS, 1.5 s; a negative control
  makes 10 fail.
- No proof depends on a computation; the numerical CC-sphere test is an illustration only.

## Independent verification runs
Three independent verification runs, all AI-assisted, each with its own programs. Runs A and B examined the first
write-up of the result; run 2 examined the text of the note.
- **Run A** (statement and proofs). Read the problem list source, Tang 1996 (arXiv source: Definition 2.1, Theorem
  2.3 and its proof, formula (2.9)), CCLO in arXiv v2 and in the accepted manuscript of the journal version
  (Sections 1–3, Theorem 1.1, Section 5.1), Cowling–Li–Ottazzi–Wu (arXiv source). Re-derived Lemmas 3.1, 3.2,
  4.1, Steps 0, 3, 4 and Remark 4.2 line by line; checked the deduction "H_f = 1 everywhere ⇒ 1-quasiconformal in
  CCLO's sense"; recomputed the Section 5 identities with its own SymPy program by a different route (all True);
  re-ran the original program (identical output). Verdict: CONFIRMED_WITH_FIXES (presentation and labelling only).
- **Run B** (independent computation and literature). Own SymPy program (33 lines of output, all as expected),
  including the Heisenberg CR structure derived from the Siegel domain, the sign rule on the boundary values of the
  inversion (z, w) ↦ (z/w, −1/w) (u = 1/|w|² > 0) and its composition with τ, controls; numerical illustration of
  the sphere distortion (τ: exactly 1; inversion and its composition with τ: 1.34, 1.075, 1.018, 1.004 for
  r = 0.2, 0.05, 0.0125, 0.003; the non-conformal contact map (2x, y, 2t): 2). Checked CCLO in arXiv v2, Tang 1996,
  CLOW, the bibliography with Crossref and zbMATH, and screened the Semantic Scholar citing lists. Verdict:
  CONFIRMED_WITH_FIXES (bibliography and related work).
- **Run 2** (the text of the note, 2026-10-20). Read the sources again: the problem list source (compiled to obtain
  the typeset numbering), Tang 1996 (arXiv source: Definition 2.1, Theorem 2.3, formula (2.9)), CCLO in the
  accepted manuscript and in the arXiv v2 source (compiled; Section 2.1, Definitions 3.1, 3.2, 3.13, Theorem 3.3
  with (3.4), Theorem 1.1, Section 5.1, Theorem 1.4(i), Proposition 2.12, the two citations of Tang 1996),
  Cowling–Li–Ottazzi–Wu (arXiv v2 source, compiled: Corollary 3.2, Theorem 3.3), the zbMATH reviews of
  Korányi–Reimann 1990 and Falbel–Veloso, and Wu–Wang in the copy of the Russian original fetched on 19 October
  (Math-Net.Ru refused the request of 20 October with HTTP 403). Re-derived Lemmas 2.1, 3.1, 3.2, 4.1, Steps 0–5
  and Remarks 4.2–4.5 line by line. Own SymPy program by routes not used before (CR structure of S^3 from its
  defining function; Hopf coordinates; Heisenberg CR structure from the Siegel embedding; σ and τ as restrictions
  of antiholomorphic maps of C^2; u = 1/|w|² (CR) for the Siegel inversion and u = −1/|w|² (anti-CR) for its
  composition with τ; controls (z̄, w) and (2x, y, 2t); Remark 4.4; Lemma 4.1 for all 6560 nonzero integer
  matrices with entries in [−4, 4]): 36 of 36 PASS. Re-ran all five programs of the package from the source
  archive: identical outputs. Repeated the searches (below). Found no mathematical error. Required corrections,
  all made: the typeset numbering of the problem list (Problems 2.0.3 and 12.3.1, not "Section 1, third problem;
  Section 11.3"); the arXiv v2 number of CCLO Section 5.1 (it was given as 6.1); the numbers Theorem 3.3 and
  Corollary 3.2 of Cowling–Li–Ottazzi–Wu; C^2 instead of C^1 in Remark 4.2 (the computation uses du); the
  description of orientation in Wu–Wang (the sign of the factor γ(p) of the hc-differential on the vertical
  direction); the credit to Wu–Fu (2014); the record of this run and of the searches.
- **None of the runs found a mathematical error.** None of them checked the proof of CCLO Theorem 1.1; it is cited.
- **Corrections required by runs A and B, all made:** the remark on the two
  definitions of the CC distance; the coorientation argument (no "non-coorientable" case); "modulo CCLO 2019
  Theorem 1.1" wherever the status is stated; CCLO numbering in both versions; one sentence in the proof of
  Lemma 3.1; the four authors of the problem list; Capogna–Le Donne's data; Falbel–Veloso added; every public
  summary says "answered for the CC-metric reading; analytic reading open; corollary of CCLO 2019".
- **Checks made when the note was written.** All DOIs verified with Crossref (CCLO; CLOW; Wu–Wang; Korányi–Reimann
  1985, 1990, 1995; Capogna–Le Donne, which Crossref dates 2021 for volume 379 (online 2020); Falbel–Veloso;
  Chow). Wu–Wang's Corollary 1.1, its "almost everywhere" and its orientation split, and the Russian page range
  396–417, were checked in the Russian original (open copy on Math-Net.Ru). One statement of run A was corrected:
  the published CCLO cites Tang 1996 twice (a list of works on Carnot groups, and after Theorem 1.1), not once;
  arXiv v2 cites it three times.

## Relation to the literature, novelty and scope
- **Credits.** The question is Puqi Tang's (problem list editors Buzzard, Fornæss, Gavosto, Krantz). Prior partial
  result: Tang, Ann. Acad. Sci. Fenn. Math. 21 (1996) 289–308, Theorem 2.3 (locally embeddable, L^1_loc horizontal
  derivatives ⇒ smooth and CR); the abstract of his 1994 Purdue thesis states the same (thesis not read). The
  whole analytic content: CCLO 2019, Theorem 1.1. Flat model: Cowling–Li–Ottazzi–Wu, Proc. AMS Ser. B 7 (2020),
  Theorem 3.3 with Corollary 3.2 (tight stratified groups; arXiv v2 numbering); earlier, Wu–Fu, The Scientific
  World Journal 2014 (Goursat groups; abstract only). Embedded, a.e.: Wu–Wang, Sib. Math. J. 53 (2012),
  Corollary 1.1.
  Related: Korányi–Reimann 1985 (C^4 conformal maps of the Heisenberg group are CR, as quoted by Tang), 1990 (the
  λ > 0 convention, from the zbMATH review), 1995 (per Wu–Wang: 1-QC maps of the Heisenberg group are CR or
  anti-CR; not read); Capogna–Le Donne, Math. Ann. 379 (2021); Falbel–Veloso, Math. Nachr. 289 (2016)
  (R-circle-preserving diffeomorphisms are CR or anti-CR; zbMATH review only); Chow–Rashevskii.
- **Neutral remark (in the paper only).** The proof of Tang's Theorem 2.3 uses K = (1 + |μ|)/(1 − |μ|) (his
  (2.9)), an implicit orientation convention; complex conjugation on S^3 shows that the convention is needed.
- **Searches (19–20 October 2026).** arXiv API, zbMATH, Crossref, Semantic Scholar citing lists (about 25 works
  citing CCLO, 17 citing Tang 1996; titles screened), OpenAlex citing lists (16 citing Tang 1996; 20 and 8 citing
  the two OpenAlex records of CCLO; screened when the result was first obtained and again by run 2 on 20 October),
  five web searches in all.
  No printed answer to the question was found in these sources, and no paper stating the conclusion for general
  strongly pseudoconvex CR 3-manifolds or naming Tang's question. A search that finds nothing is not a proof of
  novelty.
- **What the note adds.** Only the explicit statement for general (possibly non-embeddable, non-compact) strongly
  pseudoconvex CR 3-manifolds, the identification of the sign with the coorientation, and the explicit anti-CR
  counterexample to the literal question. No priority claim beyond that.
- **Sources not read in full.** Korányi–Reimann 1985/1990/1995 (not read); Wu–Wang (Section 1 and the definition
  of orientation only); Tang's thesis (abstract only); Capogna–Le Donne (arXiv abstract only); Wu–Fu (abstract
  only); Falbel–Veloso (zbMATH review only); the proof of CCLO Theorem 1.1 (cited, not re-proved).

## 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/