# Verification report — OWR-4138-001 (Nichols–Stolz Conjecture 6.2, Bernoulli displacement model)

Verification date: 2026-09-29.

**Verdict.** The conjecture is true for every coupling λ ≠ 0 and every p ∈ (0, 1). The almost-sure spectrum of the
one-dimensional Bernoulli displacement model is Σ_λ = σ(h_{ω*,λ}) ∪ σ(h_{ω¹,λ}) = {E : |E(E−λ) − 2| ∈ [0, 2] ∪
[|λ|, √(λ²+4)]}. For |λ| > 2 this set consists of exactly six bands separated by five gaps. The inclusion
σ(h_{ω,λ}) ⊂ Σ_λ even holds for every configuration ω. The proof is complete and by hand; the computations are
consistency checks only. The reduction and the outer bound are due to Nichols and Stolz. Two independent
adversarial audits (an independent verifier and a second referee) found no mathematical error. The note is unrefereed.

## Statement checked
- **Oberwolfach Report 55/2009** (Mini-Workshop "Modeling and Understanding Random Hamiltonians: Beyond Monotonicity,
  Linearity and Independence", organisers G. Stolz and I. Veselić), doi:10.4171/OWR/2009/55. The abstract of
  R. Nichols (joint with G. Stolz) is on pp. 2985–2986.
  - The local text was read.
  - The formula Σ_λ = σ(h_{ν*,λ}) ∪ σ(h_{ω¹,λ}) is shown there for 0 < |λ| ≤ 2.
  - The authors say they cannot handle |λ| > 2 and conjecture the same formula in that case (p. 2986).
  - Notation: the report describes the model by the cell length L_1 = 2 and the single-site potential q = λδ_1,
    which is the model of the paper. It writes the extremal configuration as ν*, with ν*_k = δ_1((−1)^k). This is
    ω* shifted by one cell (ν*_k = ω*_{k+1}), so σ(h_{ν*,λ}) = σ(h_{ω*,λ}). The paper now says this in §1.
- **Published version.** R. Nichols, G. Stolz, "Spectral properties of the discrete random displacement model",
  J. Spectral Theory 1 (2011), no. 2, 123–153, doi:10.4171/JST/6 (arXiv:1008.1266). The full text was read.
  - The model is in §2.6, with q = λδ_1 displaced in cells of two sites. NS define (h_0u)(n) = −Σ_{|m−n|=1} u(m)
    (p. 125), the negative of the paper's h_0 (the paper's h_0 is the adjacency operator). The unitary
    u(n) ↦ (−1)^n u(n) maps one to the other and fixes V_ω, so every σ(h_ω) is the same in both conventions.
  - Known results: the outer edges come from ω* (Thm 2.2, Cor. 2.4); the central gap is a gap for every ω (Thm 2.6,
    proved in §5.1 with eq. (32) and the positive solution (35)–(36)); the case |λ| ≤ 2 is Cor. 2.7, which states
    Σ_λ = [E−, E+] ∖ (G−, G+); the identification with σ(h_{ω*}) ∪ σ(h_{ω¹}) is in the paragraph before it (and in
    OWR p. 2986).
  - **Conjecture 6.2** (p. 151): for any λ ≠ 0, Σ_λ = σ(h_{ω*,λ}) ∪ σ(h_{ω¹,λ}). The remark after it says this would give
    exactly six bands separated by five non-vanishing gaps.
- **Corpus record.** ulamai/UnsolvedMath v1.6.0, record OWR-4138-001 (status `open`). It reads: "For |λ|>2,
  conjecturally the almost-sure spectrum satisfies Σ_λ = σ(h_{ν*,λ}) ∪ σ(h_{ω¹,λ}); is this equality true?"

## Readings
| Reading | Answer | Where |
|---|---|---|
| Corpus/OWR question: the formula for \|λ\| > 2 | yes | Theorem 1.2(c) |
| JST Conjecture 6.2: the formula for all λ ≠ 0 | yes (for \|λ\| ≤ 2 this is NS Cor. 2.7) | Theorem 1.2(c) |
| "six bands, five non-vanishing gaps" for \|λ\| > 2 | yes, explicit bands | Theorem 1.2(d), Prop. 5.1 |
| other sign of h_0 (the NS convention), or cells {2k+1, 2k+2} instead of {2k, 2k+1} | same answer (unitary (−1)^n; translation) | §1 |
| OWR/corpus notation ν* instead of ω* | same answer (ν* is ω* shifted by one cell) | §1 |
| any p ∈ (0, 1) | yes, Σ_λ does not depend on p | Theorem 1.2(c) |
| deterministic strengthening: σ(h_ω) ⊂ Σ_λ for every ω | yes | Theorem 1.2(a) |

Nichols–Stolz Conjecture 6.3 (thin tails of the IDS at the eight other band edges) is a separate question. It is
not addressed and remains open.

## Results in the paper
- **Lemma 2.1.** Floquet discriminants: tr = x(E) for ω¹, and x(E)² − 2 − λ² (= NS's D(E)) for ω*. Here
  x(E) = E(E − λ) − 2.
- **Proposition 3.1** (the computation in NS's proof of Thm 2.6). U[h_ω(h_ω − λ) − 2]U* = (Δ + s) ⊕ (Δ − s), with
  s_k = λ(ω_k − ω_{k+1}). Hence x(σ(h_ω)) = σ(Δ+s) ∪ σ(Δ−s).
- **Remark 3.2.** The converse "x(E) ∈ σ(Δ±s) ⟹ E ∈ σ(h_ω)" fails for individual ω. For the domain wall, exactly
  one of E_± and exactly one of G_± lie in σ(h_ω) (rigorous argument). Numerically, the infinite wall has the
  eigenvalues E_− and G_− for λ > 0 and E_+ and G_+ for λ < 0 (labelled as numerical in the paper).
- **Lemma 3.3.** s takes values in {−|λ|, 0, |λ|}, with no two consecutive equal nonzero entries; the nonzero entries
  alternate in sign.
- **Lemma 3.4** (contains NS eq. (32)). σ(Δ ± s) ⊂ [−√(λ²+4), √(λ²+4)], proved with the NS positive solution and the
  ground-state identity. NS state the one-sided bound they need for the central gap; the two-sided form follows
  from the (−1)^k symmetry.
- **Lemma 4.1** (standard). For a self-adjoint block operator with A ≥ a and D ≤ d < a, there is no spectrum in (d, a)
  (Schur complement). References: Langer–Markus–Matsaev–Tretter (2001), Kostrykin–Makarov–Motovilov (2004), Tretter (2008).
- **Lemma 4.2.** Let L > 2, let q ≥ L on a set S with no two neighbours, and q ≤ 0 elsewhere. Then σ(Δ + q) misses
  (2, L). Remark 4.4 shows the hypothesis is needed: Δ + L(δ_0 + δ_1) has the eigenvalue L − 1 + 1/(L−1) ∈ (2, L).
- **Corollary 4.3.** For |λ| > 2, σ(Δ ± s) misses (2, |λ|) and (−|λ|, −2).
- **Theorem 1.2.** (a) σ(h_ω) ⊂ S_λ for every ω. (b) S_λ equals the union of the two periodic spectra. (c) For every
  p, Σ_λ = S_λ, using the periodic support theorem (NS Thm 2.3). (d) For |λ| > 2 there are six bands.
- **Proposition 5.1.** The bands are I_i^± = {λ/2 ± √(2 + λ²/4 + X) : X ∈ J_i}, with J_1 = [−√(λ²+4), −|λ|],
  J_2 = [−2, 2] and J_3 = [|λ|, √(λ²+4)]. They are ordered I_3^− < I_2^− < I_1^− < I_1^+ < I_2^+ < I_3^+.
  Table 1 gives λ = 3 and λ = 5.
- **Remark 4.5** (not used). An invariant cone field gives the same gaps via uniform hyperbolicity.

## Computations (scripts and outputs in reproducibility/)
- **Finder** (`claimant/`).
  - Exact rational checks of Prop. 3.1 on rings: 2520 configurations.
  - Exact checks of the positive solution and the ground-state identity (Lemma 3.4).
  - An exact check in the original model: for all 2615 periodic configurations with period ≤ 14 cells and
    λ ∈ {3, 5, 8/3, 21/10, −3, 12}, there is no periodic spectrum in the four gaps. It is certified by Descartes' rule
    of signs with bisection, with 0 violations, 0 undecided cases, and negative controls detected.
  - Floating-point random tests, with controls.
- **Independent verifier** (`referee/`).
  - It reran all finder scripts (the necklace check with KMAX = 11): identical results.
  - An exact Sylvester/Haynsworth inertia test of Lemma 4.2 (3600 tests, 0 failures).
  - Floquet–Bloch numerics in the original model (0 hits).
  - All words with ≤ 12 cells at λ = 3 and three phases: 24,570 matrices, 0 hits.
  - The domain-wall counterexample to the converse of the spectral mapping.
- **Second referee** (`referee/referee2/`, written before any finder script was read).
  - `ref2_exact.py` (exact rational arithmetic, standard library only):
    (A) the trace identities as polynomial identities in (E, λ), including NS's D(E) and the NS sign convention;
    (B) the reduction identity of Prop. 3.1 on rings of 3–7 cells, 1240/1240;
    (C) the positive solution for λ = t − 1/t;
    (D) an exact Floquet-level test in the original model for all 2615 binary necklaces with ≤ 14 cells and
    λ ∈ {8/3, 15/4, 24/5, 21/10, −8/3, 3/2, 9/20} (two with |λ| ≤ 2). D_ω(E) is reduced modulo E² − λE − (2+x);
    Sturm sequences in x certify that no periodic spectrum has x(E) in any gap: |x| > √(λ²+4) (the central gap and
    the exterior) and 2 < |x| < |λ| (the four inner gaps). The Floquet-level reduction identities A = D₊ + D₋ and
    B = D₊D₋ hold for every necklace, and D_ω is a polynomial in x for every necklace (including all 1980 chiral
    ones). 0 failures; point-level and interval-level negative controls detected. About 7 minutes.
  - `ref2_numeric.py` (numpy): random rings of 1400 sites, 1500 random periodic words (3–40 cells, Floquet–Bloch),
    Table 1 against Floquet–Bloch spectra of ω* and ω¹, the domain wall by matching decaying solutions, and the
    Remark 4.4 eigenvalue. No eigenvalue in the forbidden set beyond round-off.
- **Lead** (`lead/`).
  - Reran all finder scripts, including KMAX = 14: identical results apart from timings.
  - `paper_checks.py` checks exactly the trace identities (as polynomial identities), the sharpness example and
    the domain-wall eigenvalue. In floating point it checks the band formulas of Prop. 5.1 and Table 1, and finite
    domain walls.

## Independent adversarial audit
Verdict of the independent verifier (2026-09-29): classification PAPER_CANDIDATE. The verifier found the proof
correct, confirmed that it answers the question as intended, and re-derived every step by hand.

| Item | Verdict |
|---|---|
| Statement fidelity (OWR p. 2986, JST p. 151, conventions) | CONFIRMED |
| Proof (reduction, structure of s, outer bound, Schur-complement lemma, periodic spectra, band count) | CONFIRMED |
| Computations (all reruns reproduced; independent checks) | CONFIRMED |
| Novelty | no resolution found (see below); expert confirmation advised |
| Presentation | CONFIRMED_WITH_FIXES |

Required fixes of the verifier and how they were applied:
1. **(★) misstated as an equivalence.** Replaced by the spectral-mapping identity x(σ(h_ω)) = σ(Δ+s) ∪ σ(Δ−s)
   (Prop. 3.1), which gives only "⟹" for a given E. Remark 3.2 proves that the converse fails for the domain wall.
   Only "⟹" is used.
2. **Band list of Theorem (iv).** It mixed x- and E-intervals. Rewritten as the two preimages x⁻¹(J_i) = I_i^− ∪ I_i^+,
   with explicit endpoints (Prop. 5.1).
3. **Lemma 2 as an instance of a standard fact.** Now Lemma 4.1 (block operator fact, with references) followed by
   Lemma 4.2. The novelty is placed only in the application to the reduced potentials via Lemma 3.3.
4. **"Re-proves Theorem 2.6".** Changed to "reproduces the Nichols–Stolz argument for Theorem 2.6 (eq. (32))". See the
   text before Lemma 3.4 and the paragraph after the proof of Theorem 1.2.
5. **Further literature checks.** Partially done; see the section on the literature below. The full text of
   Chulaevsky (2023) was still not accessible, and the authors of the conjecture were not contacted. Both remain
   open items (listed below).
6. **Framing.** The paper is a short note resolving Conjecture 6.2 and credits the reduction to Nichols–Stolz.
   Conjecture 6.3 is stated as open.

**Referee 2 (2026-09-29): ACCEPT WITH MINOR FIXES.** Not fatal. The referee checked every step of the proof by hand,
found it correct and complete, and wrote independent exact code before reading any finder script.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (fixes 1–3 are wording) |
| Proofs | CONFIRMED, no gaps |
| Computations | CONFIRMED (finder, verifier, lead, and the referee's independent exact check) |
| Novelty | no prior proof or disproof found; same three citing works; Chulaevsky (2023) still unread (paywalled) |
| Presentation / house style | CONFIRMED_WITH_FIXES |

All required fixes of referee 2 were applied:
1. **Sign of h_0.** The false sentence "Their kinetic term is the operator h_0 above" (§1) was replaced: NS define
   (h_0u)(n) = −Σ_{|m−n|=1} u(m), the negative of the paper's h_0, and the unitary (−1)^n justifies the change.
   The same error in this report ("h_0 is the adjacency operator") was corrected.
2. **OWR notation.** §1 now states that the Oberwolfach report and the corpus record write ν*, with
   ν*_k = δ_1((−1)^k), which is ω* shifted by one cell (same spectrum), and that the report's L_1 = 2, q = λδ_1 is
   the model of the paper.
3. **Third bullet of the introduction.** The identification [E−,E+] ∖ (G−,G+) = σ(h_{ω*}) ∪ σ(h_{ω¹}) is cited as
   "[NS11, Cor. 2.7 and the preceding paragraph]; see also [OWR p. 2986]".
4. **Outer bound.** The introduction now says that the NS positive solution gives the one-sided bound (32) needed
   for the central gap, that in the paper's convention it bounds Δ ± s from above, and that the two-sided bound
   follows from the (−1)^k symmetry (Lemma 3.4). The sentence before Lemma 3.4 says the same.
5. **Scope and priority.** The search count now includes referee 2 (one more web search, plus arXiv, OpenAlex,
   Semantic Scholar, Crossref and zbMATH checks; nothing new). It states that the full text of Chulaevsky (2023) is
   behind a paywall and still unread. The introduction describes [Chu23] "according to its title and available
   summary".
6. **Verification and release.** The Verification paragraph has a new item crediting the referee's independent
   exact check. The referee's scripts and outputs are in `reproducibility/referee/referee2/`, with rows in the
   reproducibility README. This section records the referee-2 verdict.
7. **Rebuild.** The paper was rebuilt with tectonic (no overfull or underfull boxes), every page was re-rendered and
   inspected, and paper.pdf, main.tex, references.bib, source.zip and the three Zenodo files were regenerated, with
   new checksums in ZENODO_METADATA.md.
8. **Further reading (not an edit).** The referee asked to keep as open items a full reading of Chulaevsky (2023)
   through library access and, if possible, of Nichols' 2010 thesis. Both are listed below; neither has been done.

Optional suggestions of referee 2 that were also applied: Remark 3.2 states the numerical domain-wall selection
(E−, G− for λ > 0; E+, G+ for λ < 0), and the Verification paragraph notes the Floquet-level check.

## Relation to the literature, novelty and scope
- **Searches (September 2026).** Queries are logged in `queries.log` of the working folder.
  - The finder searched arXiv, Crossref, OpenAlex, zbMATH and the web (1 search).
  - The verifier searched Semantic Scholar, OpenAlex and the web (1 search), and read the Klopp–Loss–Nakamura–Stolz
    survey arXiv:1107.0386, which treats the continuum model.
  - The lead ran Crossref, OpenAlex and arXiv checks and 1 web search.
  - Referee 2 repeated arXiv API searches (8 queries), OpenAlex (citations and keyword search), Semantic Scholar,
    Crossref and zbMATH checks and made 1 web search. Nothing new was found. One NSF-PAR PDF from the web search could
    not be downloaded (timeouts); its snippet did not concern this conjecture. The link to Nichols' thesis was
    unreachable.
- **Citing works.** Crossref reports 1 and OpenAlex 3 works citing the JST paper.
  - H. Krüger (two papers, skew-shift operators): a different question.
  - V. Chulaevsky, "Anderson localization in discrete random displacements models", J. Stat. Phys. 190 (2023),
    Paper 5, doi:10.1007/s10955-022-03020-3. According to its title and the available description (via the lead's web
    search), it concerns spectral and dynamical localization for lattice displacement models with Bernoulli
    displacements, not the almost-sure spectrum. Crossref, OpenAlex and Semantic Scholar have no abstract, there is
    no arXiv version, and the publisher's full text is behind a paywall (referee 2's request to the Springer page
    returned a bot challenge, which was not bypassed). The full text was **not** read.
- **Other related work, cited in the paper.**
  - Damanik–Sims–Stolz (JFA 2004): localization for 1D random word models.
  - Damanik–Gorodetski–Kleptsyn, arXiv:2504.08153: localization for block-factor potentials.

  Neither concerns the set Σ_λ.
- **Nichols' PhD thesis** (UAB, 2010, reference [18] of the JST paper) predates the JST paper (received 11 August
  2010), which still states the conjecture as open. It was not read.
- **Conclusion.** No proof or disproof of Conjecture 6.2 was found. The reduction and the outer bound are due to
  Nichols–Stolz, and Lemma 4.1 is standard. The contribution is the observation that closes the four remaining gaps,
  together with the deterministic inclusion. Experts may consider this easy once the reduction is known. This
  negative search is not a proof of priority.
- **Open items (not done for this version).**
  - The full text of Chulaevsky (2023) has not been read; it should be read through library access.
  - Nichols' 2010 thesis (reference [18] of the JST paper) has not been read. Neither is expected to contain a proof:
    the JST paper (received August 2010) still states the conjecture as open, and Chulaevsky's paper concerns
    localization.
  - Nichols and Stolz have not been asked whether Conjecture 6.2 was settled elsewhere.
- **Scope.** Conjecture 6.2 is settled for all λ ≠ 0 and p ∈ (0, 1). Conjecture 6.3 (IDS tails) is not addressed.
- **Suggested corpus status.** Solved (new proof in this note, pending expert confirmation).

## 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/