# Verification report — OWR-17288-004 (Chirvasitu: loose embeddings of compact metric spaces in a fixed Euclidean space)

Verification date: 2026-10-20.

**Verdict.** The note gives a **complete negative answer to both questions in the record's reading**; every
statement it makes is proved, and the scope is as follows.
- **Question 1 (first bullet), in the reading "loosely embeddable in the same R^N".** For every N ≥ 1 there is a
  countable compact metric space all of whose finite subspaces are loosely embeddable (LE) in R^N but which is not
  LE in R^N (Theorem A for N = 1, Theorem B for N ≥ 2).
- **Not answered.** Chirvasitu's later Question 2.1 of arXiv:2101.01242 asks under the same hypothesis only whether
  X is LE in *some* finite-dimensional Euclidean space. It remains open. The example X_1 of Theorem A is LE in R^2
  (Proposition 5.1), so it is not a counterexample to Question 2.1; for N ≥ 2 the note makes no claim about X_N.
- **Question 2 (second bullet) = Question 2.2 of arXiv:2101.01242.** Every flat torus R^n/Λ, n ≥ 2 (for example
  the square torus R^2/Z^2), with its geodesic distance has, for every N, a finite subset that is not LE in R^N
  (Theorem C). The proof is **non-constructive**: for general N it gives no explicit finite subset and no bound on
  its size. For N ≤ 2 an explicit four-point subset of the hexagonal torus is given (Proposition 6.3); no explicit
  witness is known for the square torus or for N ≥ 3.
- **Also proved.** No flat torus of dimension ≥ 2 is LE in any Euclidean space (Corollary D). Round spheres, RP^n
  and CP^n are LE (Section 7), so they satisfy the hypothesis.
- **Credit.** The definitions and both questions are Chirvasitu's (the OWR abstract; arXiv:2004.09962v1, Questions
  3.16 and 3.17; arXiv:2101.01242, Questions 2.1 and 2.2). The key Lemma 6.2 is the flat-torus analogue of a theorem
  of J. Heitzig (arXiv:math/0203233, 2002: continuous maps R^m → R^n, m ≥ 2, that preserve equality of distances
  are similarities), and its proof follows his method; Theorem C and Corollary D are not stated there.

The note is unrefereed.

## Statement checked
- **Primary source.** A. Chirvasitu, "Quantum isometries", abstract in *Mini-Workshop: Operator Algebraic Quantum
  Groups*, Oberwolfach Report 45/2019, Oberwolfach Rep. 16 (2019), no. 4, pp. 2821–2867 (abstract pp. 2860–2861;
  the questions on p. 2860), doi:10.4171/owr/2019/45 (Crossref: published 2020-11-18; organizers M. Brannan,
  M. Caspers, M. Weber, A. Wysoczańska-Kula). Read in a text extraction of the open-access report and, by
  independent verification run 2, on the rendered PDF page itself.
  - Definition (p. 2860): a loose embedding of (X, d) in R^n is a continuous map φ with
    |φx − φy| = |φx′ − φy′| ⇔ d(x, y) = d(x′, y′) for all x, y, x′, y′.
  - First bullet: a compact metric space all of whose finite subspaces are LE in R^N for the same N; is X itself
    "uniformly embeddable" in R^N? Second bullet: does every compact Riemannian manifold with its geodesic distance
    satisfy "the hypothesis of the preceding question"? Two further bullets (LE in ℓ^2; quantum isometry groups of all
    compact metric spaces) are not part of the record and are not treated.
- **Record.** UnsolvedMath (ulamai), release 1.8.0, commit d371ee9, record OWR-17288-004, title "Finite-Dimensional
  Uniform Embeddings of Compact Metric Spaces", status open. Its statement asks whether X "is itself loosely
  embeddable there" (in the same R^N) and whether every compact Riemannian manifold with its geodesic distance
  "has this property". The original statement keeps the source's "uniformly embeddable".
- **Written versions of the questions.** A. Chirvasitu, arXiv:2004.09962v1 (21 April 2020), Section 3.2:
  Questions 3.16 and 3.17 and Theorem 3.19 (weak LE); the second version (23 December 2020, with the J. Geom. Phys.
  DOI) moved this material to arXiv:2101.01242 (v1), Section 2: Question 2.1 (same hypothesis; conclusion "LE", which
  by its Definition 0.1 means LE in some finite-dimensional Hilbert space) and Question 2.2 (= second bullet).
  Definition 1.1 there: Riemannian metric spaces are connected.

## Readings
- **H(N):** every finite subset of X is LE in R^N.
- **Question 1 (record reading):** does H(N) imply that X is LE in the same R^N? "Uniformly embeddable" is not
  defined in the abstract; the only notion defined there is loose embeddability, and the hypothesis calls N
  "uniformly-chosen". The introduction of arXiv:2101.01242 uses "uniformly loosely embeddable" for finite subspaces
  that are LE in spaces of uniformly bounded dimension. The note reads "uniformly embeddable" as a slip for "loosely
  embeddable", as the record does. A coarse ("uniform embedding" in the large-scale sense) reading would make the
  question trivial for compact spaces; nothing in the source supports it.
- **Question 1 in Chirvasitu's later form (Question 2.1 of arXiv:2101.01242):** H(N) ⇒ LE in some R^M. A negative
  answer to the record's question need not answer it; the note does not answer it.
- **Question 2:** does every compact connected Riemannian manifold satisfy H(N) for some N? The other conceivable
  reading ("H(N) implies LE" for manifolds) would be satisfied vacuously by a manifold failing H(N); the source's
  wording "satisfy the hypothesis of the preceding question" rules it out.

## Results in the paper
- **Lemma 2.1, Lemma 2.2.** Basic facts on loose embeddings (restriction, similarities, the function Φ); in R^N the
  only point equidistant from N + 1 pairwise equidistant points is their centroid, at distance c_N S with
  c_N^2 = N/(2N + 2) < 1/2.
- **Theorem A (N = 1).** X_1 = {p, x_1, x_2, …}, d(p, x_k) = 2^-k, d(x_j, x_{j+1}) = 2^-j,
  d(x_j, x_k) = 2^-j + 2^-(k+1) (k ≥ j + 2). The only equalities between distances of distinct pairs are
  d(p, x_k) = d(x_k, x_{k+1}) (binary digits). Not LE in R: with f(p) = 0 these equalities force f(x_{k+1}) = 2f(x_k),
  contradicting continuity at p. H(1): the doubling map f(x_k) = 2^(k−1) is a loose embedding of every truncation
  Y_K = {p, x_1, …, x_K}, and every finite subset lies in some Y_K. Remark 3.2 gives the explicit map for an arbitrary
  finite subset (runs, base M), valid for every integer M ≥ 2^(m+3), m + 1 the longest run; the program in
  `original/` uses M = 2^(m+4).
- **Theorem B (N ≥ 2).** Gadgets (a regular N-simplex of edge 2^-k with a centre at distance 2^-(k−1)) glued by
  d(a, b) = e(a) + |h(a) − h(b)| + e(b) with heights e(a) = 2^-k(1 + θ_a) such that {1} ∪ {e(a)} is Q-linearly
  independent. Equality classes: edges of G_j together with radii of G_{j+1}; radii of G_1; singletons. Not LE in
  R^N: Lemma 2.2 forces the image edge lengths to grow like c_N^-k. H(N): the truncations Y_K are LE in R^N for
  generic rotations and translations (each unwanted coincidence is the zero set of a real-analytic function that is
  not identically zero on SO(N)^K × (R^N)^(K+1); N ≥ 2 is used for the rotation in case (3)). This half is proved by
  genericity, not by an explicit map.
- **Remark 4.2.** For N = 1 the gadget construction fails (two cross differences always coincide).
- **Proposition 5.1 (scope).** X_1 is LE in R^2 (inductive choice on circles, avoiding finitely many points).
- **Lemma 6.1.** For a compact metric space with the chain property (P1), the isosceles property (P2) and no
  isolated points (P3), H(N) gives a continuous non-constant ψ: M → R^N with |ψx − ψy| = Φ(d(x, y)) (normalisation,
  uniform bound, ultrafilter limit, packing argument).
- **Lemma 6.2.** On a flat torus of dimension ≥ 2 every such continuous ψ is constant (translations give a
  continuous orthogonal representation; the squared chord K(z) is a finite trigonometric polynomial; it is radial
  near 0, hence radial on R^n by real-analyticity; radial plus periodic forces K = 0, Step 6 written out). It is the
  flat-torus analogue of Theorem 1 of Heitzig (arXiv:math/0203233); Step 2 is his Lemma 1, and Steps 3–5 follow the
  route of his proof.
- **Theorem C, Corollary D, Proposition 6.3** as in the verdict.
- **Section 7.** S^n (inclusion), RP^n (x ↦ xx^T), CP^n (x ↦ xx^*) are LE.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program of the note** (`check_note.py`, standard library, exact arithmetic, about 3 seconds; TOTAL failures: 0):
  metric and pattern of X_1; the doubling map on Y_K, K ≤ 40; the map of Remark 3.2 with M = 2^(m+3) and with
  M = 2^(m+4) on all 2047 nonempty subsets of {p, x_1..x_10} and on 2000 random subsets of {p, x_1..x_40} (control:
  M = 2^m fails on 1542 subsets); Lemma 2.2 for N ≤ 12; the metric of p + G_1..G_6 for N = 2, 3, 4 with rational
  heights in three regimes, and the pattern (4.1) with formal heights; the maps of Theorem B exactly in Q(√2, √3)
  with rational Cayley rotations on Y_5 (N = 2, 21 points) and Y_4 (N = 3, 21 points); Remark 4.2; Proposition 5.1
  exactly in Q(i) on {p, x_1..x_14}; Proposition 6.3 exactly (with a control on the square torus); the identities of
  Section 7 on 200 rational pairs each.
- **Original programs** (`original/`): `chain_N1.py` (exact; X_1 on {p, x_1..x_14}; the map of Remark 3.2 with
  M = 2^(m+4) on 2047 subsets and 3000 random subsets) and `gadget_N.py` (mpmath, 60 digits; X_2, X_3 with six
  gadgets; 400 random finite subsets for each N and p + G_1..G_4).
- **Verification run A** (`verification_run_A/`): exact checks of the map of Remark 3.2 on all 8191 nonempty subsets
  of {p, x_1..x_12} and 5000 random subsets of {p, x_1..x_60}; Lemma 2.2 exactly (sympy) for N = 2, 3, 4; X_N
  numerically for N = 2, 3, 4 (two height choices each, 300 random subsets and the full p + G_1..G_5); Proposition 5.1
  at 60 digits (30/30 exact patterns); miscellaneous checks (the Clifford map of the square torus is not a loose
  embedding; the N = 1 gadget pictures repeat a cross difference).
- **Verification run B** (`verification_run_B/`): exact checks of X_1 (8191 + 1500 subsets; brute force: the loose
  maps of {p, x_1, x_2, x_3} into the integers in [−9, 9] with f(p) = 0 are geometric with ratio 2; self-tests); Lemma
  2.2 for N = 2..8; X_N for N = 2, 3 exactly in Q(√2, √3) with rational Cayley rotations on p + G_1..G_4 and on 250 and
  120 random subsets of p + G_1..G_8; the hexagonal-torus witness exactly in Q(√3); the identities of Section 7
  numerically.
- **Re-runs.** On 2026-10-20 all nine programs of `original/` and of the two runs were run again from a copy of the
  package; every output is identical to the recorded one (the timing line with a local interpreter path at the end
  of the outputs of run A was removed from the shipped copies). Both runs
  had also rerun the original programs from copies with identical outputs (`verification_run_B/outputs/claim_rerun_*`).
  Independent verification run 2 ran all ten programs again from an extracted copy of the source archive: every
  output byte-identical to the recorded one.
- **Independent verification run 2** (`independent_run_2/`, five programs, about 15 s; all `TOTAL failures: 0`):
  the map of Remark 3.2 exactly with M = 2^(m+3) and M = 2^(m+2) on all 8191 nonempty subsets of {p, x_1..x_12}
  (control M = 2^m fails on 6560) and on 2000 random subsets of {p, x_1..x_60}; brute force: the integer loose maps
  of {p, x_1..x_4} in [−16, 16] are exactly (g, 2g, 4g, 8g); X_2 and X_3 at 60 digits with the explicit heights
  (metric, pattern (4.1), the maps of Theorem B with random SO(N) rotations on the full Y_6 and Y_5, 5 draws each,
  and on 300 random subsets of Y_8 for each N; smallest relative gap between distinct image lengths 3.1·10^-8);
  Proposition 6.3 exactly in lattice coordinates (with controls); Proposition 5.1 exactly in Q(i) on
  {p, x_1..x_12}; Lemma 2.2 for N = 1..12, Remark 4.2 and the Section 7 identities with sympy.

## Independent verification runs
Two independent verification runs, both AI-assisted, examined the first written version of the results on
2026-10-20. Each re-derived every proof and wrote its own programs. A further independent run (run 2) then examined
the text of the note itself.
- **Run A** re-derived Theorems A and B, Lemma 6.1, Lemma 6.2, Theorem C, Corollary D, Proposition 5.1 and Section 7,
  and compared every statement about the three source texts with these texts. Verdict: confirmed; no mathematical
  error. Required corrections (all made): the scope of the answer to bullet 1 next to the result in the abstract,
  the introduction, the main statement and the conclusion; Step 6 of Lemma 6.2 written out; an unused sentence on
  t_jk removed; the case of the block {p} in the gluing triangle inequality; the statements that Theorem C answers
  Question 2.2 of arXiv:2101.01242 and that the Corollary is contrary, for flat tori, to the expectation stated in the
  abstract of that paper; the bibliographic data checked.
- **Run B** recomputed everything in exact arithmetic, re-derived the proofs, checked the bibliography against
  Crossref, zbMATH and the journal file, repeated the literature search, and found the hexagonal-torus witness.
  Verdict: confirmed; no mathematical error. Required corrections (all made): DOIs and the discrepancy in the data of
  the Palestine J. Math. paper; the arXiv numbering stated; the constant M of Remark 3.2 consistent between text and
  programs; the literature claim softened to "no citing work found that addresses the questions", with the reason
  why the check is incomplete; the explicit witness added; the unproved expectation that X_N is LE in R^(N+1) dropped.
- **Run 2 (of the written note; AI-assisted, independent).** Read the sources again from their PDF files (the OWR
  page; arXiv:2101.01242v1 and its journal version; arXiv:2004.09962 v1 and v2), re-derived every proof of the note
  line by line (in particular the metric axioms and the binary pattern of X_1, the base-M constant, the four gluing
  cases and the Q-independence argument of Theorem B, the three non-vanishing cases, the arc of Proposition 5.1,
  Lemma 6.1 with l_0 = λ_1/8 and the bound 2⌈t/t_0⌉, every step of Lemma 6.2, and the hexagonal witness), wrote its
  own programs and re-ran the package. Verdict: confirmed; no mathematical error. Required corrections (all made):
  the numbers cited from arXiv:2004.09962 are those of its v2 (the note had said v1; in v1 Theorem 3.1 and
  Proposition 3.3 are Theorem 3.13 and Proposition 3.15); the first appearance of both questions in
  arXiv:2004.09962v1 (Questions 3.16, 3.17) credited; Heitzig's preprint arXiv:math/0203233, found by this run,
  cited with its relation to Lemma 6.2 (also in the abstract); the list of journal renumberings of arXiv:2101.01242
  completed (Definitions 0.1, 1.1 and Lemma 1.2 become 1.1, 2.1, 2.2); "asks two questions" corrected to the first two
  of four; this run recorded in the note and here.
- **Changes made in writing the note.** The finite-subset halves of Theorems A and B are proved through the
  truncations Y_K and Lemma 2.1(b): for Y_K the map of Remark 3.2 is the doubling map, and the genericity argument
  of Theorem B is the one checked by the runs, applied to Y_K. The exact program of the note re-checks these cases.
- **A remark on a cited proof (not used).** Theorem 2.4 of arXiv:2101.01242 (under H(N) a compact Riemannian metric
  space is weakly LE in R^N) is not used by the note; Lemma 6.1 proves directly the weaker statement needed, with the
  conclusion "non-constant" instead of "injective". Both runs read the printed proof of Theorem 2.4 (arXiv v1,
  pp. 7–9; Theorem 3.4 of the journal version) and could not verify two steps as written: in part (b) the tip of
  the isosceles triangle depends on the index α of the net, while F_α is assumed to contain it; in part (c) the
  passage from |ψ_α x_α − ψ_α y_α| = 1 with x_α → x, y_α → y to |ψx − ψy| = 1 would need an equicontinuity of the
  maps ψ_α that is not established there. These are observations about the presentation of a proof that the note
  does not rely on; no claim is made about the truth of Theorem 2.4.

## Relation to the literature, novelty and scope
- **Searches (20 October 2026).** arXiv API (author Chirvasitu with embed*, metric, distance, isometr*; the phrases
  "loose embedding", "loose embeddings", "loosely embeddable"; run B listed all 175 arXiv entries of the author and
  read the abstracts of the later geometry-flavoured ones), Crossref, zbMATH Open, Semantic Scholar citation lists,
  OpenAlex, NSF PAR, and a few web searches. No printed answer to either question was found.
  - Semantic Scholar lists only arXiv:2101.01242 as citing arXiv:2004.09962, and no work citing arXiv:2101.01242.
  - OpenAlex counts 2 works citing the J. Geom. Phys. paper and Crossref 1. The OpenAlex list of citing works was
    refused (HTTP 429), and a repeated Semantic Scholar request was refused too; the second OpenAlex citer was not
    identified. No citing work found addresses the questions.
  - Run 2 repeated the searches (arXiv API: the three phrases, the 60 most recent entries of the author up to
    2026-10-11, "equality of distances", "distance patterns" + embed, "flat torus" + chord, "quantum isometry" +
    embed; Crossref for all DOIs; zbMATH Open, including citing-document searches for both papers, with no result;
    Semantic Scholar lists as above; OpenAlex list refused again, HTTP 429, also after one retry; one web search,
    which found only Chirvasitu's papers). It found J. Heitzig, *A characterization of similarity maps between
    Euclidean spaces related to the Beckman–Quarles theorem*, arXiv:math/0203233 (2002; zbMATH lists it as a
    preprint), and read it in full: its Lemma 1 (for a left-invariant distance function on a monoid, a
    distance-equality-preserving map into R^n is an orbit of a homomorphism into the motions) is Step 2 of Lemma
    6.2, and its Theorem 1 is the Euclidean analogue of Lemma 6.2, proved by the same route. It does not apply
    directly to tori (the torus distance is a function of |x − y| only locally) and does not state Theorem C or
    Corollary D. The note now cites it.
  - Related area, not read (bibliographic data and snippets only): helical immersions (Sakamoto 1982, 1986; Hong;
    Pak–Sakamoto), which concern smooth isometric immersions whose chord length depends only on arc length. The
    mechanism of Lemma 6.2 is classical in spirit (von Neumann–Schoenberg screw functions). Corollary D may be known
    or folklore, although it was found nowhere.
- **What was read.** The OWR abstract (also on the PDF page); arXiv:2101.01242 v1 completely (and, by run B and run
  2, the journal version in the NSF PAR copy, for the numbering and the questions); of arXiv:2004.09962 v2 the
  introduction, Sections 2 and 3 and the statements cited (Theorem 0.1, Definition 2.2, Example 2.3, Remark 2.4,
  Theorem 3.1, Proposition 3.3), and of v1 the statements cited and Section 3.2 (by run 2); Heitzig's preprint
  arXiv:math/0203233 completely (by run 2). Goswami (2015), Deza–Maehara (1990),
  Besicovitch (1940) and von Neumann–Schoenberg (1941) were not read; what is said about the first two is taken from
  Chirvasitu's papers.
- **Bibliographic data.** Checked with Crossref on 2026-10-20: doi:10.4171/owr/2019/45 (Oberwolfach Rep. 16(4),
  2821–2867); doi:10.1016/j.geomphys.2020.104089 (J. Geom. Phys. 161 (2021), Article 104089; zbMATH 1456.30098);
  doi:10.1016/j.aim.2015.03.024 (Adv. Math. 280 (2015) 340–359); doi:10.1090/S0002-9947-1990-0974513-6 (Trans.
  AMS 317(2) (1990) 661–671); doi:10.1112/jlms/s1-15.1.3 (J. London Math. Soc. s1-15 (1940) 3–6);
  doi:10.1090/S0002-9947-1941-0004644-8 (Trans. AMS 50 (1941) 226–251); doi:10.1007/BF01456411 and
  doi:10.1307/mmj/1029003415 (Sakamoto). Palestine J. Math. 11 (2022): no DOI found; zbMATH 1516.30076 gives no. 4,
  pp. 233–242, while the typeset journal file in the NSF PAR copy is headed Vol. 11(1)(2022), 260–268; the journal
  version renumbers (arXiv Definitions 0.1, 1.1, Lemma 1.2, Proposition 1.3, Theorem 1.6, Questions 2.1/2.2,
  Definition 2.3, Theorem 2.4 are journal 1.1, 2.1, 2.2, 2.3, 2.6, 3.1/3.2, 3.3, 3.4). The note cites arXiv
  numbering throughout: v2 of arXiv:2004.09962 (the version with the journal DOI) and v1 of arXiv:2101.01242.
  Heitzig, arXiv:math/0203233: preprint, no journal version found (zbMATH Open lists it as a preprint).
- **Caveats.** A search that finds nothing is not a proof of novelty; no priority is claimed beyond "no prior answer
  was found". Bullet 1 is answered only in the reading "the same R^N"; Question 2.1 of arXiv:2101.01242 is not
  answered. Theorem C is non-constructive.

## 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/