# Verification report — OWR-15951-003 (Fillman: half-lines in the spectra of separable continuum Fibonacci operators with two couplings)

Verification date: 2026-10-18.

**Verdict.** The note gives a **partial answer**; every statement is proved as stated, and two statements are
explicitly conditional or depend on inputs from Damanik–Fillman–Gorodetski (DFG21 = J. Funct. Anal. 280 (2021)
108911; page numbers of arXiv:2001.03875v1, whose text agrees with the journal version in the parts used). Three
independent AI-assisted verification runs checked it; the third examined the final text (see below).
- **Theorem A (locally constant bumps, all real couplings).** For f_j = c_j χ[0,1) the spectrum of
  L^(2)_{λ1 V, λ2 V} contains a half-line for all real λ1, λ2; equivalently Σ_{μ1} + Σ_{μ2} ⊇ [E1, ∞) for DFG's
  Σ_μ. For λ1 = λ2 this is DFG21, Theorem 3.4. The proof uses DFG21, Proposition 4.2 and DFG's construction of the
  curve pieces (p. 13) as DFG's own proof does, for two families of windows, plus the new Lemma A and Lemma E.
- **Theorem C (unconditional; new to our knowledge).** For BV pieces g_a, g_b with h = g_b − g_a absolutely continuous and h' of
  bounded variation: I(t²) = sin²t |h(1)e^{2it} − h(0)|²/(16t⁴) + R(t), |R| + |R'| ≤ C t⁻⁵. Corollary 5.4: if
  h(0) + h(1) ≠ 0, then I ≥ c t⁻⁴ and |dI/dt| ≤ C I away from πZ (DFG's hypotheses (2)–(3)).
- **Theorem B (conditional).** For BV bumps with h = f_0 − f_1 absolutely continuous, h' BV and
  f_0(0+) + f_0(1−) ≠ f_1(0+) + f_1(1−): half-line for all real couplings, **assuming statement (P)**, i.e. that
  DFG's construction of the curve pieces with property (4) of their Proposition 4.2 applies to the curves of these
  pairs. DFG assert this for general pieces in Remark 4.25 ("a modified version of property (1) and property (4)
  can be established") but do not prove it.
- **Open.** Pairs with h(0) + h(1) = 0 (among them every pair with continuous V_ω, hence all smooth Fibonacci
  potentials) and all other pairs outside the hypotheses of Theorem B: BV pairs whose difference is not absolutely
  continuous (e.g. f_1 = 0, f_0 = χ[0,1/2)) or has a derivative that is not of bounded variation, and bumps not of
  bounded variation. The method stops there because the leading term of Theorem C vanishes at points of every
  window (h(0) = −h(1) ≠ 0) or identically (h(0) = h(1) = 0). For bumps symmetric about 1/2 with f_1 = 0 and
  h(0) = h(1) = 0, such as f_0 = sin πx or x(1 − x), the invariant even vanishes inside every window computed
  (tested, not proved), so hypothesis (2) of DFG's Proposition 4.2 fails there as well.

The note is unrefereed. The status of the corpus record stays open (partial, unrefereed result).

## Statement checked
- **Primary source.** J. Fillman, "Spectral properties of continuum quasicrystal models", extended abstract in
  Oberwolfach Report 46/2017 (Spectral Structures and Topological Methods in Mathematical Quasicrystals),
  Oberwolfach Rep. 14(4) (2017) 2781–2845, pp. 2797–2798, doi:10.4171/OWR/2017/46. Read in the open PDF on the
  EMS Press site (fetched 2026-10-18 by the first version, by both verification runs and again at the writing).
  - L_V u = −u'' + V u on L²(R); Eq. (1): V_ω(x) = Σ_n [(1 − ω_n) f_0(x − n) + ω_n f_1(x − n)], f_0, f_1 ∈ L²[0,1)
    arbitrary; Eq. (2): ω_n = χ_[1−α,1)(nα mod 1), α = (√5 − 1)/2 (printed as χ_[1−α)); separable operator
    L^(2)_{V1,V2} = −∇² + V1(x) + V2(y) with σ(L^(2)) = σ(L_{V1}) + σ(L_{V2}) (p. 2798).
  - The question (p. 2798): if V_ω is defined by Eqs. (1)–(2), does σ(L^(2)_{λ1 V_ω, λ2 V_ω}) contain a half-line
    for every choice of λ1, λ2 > 0?
- **Corpus record.** ulamai/UnsolvedMath, OWR-15951-003 (dataset version 1.8.0; status `open`); its statement is
  the question above with "Eqs. (1)–(2)" of the source.
- **Conventions.** The symbol 1 of Eq. (2) has frequency α > 1/2, so it is DFG's frequent letter a; hence
  σ(L_{λV_ω}) = Σ(λ f_1, λ f_0). Condition (3) of the note is symmetric under f_0 ↔ f_1, so the letter convention
  cannot affect Theorem B. Unit cells throughout (as in the source).

## Readings
| Reading | Answer | Where |
|---|---|---|
| All bump pairs (f_0, f_1) ∈ L²[0,1)², all λ1, λ2 > 0 (the source's reading) | partial: yes for constant bumps (Theorem A); yes for the bumps of Theorem B conditionally on (P); open otherwise | Theorems A, B; Section 7 |
| Only DFG's locally constant model (a narrow reading of "continuum Fibonacci potential") | yes, for all real λ1, λ2 (Theorem A, with DFG's own inputs) | Theorem A |
| Equal couplings λ1 = λ2 | constant bumps: DFG21, Theorem 3.4 (known); bumps of Theorem B: half-line part of DFG21, Question 1, conditionally on (P), unit cells only | Section 1.3 |
| Periodic case f_0 = f_1 a.e. | not a quasicrystal; excluded by (3); for constant bumps (c_0 = c_1) trivial | Lemma 2.1, Theorem A proof |
| Unequal cell lengths | not part of the source's question (unit cells); mentioned only with DFG21, Question 1 | Sections 1.3, 4 |

## Results in the paper
- **Lemma 2.1 (aperiodicity).** g_a ≠ g_b in L² ⇒ the Fibonacci concatenation is not periodic (closed group of
  periods; irrational period impossible by connectedness; integer period contradicts aperiodicity of the
  subshift). So DFG21, Assumption 3.1 holds in Theorems A (c_0 ≠ c_1) and B.
- **Proposition 2.2** = DFG14 (Ann. Henri Poincaré 15 (2014)), Proposition 6.3: spectrum = bounded forward
  trace-map orbits (stated under aperiodicity and irreducibility; irreducibility holds with ℓ = 1 for unit cells and
  g_a ≠ g_b, and for two letters the property it is assumed for follows from aperiodicity, DFG21, Remark 3.3).
- **Lemma 3.1** = DFG21, Lemma 2.1 (proved as Damanik–Gorodetski, CMP 305 (2011), Lemma 6.2; Newhouse gap lemma).
  **Lemma 3.2**: (a) squaring multiplies thickness by at least x/y (proof given); (b) = Fillman–Tidwell, Lemma 3.2.
- **Lemma A (two-set Bethe–Sommerfeld lemma, new).** Complete proof (Steps 1–4).
- **Section 4 (input from DFG21).** Windows and Π̃ (p. 13); (C-small): m fixed with α, β < π/4, using DFG's
  statement that the cut pieces are small for large m; Proposition 4.1 = DFG21, Proposition 4.2; statement (P);
  Lemma 4.2 (hulls = windows, from the endpoint property in (P)); Proposition 4.3 (both families, the second by
  the reflection s ↦ 2π(2n+1) − s); **Lemma E** (two couplings: aligned windows from the coupling-independent free
  curve).
- **Section 5.** Lemma 5.1 (I in (q, ζ) coordinates); Dyson expansion in the rotating frame; Proposition 5.2 (a
  quantitative form of standard high-energy asymptotics, not claimed as new);
  **Theorem C**; Lemma 5.3 (property (1) for all L¹ pieces, ‖R(z)‖ = e^{|Im z|}); Corollary 5.4.
- **Section 6.** Proofs of Theorems A and B. **Section 7.** Open part and numerical illustrations (tested only).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program written with the note** (`check_note.py`, standard library, about 0.5 s; 32 checks, 0 failed): the
  polynomial identity behind Lemma 5.1 (exact) and Lemma 5.1 on 200 random SL(2,Q) pairs; the coordinates (α, ζ),
  product rule, J, n, R(φ), the rotating frame, N(σ), ζ(N(s1)N(s2)), the factorization of the Dyson integrand
  (exact at rational points of the unit circle) and formula (9) (quadrature); the algebra of Theorem C and
  Corollary 5.4 (exact, 1125 cases); DFG (3.11) against L and its logarithmic derivative (last term −2t/(t² − μ);
  DFG print 2t/√(t² − λ)); the constants of Lemma E
  (exact, 3042 cases) and an interval model of Lemma A; ‖R(z)‖ = e^{|Im z|}; an illustration of Theorem C for four
  pairs up to t ≈ 62 (Runge–Kutta in the rotating frame; I by traces and by formula (7) agree to 2·10⁻⁶;
  max |I − L| t⁵ = 0.277).
- **Programs of the first version** (`original/`): symbolic checks (SymPy) and the numerics of Section 7; re-run on
  2026-10-18 at the writing (0.8 s, 1.5 s, 125 s): outputs identical to the recorded ones.
- **Programs of the three verification runs** (`verification_run_A/`, `verification_run_B/`,
  `independent_run_2/`), see below. The third run re-ran 24 programs of the package from `source.zip`: all recorded
  outputs reproduced up to elapsed times (`independent_run_2/rerun_of_package.md`).
- Numerical values in the note are labelled as tested: |I − L| t⁵ and |I' − L'| t⁵ bounded for ten pairs with
  h(0) + h(1) ≠ 0 (Taylor arithmetic with about 78 correct digits, t = 25…286), for four pairs up to t ≈ 1011 (exact
  Airy/trigonometric matrices) and for six pairs up to t ≈ 410 (third run, 60 digits); wrong leading terms grow like
  t; for h(0) + h(1) = 0: for (0, 1 − 2x) the minimum of I is ≈ t⁻⁸/16 at t − t* ≈ −1/t and sup |d log I/dt| ≈ 3.2·10⁴
  on the window [40π + 0.15, 41π − 0.15]; for (0, sin πx) and (0, x(1 − x)) the invariant has zeros inside the
  windows (third run), so |d log I/dt| is unbounded there (the first version's coarse-grid values 256 and 846 are not
  used); the double-precision trace formula is noise-limited (absolute error up to ≈ 4·10⁻¹³ for t ≈ 130–250).

## Independent verification runs
The results were first obtained, with proofs, together with the programs in `original/`. Two independent
verification runs, both AI-assisted, followed (2026-10-18); each read the source and the closest papers and wrote
its own programs; neither saw the other's report.

| Item | Run A | Run B |
|---|---|---|
| Statement, conventions, letters, scope against the source | CONFIRMED (OWR PDF read) | CONFIRMED (OWR PDF read; typo χ_[1−α) noted) |
| Lemma A, Lemma 3.2 | CONFIRMED (re-derived step by step) | CONFIRMED (by reading) |
| Lemma E, alignment of windows | CONFIRMED | CONFIRMED given Fact D |
| Three implicit points of DFG's proof (hulls, second family, α, β < π/4) | CONFIRMED (second family: arithmetic and reflection checked; hulls given the construction) | second family CONFIRMED; hulls and α, β < π/4 plausible, NOT re-proved |
| Lemma 5.1 | CONFIRMED (two symbolic derivations) | CONFIRMED (exact on 200 SL(2,Q) pairs; wrong sign detected) |
| Proposition 5.2, Theorem C, Corollary 5.4, Lemma 5.3 | CONFIRMED (re-derived; numerics by two independent methods) | CONFIRMED (by reading; algebra by own SymPy; asymptotics by exact arithmetic, 12 cases) |
| Theorem A | CONFIRMED modulo DFG's inputs | CONFIRMED conditional on DFG's own inputs |
| Theorem B | CONFIRMED conditional on (P) | CONFIRMED conditional on (P) |
| DFG21, Proposition 4.2: proof uses only (1)–(4) | read; no other use found; hyperbolic estimates not re-proved | NOT CHECKED |
| Literature and bibliography | citation lists, sources re-read | arXiv, OpenAlex, Semantic Scholar, zbMATH, Crossref; 2 web searches |

**Corrections required by the runs, all made:**
run A — exact quotation of DFG21, Remark 4.25 with the word "modified", and why unit cells need no modification;
aperiodicity (Lemma 2.1); conditionality and dependence on DFG visible in statements, abstract and scope; the open
part includes BV pairs with non-absolutely-continuous difference (the interior-jump formula is not stated, since it
is not proved); ‖R(z)‖ ≤ e in Lemma 5.3 and labels of numerical values with the double-precision caveat. Run B —
the coarse-grid values 256 and 846 replaced by the lower bounds 3.2·10⁴ and 1.6·10⁵ and the dip described; the DFG
inputs and the hull and smallness points in the statements; the JFA version of DFG21 (compared later, see the third
run) and the open-problem list of Damanik–Fillman, GSM 249, declared not accessible; unequal cell lengths only with
DFG's Question 1.

### Third run: independent verification of the final text (2026-10-18)
A third independent verification run, also AI-assisted and with its own programs (`independent_run_2/`), examined the
text of the note, not the earlier claim file. It re-derived all proofs line by line (Lemma 2.1; Lemma 3.2; Lemma A
with its constants and the hypotheses of the gap lemma; the inputs of Section 4, statement (P), Lemma 4.2 and
Proposition 4.3 with the second family by reflection; Lemma E and the alignment of the windows; Lemma 5.1; the Dyson
expansion; Proposition 5.2; Theorem C with the O(t⁻⁵) remainder for I and I'; Lemma 5.3; Corollary 5.4; Theorems A
and B with the conditional statement (P)) and found no error. It re-read the source and DFG14, Fillman–Tidwell,
Fillman–Mei and the Damanik–Fillman survey, and compared DFG21 arXiv v1 with the journal version in all parts the note
uses (Lemmas 2.1 and 2.3, Section 3, the construction preceding Proposition 4.2, Proposition 4.2, Section 4.5, Remark
4.25, Question 1): the texts agree and the numbering is the same (the journal version adds Remark 2.4 on f(x) = x^m and
renumbers a remark of Section 5); both versions print the last term of the logarithmic derivative in Section 4.5 as
2t/√(t² − λ) instead of −2t/(t² − λ). Its programs: SymPy checks (38, all pass) of Lemma 5.1, the coordinates of
Section 5, (9), (3.11) and the identities of Theorem C and Corollary 5.4; a 60-digit piecewise Taylor solver computing I
from its trace definition (24 checks, all pass): Theorem C and Corollary 5.4 for six pairs on both window families up
to t ≈ 410 (flat |I − L| t⁵ and |I' − L'| t⁵; t⁴ I about twice the bound of Corollary 5.4), controls growing like t,
Proposition 5.2, Lemma 5.3 at complex k, and the degenerate pairs; elementary checks (13, all pass) of the coding (2),
Lemma 3.2 and Lemma E on model windows. It re-ran 24 programs of the package from `source.zip` (all recorded outputs
reproduced up to elapsed times). Its new finding: for f_1 = 0 and f_0 = sin πx or x(1 − x) (symmetric about 1/2,
h(0) = h(1) = 0) the transfer matrix of f_0 has equal diagonal entries, ζ_b is purely imaginary, and Im ζ_b changes
sign near t* in every window examined (t from 33 to 256), so the invariant vanishes there; the earlier statement
"sup |d log I/dt| ≥ 1.6·10⁵ for (0, sin πx)" was true but understated this. Like the first two runs it did not re-prove
the hyperbolic estimates of DFG21, Proposition 4.2, the construction of the pieces for other curves, or the smallness
of α, β.

**Corrections required by the third run, all made:** the comparison with the journal version of DFG21 (note, this
report, the HF notice); the misprint in DFG's logarithmic derivative (note and check C7 of `check_note.py`); Section
7 (ii): the zeros of I for symmetric bumps, the refined values for (0, 1 − 2x); Section 7 (i) and (iii): wording
("high-precision" instead of "exact high-precision"), t ≈ 1011, the size of the double-precision error; Section 8:
(9) is checked by quadrature; the complete list of open cases (abstract, Sections 1.2 and 7); the remark on
irreducibility after Proposition 2.2; the record of the third run (note, this report); the README (exact parameters
of the recorded window scans, `independent_run_2/`); Proposition 5.2 no longer claimed as new.

## Relation to the literature, novelty and scope
- **Searches (2026-10-18).** First version: arXiv API (Fillman, Damanik, Gorodetski, Mei, Yessen, Takahashi,
  Tidwell with Fibonacci, half-line, Bethe–Sommerfeld, separable, Minkowski sum, sums of Cantor sets,
  Fricke–Vogt, square Fibonacci), OpenAlex (4 works citing DFG21), Semantic Scholar (5 citing DFG21), zbMATH, two
  web searches. Run B: OpenAlex and Semantic Scholar citations of DFG21 (5) and DFG14 (25), further arXiv queries,
  zbMATH, Crossref, two web searches. Writing stage: three arXiv API queries (latest papers of Fillman; Fibonacci
  with half-line; Fibonacci with separable) and Crossref for the bibliography; no web search. Third run: nine
  sorted arXiv API queries (Fibonacci with half-line; Fricke–Vogt; Fillman; Gorodetski; Damanik; continuum
  Fibonacci; Bethe–Sommerfeld with Cantor; separable with Fibonacci; sums of Cantor sets with spectrum) and id
  lookups, Crossref (all DOIs of the bibliography), OpenAlex (DFG21 cited by 4; the list of citing works could not be
  retrieved), zbMATH (three queries), one web search. Newest items seen: Gorodetski–Luna–Wood, arXiv:2610.17005
  (sums of small Cantor sets), Fillman–Lim, arXiv:2609.21029; neither concerns this question.
  - No printed answer to the question was found. Closest: DFG21 (equal couplings, constant pieces), Fillman–Tidwell
    (RMJM 53 (2023), sums g[F] + g[F] of one set), Fillman–Mei (AHP 19 (2018), I(E) = O(E^{−1/2}) for general
    pieces, no leading term), DFG14, Damanik–Fillman survey (arXiv:2007.01402, Theorem 8.4: equal couplings),
    Damanik–Embree–Fillman–Gorodetski–Mei (arXiv:2603.24462, strongly coupled regime). Quasi-periodic
    Bethe–Sommerfeld results (Karpeshina–Parnovski–Shterenberg; Parnovski–Shterenberg) do not apply to
    Fibonacci-type potentials. A search that finds nothing is not a proof of novelty.
- **What was read.** The source (OWR PDF); DFG21 in arXiv v1 (12 January 2020), Sections 2–4 and 6, whose
  page numbers are used, and in the journal version (J. Funct. Anal. 280 (2021) 108911), compared with v1 in the
  parts used by the note (same text and numbering); DFG14 (arXiv v1) Section 6; Fillman–Tidwell (arXiv v1) Section 3, Theorem 1.6,
  Corollary 1.9; Fillman–Mei (arXiv v1) proof of Theorem 1.3; the survey's Theorem 8.4; the abstract of DEFGM.
- **Not accessible to us.** The list of open problems in Damanik–Fillman, One-Dimensional Ergodic Schrödinger
  Operators (GSM 249, pp. 1035–1037). The note says so. (The journal version of DFG21, earlier not accessible to us,
  was later obtained from the publisher's site and compared, see above.)
  Newhouse (1970, 1979), Damanik–Gorodetski (2011) and Pöschel–Trubowitz (1987) are used as quoted in DFG21.
- **Scope.** Answered: constant bumps (Theorem A, unconditional modulo DFG's own inputs); the bumps of Theorem B
  conditionally on (P). New to our knowledge: Lemma A, Lemma E (two couplings), Theorem C, Corollary 5.4,
  conditional Theorem B; Proposition 5.2 is a quantitative form of standard high-energy asymptotics and is not
  claimed as new. No priority is claimed.

## 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/