# Verification report — AMR-066-0018 (Gromov's conjecture [?18] on manifolds isometric at infinity to flat manifolds)

Verification date: 2026-10-03. Paper: "Nonflat Metrics with Nonnegative Scalar Curvature That Are Flat at Infinity:
A Counterexample to a Conjecture of Gromov" (12 pages).

**Verdict.** The conjecture is false in every dimension n ≥ 3. On R² × S^{n−2} there is an explicit complete metric
g = dr² + f(r)² dτ² + h(r)² g_{S^{n−2}} with Sc ≥ 0, with Sc > 0 on an open shell, which outside a compact set is
isometric to the complement of a compact set in the flat manifold S¹ × R^{n−1} = R^n/Z. For n ≥ 4 the manifold is
simply connected and spin. Quotients of products with Euclidean spaces give the same for all flat ends T^k × R^N
with N ≥ 2, and a cut-and-paste gives examples that are isometric at infinity to T^{n−1} × R for a class of flat
tori which includes the standard ones. The phenomenon is not new. It is the Kaluza–Klein bubble mechanism
(Witten 1982). The hyperbolic analogue of the conjecture was refuted by Hao, Hu, Liu and Shi (SIGMA 2023) by a
gluing construction of the same kind. For n = 4 a counterexample also follows by combining published results
(Chen–Liu–Shi–Zhu, or Dai–Sun, with the Euclidean Reissner–Nordström metric of negative mass). What is new is the
explicit elementary construction for all n ≥ 3, including n = 3, the simply connected spin examples for n ≥ 4, and
the explicit statement that [?18] is false. The note is unrefereed. Two independent verification runs, both
AI-assisted, found no mathematical error.

## Statement checked
- **Primary source.** M. Gromov, *101 Questions, Problems and Conjectures around Scalar Curvature* (incomplete and
  unedited version of October 1, 2017), IHES,
  https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf
  (1,430,132 bytes; sha256 1478b4025eb55e7237ab3f066e8924938a1de4cd407d74893b49f2b9007d9645; three downloads, by the
  finder and by the two verification runs, gave identical copies). Section 8, paragraph "Semiperiodic Metrics",
  printed p. 24. In paraphrase:
  - Proposition: if X is complete with Sc(X) ≥ 0 and isometric at infinity to a complete flat manifold X_fl, and
    there is a homomorphism π₁(X) → π₁(X_fl) compatible with the isometry at infinity, then X is flat.
  - Footnote 35: the homomorphism may be taken to be induced by a continuous map X → X_fl which is isometric at
    infinity. Footnote 36: the proof needs X to be spin, or minimal hypersurfaces in dimensions n ≤ 8. The proof
    reduces the statement to the non-existence of metrics with Sc > 0 on closed manifolds that admit maps of
    non-zero degree to tori (p. 22 of the list).
  - Item [?18]: Gromov writes that the homomorphism may in general be essential, but conjectures that if
    Γ = π₁(X_fl) acts on R^n by parallel translations, then every X with Sc(X) ≥ 0 that is isometric to X_fl at
    infinity is flat. He adds, in parentheses, that these X can be simply connected.
  - Item [?19] asks for the assumptions on π₁(X) needed for each X_fl. Item [?20] asks to relax the isometry at
    infinity to asymptotic flatness with a negativity condition on an energy at infinity.
- **Meaning of "isometric at infinity".** In *Four Lectures on Scalar Curvature* (arXiv:1908.10612v6, statement of
  Min-Oo's hyperbolic rigidity theorem) and in arXiv:1811.04311 Gromov glosses the phrase as meaning outside a
  compact subset. The note uses this meaning: complements of compact sets are isometric.
- **Later texts.** The IHES page lists only the version of October 1, 2017. Gromov's later texts on scalar
  curvature that were examined (*A dozen problems*, *Metric inequalities with scalar curvature*, *Scalar
  curvature of manifolds with boundaries*, *Mean curvature in the light of scalar curvature*, *Four Lectures*)
  do not restate, correct or withdraw [?18].
- **Corpus record.** ulamai/UnsolvedMath, AMR-066-0018 (upstream status `partially_solved`). The statement asks
  whether a complete X with Sc(X) ≥ 0 that is isometric at infinity to X_fl = R^n/Γ, with Γ acting by parallel
  translations, must be flat. It is a faithful restatement of [?18].

## Readings
| Reading | Answer | Where |
|---|---|---|
| [?18] as stated, Γ ≅ Z^k a translation group with 1 ≤ k ≤ n − 2, n ≥ 3 | false | Theorem 1.1 (k = 1), Proposition 5.1, Corollary 1.2 |
| restricted to simply connected X (Gromov's remark), k = 1, n ≥ 4 | false | Theorem 1.1(d): X = R² × S^{n−2} |
| restricted to spin X | false | X is parallelizable |
| k = n − 1 (X_fl = T^{n−1} × R, two ends), lattices with a basis w_1, …, w_{n−1} such that the distance from w_{n−1} to the span of the others is at least \|w_1\| (for example Z^{n−1}) | false | Proposition 5.2, Corollary 1.2 |
| k = n − 1, arbitrary lattices (the condition above fails, for example, for the hexagonal lattice when n = 3) | false, at remark level only | Remark 5.3, relying on the connected sum theorem of Gromov–Lawson and Schoen–Yau |
| simply connected X for k ≥ 2, n ≥ 4 | false, at remark level only | Remark 5.3, relying on the surgery theorem of Gromov–Lawson and Schoen–Yau |
| k = 0 (X_fl = R^n) | true | rigidity case of the positive mass theorem (Remark 5.4): spin X by Witten's argument; n ≤ 7 by Schoen–Yau; all dimensions by Schoen–Yau, Surv. Differ. Geom. 24. Covered by Gromov's proposition |
| n ≤ 2 | true | Remark 5.4 (Gauss–Bonnet) |
| k = n (X_fl compact) | the hypothesis is empty | not used in the note |
| with a compatible homomorphism (Gromov's proposition) | not affected | Theorem 1.1(c): no compatible homomorphism exists for the examples |
| n = 3, simply connected X | no such X exists at all, for k = 1, 2 | Remark 5.5; the example for n = 3 has π₁ = Z |

## Results in the paper
- **Lemma 2.1.** For g = dr² + f² dτ² + h² g_{S^m} the curvature operator is diagonal in the frame
  ∂_r, f^{−1}∂_τ, h^{−1}ē_i, with sectional curvatures −f″/f, −h″/h, −f′h′/(fh), (1−h′²)/h². Hence formula (3) for
  Sc, its form (4), and the key identity (5): if h′ = φ and h^m f′ = 1 − φ, then
  (f h^m/2)·Sc = (1 − m f h^{m−1}) φ′ + m(m−1)/2 · f h^{m−2}(1 − φ²). The proof uses Cartan's structure equations
  and is written out.
- **Lemmas 3.1 and 3.2.** The step function χ(t) = ψ(t)/(ψ(t)+ψ(1−t)), ψ(t) = e^{−1/t}, and the profiles
  φ(r) = χ((r−a)/(b−a)), h = 1 + ∫φ, f = ∫(1−φ)h^{−m}, with a = 1/(4m), b = 1/(2m). Then f = r and h = 1 on [0, a];
  f = F and h = r − c on [b, ∞); f ≤ r; and m f h^{m−1} ≤ (1/2)(1+1/(8m))^{m−1} < (1/2)e^{1/8} < 0.567 on [0, b].
- **Theorem 1.1.** On X = R² × S^m, n = m + 2 ≥ 3:
  - the metric is smooth (it is the product of a flat disc and the unit sphere for r < a) and complete;
  - Sc ≥ 0, and Sc > 0 for a < r < b, by (5);
  - Φ(r, τ, ω) = (Fτ, (r−c)ω) is an isometry of {r > b} onto S¹(2πF) × (R^{n−1} ∖ closed ball of radius 1+1/(8m));
  - the circle at infinity bounds a disc in X, and its image generates π₁(X_fl), so no compatible homomorphism
    exists;
  - X is parallelizable, and simply connected for n ≥ 4.
- **Proposition 5.1.** For every lattice Λ ⊂ R^k and N ≥ 2, the quotient of X × R^{k−1} by a group Z^{k−1} of
  isometries (rotations of X combined with translations) is complete, has Sc ≥ 0, is not flat, and is isometric
  at infinity to T_Λ × R^N.
- **Proposition 5.2.** For N = 2 the example of Proposition 5.1 is flat outside T_Λ × (disc of radius ρ₁), and
  ρ₁ < |w_1|/2 (because F ≥ 13/36 for m = 1). Cutting it along two parallel flat hypersurfaces {y₁ = ±A},
  A ≥ |w_1|/2, and identifying them by a translation gives a complete non-flat manifold with Sc ≥ 0 which is
  isometric at infinity to T^{n−1} × R. The proof gives the collars on which the gluing translation is an
  isometry, and the completeness of the result. This is the flat version of the passage from Z^{n−2} to Z^{n−1} in
  Hao–Hu–Liu–Shi, Section 2.4.1.
- **Corollary 1.2.** [?18] fails for every translation group of rank 1 ≤ k ≤ n − 2 and for the groups of rank
  n − 1 covered by Proposition 5.2.
- **Remarks.** Remark 5.3 (arbitrary tori for k = n − 1 by a connected sum; simply connected examples for k ≥ 2 by
  surgery) rests on a cited theorem and is labelled as such. Remark 5.4: the conjecture holds for n ≤ 2 and for
  k = 0. Remark 5.5: for n = 3 no simply connected example exists. Remark 5.6: the examples have mass zero and
  violate the hypotheses of the known positive mass theorems for such ends; the three-dimensional example shows
  that the π₁ hypothesis in the conical theorem of Chen–Liu–Shi–Zhu (Theorem 1.10 of the arXiv version) and of
  Liu–Shi–Zhu (Theorem 1.2) is necessary.

## Computations (scripts and outputs in reproducibility/)
The proofs do not depend on these computations.
- **Lead** (`lead/lead_checks.py`, numpy and sympy, about 5 seconds; 32 reported checks, all pass).
  - Exact rational arithmetic: e < 1.134⁸; (1/2)(1+1/(8m))^{m−1} < 0.567 for m ≤ 3000; the constants 13/36 and
    81/(52π) < 1/2 of Proposition 5.2.
  - Nested Gauss–Legendre quadrature for m = 1..12, 25, 100: Lemma 3.2; Sc from (3) is positive at the 399 interior
    grid points of (a, b) and agrees with (5) to relative error below 10⁻¹⁵; F = 0.374182… (m = 1) and
    0.187088… (m = 2).
  - 40 random admissible parameters (m, a, b): Sc > 0.
  - Negative controls: three parameter choices violating the condition of Remark 3.3 give Sc < 0 somewhere
    (minimum −1.67, −6.30, −10.1).
  - Φ pulls the flat metric back to g, symbolically for m = 1, 2, 3; the lattice decomposition of Proposition 5.1
    on 200 random rational lattices; for m = 1, F ≥ 13/36 and ρ₁/|w_1| = 0.4785 < 1/2.
- **Finder** (`finder/verify_sc.py`, `finder/profile_numeric.py`).
  - (3) agrees with a direct computation from the Christoffel symbols for m = 1..5; (4) and (5) hold for symbolic
    m; the Riemann tensor of the end metric vanishes for m = 1, 2, 3; negative controls.
  - Double-precision integration (DOP853) for m = 1, 2, 3, 4, 6, 10, 20 on a 4001-point grid: Lemma 3.2 and
    Sc ≥ 0. The maximum of m f h^{m−1} is between 0.37 and 0.43.
- **First independent verification run** (`independent_run/`, AI-assisted, written without reusing the finder's
  code).
  - A general, non-diagonal Levi-Civita and Riemann routine with S^m in a graph chart: (3) symbolically for
    m = 1, 2, 3 and at a rational point for m = 4; (5) for symbolic m and for m = 1..8; flatness of the end for
    m = 1, 2, 3; negative controls.
  - mpmath with 25 digits, by quadrature, for m = 1, 2, 3, 4, 6, 10, 50: Sc from (3) is positive at all 399
    interior grid points; h(b) = 1 + (b−a)/2 to 18 digits; the same values of F.
  - Re-runs of the finder's two scripts reproduced the recorded outputs.
- **Second independent verification run** (`independent_run_2/`, AI-assisted, written from the text of the note
  without reusing any earlier code; 104 reported checks, all pass).
  - `run2_curvature.py` (41 checks): a curvature routine for arbitrary metric matrices with S^m in stereographic
    coordinates, m = 1, 2, 3. It confirms the sectional curvatures (2) and the vanishing of all other curvature
    components, formula (3), and the key identity (5) with the scalar curvature computed from the Christoffel
    symbols; the same scalar curvature in a non-diagonal chart (m = 1, 2); the core and end pieces, with a
    negative control; the pull-back under Φ; (4) and (5) for a symbolic exponent m; the mean curvature
    f′/f + m h′/h of the level sets and the singular picture described in the introduction; and that the Euclidean
    Reissner–Nordström metric is scalar-flat and closes up smoothly also for negative mass parameter.
  - `run2_profile.py` (33 checks): the inequality chain of Lemma 3.2(iv) in exact rational arithmetic. Rigorous
    enclosures by interval arithmetic and monotone Riemann sums: for m = 1, 2, 3, 4, 6, 10, 25, 100 the supremum of
    m f h^{m−1} over [a, b] is at most 0.424, and F ∈ [0.3741759, 0.3741888] for m = 1. A 50-digit quadrature for
    eight values of m ≤ 40: Sc from (3) is positive at 249 interior points and agrees with (5). Five further
    admissible parameter choices, and three negative controls.
  - `run2_torus_twoends.py` (30 checks): Proposition 5.1 on 240 random rational lattices in exact arithmetic; the
    constants of Proposition 5.2, with ρ₁/|w_1| ∈ [0.47850, 0.47852] from the rigorous enclosure of F; the collars,
    the gluing translation and the connectedness in the cut-and-paste; the identity 2A = dist(w_{n−1}, H) on 160
    random lattices; the lattice condition holds for Z^{n−1} and fails for the hexagonal lattice.
  - All five earlier scripts were re-run from an extracted copy of the source archive and reproduced the recorded
    outputs, up to the printed run times (`independent_run_2/rerun_packaged_summary.txt`).
- All outputs were regenerated when the package was assembled. They agree with the recorded ones, byte for byte
  for the two profile scripts and up to the printed run times for the two symbolic scripts. In the script
  `independent_run/indep_curv_b.py` the print label of the tested formula was later changed to "formula (3)", and
  its output was regenerated once more; no computation was changed.

## Independent verification runs
Both runs were AI-assisted verification runs. They are not independent human peer review.

### First run
The first run (2026-10-03) re-derived the curvature formulas and every inequality, re-downloaded the source, read
the relevant parts of the cited papers in their arXiv sources, and wrote the separate programs listed above. It
examined the construction of Theorem 1.1, the torus quotients of Proposition 5.1 and the remarks, in the form of
the working write-up that preceded the note.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (the counterexample answers [?18] as posed and as intended; Gromov's remark on simply connected X is covered) |
| Proofs | CONFIRMED (every step re-derived; no mathematical error) |
| Computations | CONFIRMED (own symbolic and 25-digit checks pass; recorded outputs reproduced) |
| Answer as posed | CONFIRMED (false for every n ≥ 3) |
| Novelty | CONFIRMED_WITH_FIXES (the mechanism and the case n = 4 must be credited to earlier work) |
| Presentation | CONFIRMED_WITH_FIXES |

Required fixes, all applied:
1. **Novelty framing.** The abstract and the introduction say that for n = 4 the existence also follows by
   combining the Lohkamp-type compactification step of Chen–Liu–Shi–Zhu (Proposition 4.11 and the first step of
   the proof of Theorem 1.8 in arXiv:2112.14442v1), which does not use incompressibility, with the Euclidean
   Reissner–Nordström metric of negative mass (Brill–Horowitz 1991). The Kaluza–Klein bubble mechanism is
   credited to Witten (1982). Novelty is claimed only for the explicit elementary construction for all n ≥ 3
   including n = 3, the simply connected spin examples, and the explicit statement that [?18] is false.
2. **Citations added.** Alaee–Khuri–Kunduri (arXiv:2605.12352); Dahl–Kröncke (Math. Ann. 388 (2024)), with the
   remark that the change of topology is essential; the conical theorem of Chen–Liu–Shi–Zhu, with the remark that
   the three-dimensional example shows that its π₁ hypothesis is necessary (Remark 5.6).
3. **Further novelty checks.** Repeated when the note was written and again in the second run; see the last
   section for what was found and for what could not be done.
4. **Remarks.** The connected sum for arbitrary tori with k = n − 1 and the surgery statement are in Remark 5.3,
   which is labelled as resting on a cited theorem. Corollary 1.2 does not use it.
5. **Optional items.** The README notes that Taylor-type ODE solvers mishandle the flat step function. The
   corner-model remark of the working write-up was first replaced by the comparison with the gluing construction
   of Hao–Hu–Liu–Shi; after the second run the note has both.

Written after the first run and not examined by it: Proposition 5.2 and the corresponding part of Corollary 1.2;
the comparison with Hao–Hu–Liu–Shi and with Dai–Sun; Remark 5.5 in its present form. The second run examined
all of these.

### Second run
The second run (2026-10-03) worked from the note as packaged after the first run. It fetched Gromov's file again
and read p. 24 on the rendered page. It read the arXiv sources of Hao–Hu–Liu–Shi, Dai–Sun, Chen–Liu–Shi–Zhu,
Liu–Shi–Zhu, Khuri–Wang, Alaee–Khuri–Kunduri, Dahl–Kröncke, Horowitz–Lu and Minerbe, and the relevant parts of two
later texts of Gromov. It checked every proof line by line, wrote the programs in `independent_run_2/`, re-ran all
packaged programs, and read every page of the PDF.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED |
| Lemma 2.1, Lemmas 3.1–3.2, Theorem 1.1 | CONFIRMED (re-derived; own symbolic checks; rigorous enclosures of the profile) |
| Proposition 5.1, Corollary 1.2 | CONFIRMED |
| Proposition 5.2 | CONFIRMED completely: smoothness of the glued metric, Sc ≥ 0, completeness, flat ends, non-flatness, the lattice condition. It stays a proposition |
| Remarks 5.3–5.6 | CONFIRMED at the level claimed |
| Cited results | CONFIRMED against the arXiv sources; three statements made more precise |
| Hao–Hu–Liu–Shi and Dai–Sun | Neither contains, states or directly implies a nonflat metric with Sc ≥ 0 that is flat at infinity, and neither mentions [?18]; the note credits both accurately |
| Novelty | CONFIRMED_WITH_FIXES (one more credit: the singular picture and the remark of Liu–Shi–Zhu on fill-ins) |
| Computations | CONFIRMED |
| Presentation | CONFIRMED_WITH_FIXES |

Required fixes, all applied:
1. **Verification paragraph.** Both independent checks are described as AI-assisted verification runs; the second
   run is recorded.
2. **Title.** "Flat Cylindrical Ends" was replaced by "That Are Flat at Infinity". The ends of S¹ × R^{n−1} are
   not cylindrical ends in the usual sense; only the two-ended examples have cylindrical ends.
3. **Two-ended examples.** The abstract says that they exist for a class of flat tori which includes the standard
   ones; the introduction says that the condition fails for the hexagonal lattice when n = 3.
4. **The case k = 0.** Remark 5.4 says for which X the rigidity is known (spin; n ≤ 7; all dimensions by
   Schoen–Yau, Surv. Differ. Geom. 24 (2019), 441–480).
5. **Gromov's proof.** Described as a reduction to closed manifolds that admit maps of non-zero degree to tori.
6. **Credit.** A paragraph on the singular picture was added: the flat product of a disc of radius l and a sphere
   of radius R, glued to the flat exterior, has mean curvature 1/l from inside and m/R from outside; the
   condition m f h^{m−1} < 1 is the smooth form of the inequality ml < R; Liu–Shi–Zhu (Remark 1.8 of the arXiv
   version) note that fill-ins Σ₀ × D²(l) have negative Brown–York mass for small l.
7. **Known positive mass theorems.** The note says that their rigidity statements are proved under decay
   conditions, which hold trivially for exactly flat ends.
8. **Proof of Proposition 5.2.** The collars of the two cuts and the completeness argument were written out.
9. **Sources and searches.** The scope paragraph records that Minerbe's paper was read in its arXiv version, that
   the published version of Chen–Liu–Shi–Zhu could not be accessed, and what was searched.
10. **Bibliography.** arXiv:0803.2873 added to Minerbe's paper; the Schoen–Yau reference of item 4 added. All
    DOIs were checked against Crossref.
11. **Package.** `reproducibility/independent_run_2/` added; the label of the tested formula in the first run's
    symbolic script changed to "formula (3)"; this report updated.

No theorem and no step of a proof was changed by these fixes.

## Relation to the literature, novelty and scope
- **Earlier work credited in the note.**
  - Witten (1982): the Kaluza–Klein bubble; its time-symmetric slice is a zero-energy, scalar-flat, non-flat
    metric on R² × S² that is asymptotic, but not equal, to the flat metric at infinity.
  - Brill–Pfister (1989), Brill–Horowitz (1991), Horowitz–Lu (2025): negative energy; the Euclidean
    Reissner–Nordström metrics.
  - Positive mass theorems for ends S¹ × R^{n−1} under extra hypotheses: Minerbe (2009), Liu–Shi–Zhu (arXiv 2021),
    Chen–Liu–Shi–Zhu (Math. Ann. 2025), Khuri–Wang (arXiv 2025); the comparison theorem of
    Alaee–Khuri–Kunduri (arXiv 2026).
  - Lohkamp (1999) and its adaptations to these ends by Chen–Liu–Shi–Zhu and by Dai–Sun (J. Geom. Anal. 2023).
  - Hao–Hu–Liu–Shi (SIGMA 19 (2023), 083): the hyperbolic counterpart.
  - Liu–Shi–Zhu, Remark 1.8: fill-ins Σ₀ × D²(l) with negative Brown–York mass (the singular picture).
  - Dahl–Kröncke (Math. Ann. 2024): R^{n−1} × S¹ is scalar curvature rigid under compactly supported
    deformations on the same manifold.
- **The closest earlier work.**
  - Hao, Hu, Liu and Shi, "Rigidity and non-rigidity of H^n/Z^{n−2} with scalar curvature bounded from below"
    (arXiv:2303.15752; SIGMA 19 (2023), 083). Gromov's generalised Min-Oo rigidity statement for parabolic
    quotients H^n/Γ is the hyperbolic counterpart of [?18]. The paper shows that it fails for H^n/Z^{n−2}
    (Theorem 1.2) and for H^n/Z^{n−1} (Section 2.4.1). The construction removes D² × T^{n−2}, glues in
    S¹ × (spherical cap) × T^{n−3} along a boundary with a strict mean curvature inequality, and smooths the corner
    with a theorem of Brendle–Marques–Neves. So a circle of the torus at infinity is filled by a disc, as in the
    note. The paper proves rigidity when the torus stays incompressible (Theorem 1.4). All its curvature bounds
    are Sc ≥ −n(n−1) and all its ends are hyperbolic; it does not mention flat ends, [?18] or Gromov's list. Its
    method carries over to flat space with new work, and would give ends R² × T^{n−2}. The note says that
    Theorem 1.1 is a flat counterpart of this construction, and that Proposition 5.2 uses its passage from Z^{n−2}
    to Z^{n−1}.
  - Dai and Sun (arXiv:2211.14713; J. Geom. Anal. 33 (2023), Paper No. 275), Section 4, contains the same
    compactification step as Chen–Liu–Shi–Zhu, for ends R^k × X with k ≥ 3: from negative mass and Sc ≥ 0 to a
    metric with Sc ≥ 0 that is the product metric near infinity, without topological hypotheses. The paper gives
    no example of negative mass and does not mention Gromov's list. With the Reissner–Nordström metric of negative
    mass it gives a counterexample for n = 4 only. The note cites it next to Chen–Liu–Shi–Zhu.
  - Chen–Liu–Shi–Zhu (arXiv:2112.14442v1): the introduction recalls the Reissner–Nordström metric of negative
    mass on R² × S², and the first step of the proof of Theorem 1.8 produces, from negative mass, a complete
    metric with Sc ≥ 0 that coincides with the flat product metric near infinity. The two are not combined in the
    paper, and Gromov's list is not cited. This is the derivation for n = 4 stated in the note.
- **Searches (October 2026; all requests anonymous and logged).**
  - arXiv API: "isometric at infinity"; "flat at infinity"; semiperiodic; Gromov with flat and infinity; positive
    mass for ALF, fibred or Kaluza–Klein ends; Kaluza–Klein bubbles with scalar curvature; "flat outside a
    compact"; fill-ins of tori; counterexamples for circle ends; non-rigidity with scalar curvature; compactly
    supported deformations with scalar curvature; rigidity with flat, cylindrical or periodic ends; papers of
    J. Zhu, of Dai, of Kröncke, of Shi.
  - Crossref (bibliographic data and DOIs of all references), zbMATH Open, INSPIRE full-text search
    ("isometric at infinity"; "flat at infinity"; "flat outside a compact"; "101 questions"; semiperiodic or
    Kaluza–Klein with scalar curvature).
  - Semantic Scholar: the papers citing Chen–Liu–Shi–Zhu (20), Liu–Shi–Zhu (6), Dai–Sun (3) and
    Hao–Hu–Liu–Shi (none listed), with the citation contexts. OpenCitations: the works citing
    Hao–Hu–Liu–Shi (1), the published version of Chen–Liu–Shi–Zhu (2) and Dai–Sun (1).
  - Four general web searches in all (one by the finder, one by the first run, one when the note was written, one
    by the second run). They returned the source itself, *Four Lectures*, physics papers on bubbles, and
    unrelated papers.
  - Nothing found states that [?18] is false, or discusses it, and nothing found contains an explicit nonflat
    metric with Sc ≥ 0 that is flat at infinity with a flat end of this kind.
- **Checks that could not be completed.**
  - The published Math. Ann. version of Chen–Liu–Shi–Zhu could not be read: the Springer pages returned a
    bot-protection page or a redirect to an identity provider, in both verification runs and when the note was
    written. What was compared instead: the Crossref and zbMATH records (Math. Ann. 393 (2025), no. 1, 1241–1320)
    and the reference list deposited there. It has 75 items, against 67 in arXiv v1. The additional items include
    Dai–Sun (2023) and Witten, "Positive energy and Kaluza–Klein theory" (1984). Gromov's list is not among the
    75 items. So the published introduction may differ from arXiv v1, and its theorem numbers were not checked.
    The note cites theorem numbers of the arXiv version and says so. Alaee–Khuri–Kunduri (2026) also cite
    "Theorem 1.8" of the arXiv version.
  - A systematic "cited by" search for Gromov's list could not be made. Semantic Scholar has no record for the
    list (only for *A dozen problems*); its search endpoints returned 429 on eight attempts in all. OpenAlex
    returned 429 on every attempt, in both runs and when the note was written (shared anonymous daily budget
    exhausted), so OpenAlex was not searched at all. zbMATH has no record of the list. Partial substitutes:
    - INSPIRE full-text search for "101 questions" gave six hits, one of them relevant: Lesourd–Unger–Yau
      (arXiv:2009.12618), which cites the list for a different question.
    - A web search led to Chodosh's survey (arXiv:2510.04481), which cites the list for the aspherical
      conjecture and for Conjecture 34 only, and to Weinberger–Xie–Yu (arXiv:2112.13897), on a different
      question of Gromov.
    - The arXiv sources of Liu–Shi–Zhu, Chen–Liu–Shi–Zhu, Dai–Sun, Hao–Hu–Liu–Shi, Dahl–Kröncke, Khuri–Wang,
      Alaee–Khuri–Kunduri, Horowitz–Lu and Minerbe do not cite the list. (Khuri–Wang have it in their
      bibliography database, but not in the reference list of the paper.)
  - MathSciNet and Google Scholar were not searched.
- **Sources not read in the original.** Witten (1982), Brill–Pfister, Brill–Horowitz and Lohkamp (1999) are
  described from their abstracts and from the accounts in the papers that were read. Minerbe (2009) was read in
  its arXiv version (arXiv:0803.2873) in the second run. The surgery and connected sum theorems of Gromov–Lawson
  and Schoen–Yau are used, in Remark 5.3 only, in their local form, in which the metric is changed only near the
  surgery sphere.
- **Scope.** The note refutes [?18] for every n ≥ 3 and every translation group of rank 1 ≤ k ≤ n − 2, and for
  the lattices of rank n − 1 in Proposition 5.2, with complete written proofs. Arbitrary lattices of rank n − 1
  are at remark level. The conjecture holds for n ≤ 2; for k = 0 it is the rigidity case of the positive mass
  theorem. Gromov's proposition with a compatible homomorphism is not affected. In view of the paper of
  Hao–Hu–Liu–Shi, experts may regard the flat counterexample as a routine variation or as folklore. This negative
  search is not a proof of priority.
- **Suggested corpus status.** Answered in the negative for n ≥ 3 (solved), in place of "partially solved".

## 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/