# Verification report — OWR-1195-004 (kernels of signature operators twisted by almost flat bundles)

Verification date: 2026-09-30.

**Verdict.** The answer is no. On the flat square torus T^2, with f(x,y) = (x + (c/2π) sin 2πy, y) (a
diffeomorphism isotopic to the identity) and E_ε the trivial Hermitian line bundle with the connection
d + iε cos(2πy) dx (curvature norm 2π|ε|), the (ungraded) twisted signature operator D = d_∇ + d_∇\* has a
4-dimensional kernel for E_ε for every ε, and a zero kernel for f\*E_ε whenever εc ∉ 4πℤ; for its chiral half
D^+ the dimensions are 2 and 0. So no δ > 0 as in Schick's question exists. The same holds on T^4 (16 against 0
for D, 8 against 0 for D^+), in every dimension ≥ 2, for the underlying Euclidean rank-2 bundle (8 against 0 for
D, 4 against 0 for D^+), for the spin Dirac operator (2 against 0), and for the natural readings of the twisted
Betti numbers of Sauer and Schick. The note is unrefereed.

## Statement checked
- **Primary source.** Oberwolfach Report 13/2006, *Analysis and Topology in Interaction*, Oberwolfach Rep. 3 (2006),
  no. 1, 729–804, doi:10.4171/OWR/2006/13. Problem session, item of T. Schick, p. 798.
  - The published report was read (the page was also rendered and checked by eye).
  - Schick first asks: "Is the kernel a homotopy invariant as well?"
  - The precise version: (M,g) and (M',g') are fixed, and so is a homotopy equivalence f: M' → M. Is there δ > 0,
    depending on these data, such that the kernels of D_E and D'_{f\*E} are isomorphic for every bundle (E,∇) over M
    whose curvature is bounded by δ in sup-norm? Here D_E is the signature operator twisted by (E,∇).
  - The notes record that Goette and Braverman expected invariance to be unlikely.
- **Restatement.** R. Sauer and T. Schick, *Homotopy invariance of almost flat Betti numbers*, Enseign. Math. (2) 54
  (2008) 162–164 (Guido's book of conjectures, Problem 57), doi:10.5169/seals-109926 (a DataCite DOI; Crossref has no
  record of it).
  - Read in the e-periodica PDF (generated 2022-01-02), in a copy retrieved on 2026-09-30 through a web archive by
    the first verification run. On the same day e-periodica was unreachable from our network, and the Wayback
    Machine answered "temporarily offline" (HTTP 503) to the lead's own attempt, so the lead read that retrieved
    copy. The second verification run retrieved the same Wayback snapshot; it is byte-identical to this copy.
  - Theorem 57.1 is the Hilsum–Skandalis theorem (index invariance for Euclidean bundles with small curvature, where
    the norm is the supremum of the operator norm over the unit sphere bundle of Λ^2 TM). The constant depends only
    on the two manifolds and f, not on the bundle.
  - Question 57.2 regards the kernels as graded by form degree, calls the dimensions twisted Betti numbers b_k, and
    asks whether b_k(E) = b_k(f\*E) for sufficiently small curvature.
  - The note adds three remarks: the twisted Euler characteristic is an index and hence invariant; the flat case
    holds by de Rham's theorem; analysts tend to doubt the general statement. The paper points out that the first
    remark holds for flat bundles and for the parity numbers b_ev − b_odd, but not for degree-wise numbers of
    non-flat bundles (Prop. 4.4: −2 against 0).
- **Corpus record.** ulamai/UnsolvedMath, OWR-1195-004, status `open`.
  - The statement asks, for homotopy-equivalent closed Riemannian manifolds f: M' → M, for δ > 0 such that every
    bundle (E,∇) on M with curvature norm at most δ has ker D_E ≅ ker D_{f\*E}. It never says that D_E is the
    signature operator, and the record's title speaks of twisted Dirac kernels.
  - The note follows the source (signature operator) and treats the spin Dirac operator as well.

## Readings
Here D = d_∇ + d_∇\* is the (ungraded) twisted signature operator and D^± are its chiral halves (restrictions to
the ±1-eigenspaces of τ = i^{p(p−1)+n/2} ∗); the index of D^+ is the twisted signature.

| Reading | Refuted? | Witness |
|---|---|---|
| ker D_E ≅ ker D'_{f\*E} for the ungraded twisted signature operator D = d_∇ + d_∇\* (OWR 2006) | yes | T^2, E_ε: dim 4 against 0 (Theorem 1.2) |
| the same for its chiral half D^+: Ω^+ → Ω^- | yes | 2 against 0 (Theorem 1.2(iii),(iv)) |
| dimension 4, where the signature operator is the classical one | yes | T^4 = T^2 × T^2, pr_1\*E_ε and f × id: 16 against 0 for D, 8 against 0 for D^± (Prop. 4.1) |
| any dimension n ≥ 2 | yes | T^2 × N: 4·Σ_q b_q(N) against 0 for D (Prop. 4.1) |
| Euclidean (real) bundles, as in Hilsum–Skandalis and Sauer–Schick | yes | rank-2 real bundle E^ℝ_ε: 8 against 0 for D, 4 against 0 for D^+ (Prop. 4.2) |
| spin Dirac operator (the corpus title) | yes | periodic spin structure: 2 against 0 (Prop. 4.3) |
| Sauer–Schick b_k := dim(ker D_E ∩ Ω^k) = dim ker Δ_k | yes | (0,2,0) against (0,0,0) (Prop. 4.4) |
| b_k := dim of the projection of ker D_E to Ω^k | yes | (2,2,2) against (0,0,0) |
| parity numbers b_ev, b_odd (the grading that ker D_E always has) | yes | (2,2) against (0,0) |
| f an isometry | no | the answer is then trivially yes (unitary equivalence) |
| number of eigenvalues of D in [−r, r], r = \|ε\|(1+c^2)^{1/2} < π (a modified question) | not refuted | 4 against 4 (Prop. 5.1(a)); a general version is left open (Question 5.2) |
| number of eigenvalues of Δ_0, Δ_1, Δ_2 in [0, r^2], r + \|ε\| < π | not refuted | 1, 2, 1 for both bundles (Prop. 5.1(c)) |

## Results in the paper
- **Lemma 2.1 (kernel decomposition).** Let d + iα be any unitary connection on T^2 × ℂ. Then D|even is unitarily
  equivalent to 2∂̄_∇ ⊕ 2∂_∇. The kernel is spanned by s(1 + i vol), s(dx + i dy) with s ∈ ker ∂̄_∇ (in Ω^+) and by
  t(1 − i vol), t(dx − i dy) with t ∈ ker ∂_∇ (in Ω^-). So dim ker D = 2(dim ker ∂̄_∇ + dim ker ∂_∇).
  - This corrects the typo "H^0(E,d_A)" of the triage row: the second summand is ker ∂_∇.
- **Lemma 2.2 (kernel criterion).** Let A = ∫a and B = ∫b be the harmonic part of α = a dx + b dy.
  - If (A,B) ∈ (2πℤ)^2, then ker ∂̄_∇ and ker ∂_∇ are 1-dimensional; otherwise both vanish.
  - Proof: a complex gauge transformation e^{-h} with ∂_z̄ h = ρ − ρ_0 reduces to constant coefficients (the device
    of Aharonov and Casher); then use Fourier modes.
  - Corollary 2.3: dim ker D ∈ {4, 0} accordingly. Remark 2.4: the dimension of the kernel depends only on the
    harmonic part (the kernel itself depends on α).
- **Theorem 1.2.** f is a diffeomorphism isotopic to the identity. ‖F‖ = 2π|ε|. dim ker D_{E_ε} = 4 for all ε, with
  kernel basis s_±(1 ± i vol), s_±(dx ± i dy), s_± = exp(∓(ε/2π) sin 2πy); dim ker D^±_{E_ε} = 2.
  - f\*α_ε = ε cos(2πy) dx + εc cos^2(2πy) dy, with harmonic part (0, εc/2).
  - dim ker D_{f\*E_ε} = 4 if εc ∈ 4πℤ and 0 otherwise. So for c ≠ 0 and 0 < |ε| < min(δ/2π, 4π/|c|) the kernels
    are 4 and 0 (2 and 0 for D^+).
- **Remark 3.1 (holonomy and flux).** The integral of α_ε over f∘(y-circle) is εc/2, the curvature flux through the
  isotopy. For flat connections it vanishes. In general the harmonic part moves by O(‖dα‖), but the lattice
  condition is not open. Also: the kernel of D_{E_ε} changes when g is replaced by the flat metric (f^{-1})\*g.
- **Propositions 4.1–4.4.**
  - Products: ker D = ker D_1 ⊗ ker D_N, from D^2 = D_1^2 ⊗ 1 + 1 ⊗ D_N^2. On T^4: 16 against 0 for D, and
    dim ker D^± = 8 against 0.
  - Real rank 2: E^ℝ ⊗ ℂ = L ⊕ L̄; 8 against 0 for D, 4 against 0 for D^+.
  - Spin Dirac: D = [[0, −2∂_∇], [2∂̄_∇, 0]].
  - Degree-wise readings: for non-flat bundles D^2 = ⊕Δ_k + F∧ + (F∧)\*, so ker D is in general graded only by
    parity. For E_ε, Σ(−1)^k dim ker Δ_k = −2 and Σ(−1)^k b^π_k = 2, while the twisted Euler characteristic is 0.
    On T^4 the degree-wise dimensions are (0,2,4,2,0) against 0 (with a one-line proof).
- **Proposition 5.1 (small eigenvalues).** Let r = |ε|(1+c^2)^{1/2} < π.
  - (a) Both operators have exactly four eigenvalues in [−r, r] and none with r < |λ| < 2π − r (Kato's perturbation
    bound and Riesz projections; two projections at distance < 1 have the same rank).
  - (b) The four are 0 for E_ε. For f\*E_ε they are ±σ, each with multiplicity 2, where e^{−|ε|/π} d ≤ σ ≤ d and
    d = dist(εc/2, 2πℤ).
  - (c) New in this version. If r + |ε| < π (equivalently r^2 + 4π|ε| < (2π − r)^2), then for both bundles
    Δ_0, Δ_1, Δ_2 have exactly 1, 2, 1 eigenvalues in [0, r^2 + 2π|ε|], all of them in [0, r^2], and ⊕Δ_k has
    no other eigenvalue below (2π − r)^2 − 2π|ε|. Proof: ⊕Δ_k = D^2 − K with ‖K‖ = ‖F∧ + (F∧)\*‖ = 2π|ε|, (a),
    and the min-max principle (the lower counts 1, 2, 1 come from the forms with constant coefficients, whose
    Rayleigh quotients are at most r^2). This replaces the remark of version 1 that each Δ_k has 1, 2, 1
    eigenvalues in [0,1], which was tested only at (ε, c) = (0.01, 1) and (0.2, 0.7) and fails at other
    parameters with r < π.
- **Question 5.2** (left open): a dilatation-type comparison of the counting functions of small eigenvalues, with an
  additive error of the order of the curvature. Context: Gromov–Shubin, and Gromov 1996 (pp. 80 and 141).

## Computations (numerical checks, not proofs; scripts and outputs in reproducibility/)
- **Lead** (`lead/check_example.py`, numpy, a few seconds, 92 checks, all passed). It checks:
  - the formulas for f, f\*α_ε, the curvature, the holonomy and the flux (quadrature);
  - the kernels and splittings for six pairs (ε, c), with two truncations;
  - the bounds of Proposition 5.1(a),(b), and the count of four eigenvalues in [−r, r];
  - Lemma 2.2 for an (x,y)-dependent connection with eight harmonic parts;
  - the condition εc ∈ 4πℤ;
  - the twisted Laplacians at (ε, c) = (0.01, 1) and (0.2, 0.7) (kernels (0,2,0) against 0; counts 1, 2, 1 in
    [0,1] for both bundles);
  - the real rank-2 bundle, the spin Dirac operator and T^4;
  - the explicit kernel forms (spectral differentiation).
- **Lead, revision** (`lead/check_laplacians.py`, numpy, about a second, written for this revision). Exact
  Rayleigh–Ritz compressions of Δ_0, Δ_1, Δ_2 from the quadratic forms of d_∇ and d_∇\*, two truncations, twelve
  pairs (ε, c) with r + |ε| < π, both bundles: counts 1, 2, 1 in [0, r^2] and in [0, r^2 + 2π|ε|]; fifth
  eigenvalue of ⊕Δ_k above (2π − r)^2 − 2π|ε|; all 72 checks passed. The window [0,1] of version 1 gives other
  counts at several of these pairs, for example (0, 2, 0) for f\*E_ε at (1.25, 1).
- **First verification run** (`verifier/`, AI-assisted, written independently of the lead's code; re-run for the
  release):
  - `sig_torus.py`: a 2D Fourier discretization with generic convolution, 4 against 0 for four pairs (ε, c) and two
    truncations, with smallest |λ| equal to εc/2 to the printed precision;
  - `sig_clifford.py`: the Clifford-module form Σ c(e_j)∇_j for n = 2, 4: 4 against 0, and 16 (8+8) against 0;
  - `sig_degrees.py`: the degree-wise Laplacians, (0,2,0) against (0,0,0), parity 2+2 against 0+0.
- **Second verification run** (`independent_run_2/`, AI-assisted, written from the text of version 1 before
  reading the other scripts; outputs regenerated for the release under neutral file names):
  - `torus_checks.py`: an independent construction (exterior algebra of R^n from scratch, matrix-valued
    connections, Fourier–Galerkin truncation). Theorem 1.2 for 14 pairs (ε, c), two truncations each (E_ε always
    4 = 2 + 2 with next eigenvalue ≥ 5.3; f\*E_ε 0 off the resonances, 4 at εc ∈ {4π, −8π}); eq. (1), τ and
    Lemma 2.1 as matrix identities; Lemma 2.2 for eight harmonic parts; the kernel forms; T^3 (8 against 0) and
    T^4 (16 against 0, degree-wise (0,2,4,2,0), projections (2,6,8,6,2), parity (8,8)); real rank 2; spin Dirac
    (2 against 0); Prop. 4.4; Prop. 5.1(a),(b) for nine pairs with r < π.
  - `supp_checks.py`, `delta_window.py`, `real_rank2_resonant.py`, `sigma_crosscheck.py`: resonant kernels for
    large c; the Δ_k windows (this found the failure of the version-1 remark at, for example, (1.25, 2) and
    (2, 1)); the real rank-2 bundle at εc = 4π (8 against 8); σ against the lead's printed values (agreement to
    4e-9).
  - `release_rerun/`: the four release scripts rerun from a fresh extraction of the version-1 `source.zip`;
    outputs byte-identical to the recorded ones.

## Independent adversarial audit
An independent verification run (AI-assisted, 2026-09-30) re-derived the whole argument and checked it with
separate code. Its verdict was `correct: true` (classification PAPER_CANDIDATE, suggested status `solved`), with
eight required fixes. All were applied:

| Required fix | Where applied |
|---|---|
| Cite and answer Sauer–Schick 2008, Question 57.2; cite Hilsum–Skandalis for index invariance | Introduction; Section 4 (Prop. 4.4); Scope |
| Explain that ker D_E is graded only by parity for non-flat bundles; answer every natural definition of b_k | "Twisted Betti numbers" paragraph and Prop. 4.4 (ker Δ_k = ker D ∩ Ω^k, projections, parity) |
| Fix "H^0(E, d_A)" to ker ∂_A | Lemma 2.1 |
| Replace "Times T^2 gives dim 4" by T^4: 16 against 0 | Prop. 4.1 |
| State the parameter range | Theorem 1.2(iv): 4 iff εc ∈ 4πℤ, else 0; explicit choice of ε |
| Add the real rank-2 version (8 against 0) | Prop. 4.2 |
| Citation search on Sauer–Schick 2008 and Hilsum–Skandalis; claim only that no prior answer was located | Scope; this report |
| Note that the corpus statement drops "signature operator"; add the spin Dirac operator | Introduction; Prop. 4.3 |

Further changes made while writing:
- a general kernel criterion (Lemma 2.2) for all unitary connections, not only y-dependent ones;
- the product construction in all dimensions;
- a proved statement on the small eigenvalues (Prop. 5.1), used in the paragraph on stability;
- a modified open question (Question 5.2).

## Second independent verification run (AI-assisted, 2026-09-30)
A second, independent AI-assisted verification run examined version 1 of the note (7 pages). It fetched the
primary sources again (anonymous requests), re-derived all proofs, wrote new code from the text of the paper,
reproduced every numerical claim, reran the four release scripts from a fresh extraction of `source.zip` (outputs
byte-identical), checked the checksums in `zenodo/ZENODO_METADATA.md`, and repeated the literature searches
(arXiv API, zbMATH Open, Crossref, DataCite, OpenCitations, Semantic Scholar, OpenAlex).

**Its verdict.** Statement fidelity: faithful. Proofs: correct, no mathematical gap. Computations: every claim
confirmed; one remark too broad. Novelty: no prior answer found. No mathematical error (`fatal = false`). Minor
revision, with eight required fixes; none of them touches the main theorem. All were applied:

| Required fix | Where applied |
|---|---|
| R1. Consistent terminology: the dimensions 4, 8, 16 are those of D = d_∇ + d_∇\*; for D^+ they are 2, 4, 8 | Section 2 now calls D_E the (ungraded) twisted signature operator and D_E^+ its chiral half (whose index is the twisted signature, the index in Hilsum–Skandalis); abstract, sentence after Theorem 1.2, Props. 4.1 and 4.2, Scope paragraph; this report (verdict, readings); Zenodo description |
| R2. δ in Hilsum–Skandalis depends only on (M,g), (M',g') and f, not on (E,∇) | Introduction, first paragraph |
| R3. Qualify Sauer–Schick's remark that Σ(−1)^k b_k is an index | Introduction: it holds for flat bundles and for b_ev − b_odd, and fails for degree-wise numbers of non-flat bundles (Prop. 4.4: −2 against the index 0), as Azzali–Wahl Remark 8.3 warned; forward reference to Prop. 4.4 |
| R4. Remark 2.4: the dimension of the kernel, not the kernel itself, depends only on the harmonic part | Remark 2.4 (and Remark 3.1) |
| R5. The remark "each Δ_k has numerically 1, 2, 1 eigenvalues in [0,1]" was tested only at two parameter pairs and fails at others with r < π | Replaced by Prop. 5.1(c), with proof (curvature-scaled window [0, r^2 + 2π\|ε\|] under r^2 + 4π\|ε\| < (2π − r)^2, i.e. r + \|ε\| < π; the distribution 1, 2, 1 follows from the min-max principle on forms with constant coefficients rather than from continuity in ε); checked by the new `lead/check_laplacians.py` |
| R6. Credit Gromov 1996 as context for Question 5.2 | Section 5, next to Gromov–Shubin: M. Gromov, *Positive curvature, macroscopic dimension, spectral gaps and higher signatures*, Functional Analysis on the Eve of the 21st Century, Vol. II, Progr. Math. 132, Birkhäuser, Boston, 1996, pp. 1–213, doi:10.1007/978-1-4612-4098-3_1 (verified on Crossref); pp. 80 and 141 read |
| R7. Release report: describe the first check as an AI-assisted verification run, consistently with the paper; carry R1 into the verdict, the readings and the Zenodo description | This report (section "Independent adversarial audit" and above); `source.zip`, the Zenodo copies and the checksums in `zenodo/ZENODO_METADATA.md` rebuilt |
| R8. Identify the two works that OpenAlex counts as citing Sauer–Schick | Both are records of Azzali–Wahl (see below); Scope paragraph updated, "(rate limit)" removed |

Optional suggestions applied: Hanke–Schick, Geom. Dedicata 135 (2008) 119–127 (arXiv:0705.2578), whose abstract
states a new proof of the Hilsum–Skandalis theorem; Connes–Gromov–Moscovici (1990) for the almost flat approach to
the Novikov conjecture; Aharonov–Casher (1979) for the complex gauge argument in Lemma 2.2; the rank argument in
Prop. 5.1(a) given in one line (the earlier reference was to Kato's chapter on finite-dimensional spaces); one line
of proof for the degree-wise dimensions on T^4; the capitals of the Oberwolfach title protected in the
bibliography; distinct labels [HS92] (Hilsum–Skandalis) and [HaS06], [HaS08] (Hanke–Schick); "in general graded
only by parity, not by degree"; the arXiv-v3 numbering of Azzali–Wahl stated in the Scope paragraph; the Part E
parameters stated in `reproducibility/README.md`. Not applied: a discussion of Azzali–Wahl's alternative
degree-preserving definition (their Section 8 was not re-read for this revision); a sentence on the odd signature
operator in odd dimensions; changes to `lead/check_example.py` (the script and its recorded output are unchanged);
a sentence on Benameur–Mathai (recorded in this report instead).

## Relation to the literature, novelty and scope
- **Searches (2026-09-30; anonymous requests, logged in `queries.log` of the working folder).**
  - Crossref: DOIs verified for the report, Hilsum–Skandalis (Crossref gives pages 73–100; zbMATH and Sauer–Schick
    give 73–99), Hanke–Schick 2006 and 2008, Azzali–Wahl, Gromov–Shubin, Gromov 1996 (a book chapter, pp. 1–213,
    Birkhäuser Boston 1996; the volume record lists the editors S. Gindikin, J. Lepowsky and R. L. Wilson),
    Aharonov–Casher, Kato and Lawson–Michelsohn. Crossref counts 13 citing records of Hilsum–Skandalis but gives no
    public list; it has no record of the Sauer–Schick DOI.
  - DataCite: the Sauer–Schick DOI.
  - OpenAlex: Hilsum–Skandalis has 72 citing works. Sauer–Schick (W2137841298) has 2, one each in 2018 and 2019.
    The listing endpoint (`filter=cites:W2137841298`) refused anonymous requests again (HTTP 429; the free daily
    budget for requests without an API key was used up), so the two works were identified as follows. The
    autocomplete endpoint returns two OpenAlex records
    for Azzali–Wahl, W1892619887 (publication year 2018) and W2962711392 (publication year 2019), both with DOI
    10.1017/S0305004118000427. Single-record look-ups show that both list W2137841298 among their references.
    So the two citing works are two records of the same paper, Azzali–Wahl (online 2018, volume 2019).
  - Semantic Scholar: 92 records citing Hilsum–Skandalis (scanned by title) and one citing Sauer–Schick, namely
    Azzali–Wahl.
  - OpenCitations (second verification run): 11 records citing Hilsum–Skandalis, none citing Sauer–Schick.
  - arXiv API: six queries by the lead ("almost flat" with Betti, kernel, signature or homotopy invariance;
    "twisted Betti numbers"; "almost flat bundles"), and fifteen by the second verification run. They returned
    only papers on indices, K-theory or unrelated Betti numbers. Two candidates from 2018–2019 read by the second
    verification run, Carrión–Dadarlat (arXiv:1503.06170) and Hunger (arXiv:1607.07820), do not cite Sauer–Schick.
  - zbMATH Open: Hilsum–Skandalis is Zbl 0731.55013. Sauer–Schick is indexed only as part of Guido's book.
    Connes–Gromov–Moscovici is Zbl 0693.53007 (C. R. Acad. Sci. Paris Sér. I 310 (1990), no. 5, 273–277), and
    Gromov 1996 is Zbl 0945.53022 (Progr. Math. 132).
  - One web search (by the lead). Hits: the ResearchGate page of Sauer–Schick; Azzali–Wahl; Benameur–Heitsch,
    *The twisted higher harmonic signature for foliations* (arXiv:0711.0352; abstract read: leafwise homotopy
    invariance for leafwise flat bundles, not the almost flat kernel question).
- **Related work found.**
  - S. Azzali and C. Wahl, *Two-cocycle twists and Atiyah–Patodi–Singer index theory*, Math. Proc. Cambridge
    Philos. Soc. 167 (2019) 437–487, doi:10.1017/S0305004118000427 (Section 8 read in arXiv v3, whose numbering is
    used). Remark 8.3 observes that the twisted Laplacian does not preserve degree, so the alternating sum of the
    degree-wise Betti numbers may differ from the twisted Euler characteristic. The note credits this; Prop. 4.4 is
    an explicit instance (−2 against 0). Question 8.2 asks about metric dependence and homotopy invariance of
    even/odd Betti numbers for twists by 2-cocycles of the fundamental group. That setting is not addressed.
  - M. Gromov (1996), pp. 80 and 141: the numbers of eigenvalues of d + d\* in [−a, a] on λ-bi-Lipschitz equivalent
    manifolds are compared up to the dilatation of a by λ^{±n}; the passage from flat to ε-flat bundles in the
    Hilsum–Skandalis theorem is related to the homotopy invariance of the spectrum of d + d\* near zero on
    coverings (Novikov–Shubin, Gromov–Shubin). Credited as context for Question 5.2.
  - The corpus record's literature triage lists Benameur–Mathai (arXiv:1202.0272) for twisted signature
    complexes. According to the second verification run, it concerns flux-twisted complexes with flat bundles and
    does not address the question.
- **No earlier answer located.** We located no record that answers either question. The example is elementary and
  may be known to experts, and this negative search is not a proof of priority.
- **Caveats.**
  - Hilsum–Skandalis and Hanke–Schick 2006 were not consulted; their results are quoted from Sauer–Schick. Of
    Hanke–Schick 2008 only the abstract (arXiv v1) was read, and of Gromov 1996 only pp. 80 and 141 (in a scan).
  - Aharonov–Casher and Connes–Gromov–Moscovici are cited for context; only their bibliographic data were checked.
- **Scope.** The note answers both questions negatively, in the strongest form: M = M' with one flat metric and f
  isotopic to the identity. Three things remain open:
  - the modified question on small eigenvalues (Question 5.2);
  - a version of the question under a uniform spectral gap assumption;
  - the 2-cocycle questions of Azzali–Wahl.

## 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/