# Verification report — OWR-17474-010 (Bäumler, Problem 1: is |G(J)| always 2 or ∞?)

Verification date: 2026-09-29.

**Verdict.** The answer is no. On the connected, locally finite "thickened line" L(m) with m_k = (|k|+1)³ (consecutive
integers joined by m_k internally disjoint paths of length two), the zero-temperature Edwards–Anderson spin glass has
exactly four ground states almost surely, for every absolutely continuous coupling law, in particular for every
absolutely continuous law of linear growth. The note also gives a bounded-degree planar recurrent example (couplings
uniform on (0,1)), a recurrent example with couplings of both signs, and a recurrent example on which |G(J)| is a
non-degenerate random variable. The note is unrefereed. This version was revised after a second internal referee
report (2026-09-29); all eight required fixes of that report were applied (see "Independent adversarial audit").

## Statement checked
- **Primary source.** J. Bäumler, "Uniqueness and non-uniqueness for spin-glass ground states on trees", in
  Mini-Workshop: One-sided and Two-sided Stochastic Descriptions, Oberwolfach Report 11/2020, pp. 633–635,
  doi:10.4171/OWR/2020/11. The relevant pages were read in the PDF fetched from EMS Press.
  - Setting (pp. 633–634): i.i.d. absolutely continuous couplings, possibly of both signs; σ is a ground state if
    the sum of J_xy σ_x σ_y over ∂B is ≥ 0 for every finite B. Linear growth means ν((−ε,ε)) = Θ(ε).
  - Tree theorem: uniqueness ⟺ MaxFlow(0 → ∞, |J|) = 0 ⟺ SRW recurrent; |G(J)| = ∞ otherwise.
  - Problem 1 (p. 635), verbatim: "Does |G(J)| ∈ {2, ∞} hold for all graphs and all distributions of linear growth?"
  - Problem 2 (p. 635): is there a connection between random walks and ground states on more general graphs?
  - The report contains no answer. Its only 4-ground-state example is a tree with couplings uniform on (1,3), which
    is not of linear growth.
- **Broader form.** J. Bäumler, Electron. J. Probab. 24 (2019), paper no. 77, Section 4 (read in arXiv:1812.02469v2),
  on locally finite graphs. It asks whether some graph and some ν with ν((−ε,ε)) > 0 for all ε give |G(J)| not
  a.s. constant, or |G(J)| ∈ ℕ∖{2} with positive probability.
- **Corpus record.** ulamai/UnsolvedMath, OWR-17474-010 (status `open`). The record merges Problem 1 and the
  open-ended Problem 2.

## Readings
| Reading | Answer | Witness |
|---|---|---|
| all graphs, disconnected allowed | no (trivial) | two disjoint copies of ℤ: \|G(J)\| = 4 |
| connected, locally finite graphs (the setting of the EJP paper), symmetric law U(−1,1) | no | L(m), m_k = (\|k\|+1)³: \|G(J)\| = 4 a.s. (Theorem 1.3, Corollary 1.4) |
| the same, any absolutely continuous law of linear growth (indeed any absolutely continuous law) | no | the same graph (Kolmogorov–Rogozin, Theorem 1.3) |
| bounded degree, ferromagnetic law U(0,1) | no | strip Γ ⊂ ℤ², max degree 4, planar, recurrent (Theorem 1.5) |
| bounded degree, symmetric law | open | — |
| quasi-transitive graphs (e.g. ℤ^d) | open | a heuristic sketch (Remark 6.2) indicates that finite domain walls cannot occur there |
| half-plane ℤ×ℕ (not quasi-transitive; for comparison) | yes for continuous laws with support ℝ; open otherwise, e.g. for U(−1,1) | Arguin–Damron (2014) |
| EJP form, alternative "\|G(J)\| ∈ ℕ∖{2} with positive probability" | yes, occurs | Corollary 1.4 |
| EJP form, alternative "\|G(J)\| not a.s. constant" | yes, occurs | Theorem 1.6(b): P(\|G\| = 4) ≥ 47/90, P(\|G\| = 2) ≥ 0.105 |

## Results in the paper
- **Proposition 2.3 (reduction).** On any thickened line L(m,n), under generic conditions that hold a.s. for absolutely
  continuous laws, restriction to the hubs is a bijection from G(J) to the ground states of the chain ℤ with effective
  couplings K_k = Σ_i sgn(Π_{e∈P} J_e) min_{e∈P} |J_e|. So |G(J)| = 2 + 2·1{inf_k |K_k| is attained}
  (Lemmas 2.1 and 2.2).
- **Theorem 1.3.** For ν = U(−1,1) on L(m): |G(J)| = 4 a.s. if Σ m_k^{−1/2} < ∞, else 2. SRW is transient iff
  Σ_{k≥0} 1/m_k < ∞ or Σ_{k<0} 1/m_k < ∞. If Σ m_k^{−1/2} < ∞, then |G(J)| = 4 for every absolutely continuous law.
  The proof uses the exact Irwin–Hall law of K_k, a Fourier bound on its density at 0, unimodality with Chebyshev, and
  the Kolmogorov–Rogozin inequality for general laws.
- **Corollary 1.4.** m_k = (|k|+1)³ answers Problem 1 negatively; m_k = (|k|+1)² is transient with |G(J)| = 2.
- **Theorem 1.5 (strip).** Γ = {0 ≤ y ≤ ⌈(|x|+1)^{1/3}⌉} ⊂ ℤ², U(0,1): |G(J)| = 4 a.s. The proof has these steps:
  - interval structure of the columns;
  - a window bound with Hoeffding (analytic for a ≥ 10^6);
  - separation of the ends;
  - bonds are simple BOTTOM→TOP dual paths through unit squares (plane duality);
  - a length bound and a path-counting Borel–Cantelli argument.
- **Theorem 1.6.**
  - (a) U(−1,2), m_k = ⌈730 log(|k|+2)⌉: recurrent, infinitely many frustrated cycles, |G(J)| = 4 a.s.
  - (b) U(0,1), gadget 0 = K_{2,3}, gadget k ≠ 0 = j⁴+j³+1 paths of length j⁴ (j = |k|+1): recurrent,
    P(|G(J)| = 4) ≥ 47/90, P(|G(J)| = 2) ≥ 0.105.
- **Proposition 6.1.** m_k = |k|+1, U(−1,1): recurrent, |G(J)| = 2, and MaxFlow(|J|) > 0 a.s.
- **Consequences for Problem 2.** On general graphs, uniqueness does not imply recurrence; recurrence does not imply
  uniqueness (with non-symmetric laws); neither implies MaxFlow = 0.
- **Remark 6.2** is a heuristic sketch for quasi-transitive graphs; it is not claimed as a theorem.

## Computations (scripts and outputs in reproducibility/)
- **Finder** (`claimant/`, standard library):
  - Lemma 2.1 (3000 random paths), Proposition 2.3 (small windows, including literal enumeration of all flips) and
    Lemma 2.2 (all subset flips, ≤ 10 bonds), in exact rational arithmetic: 0 errors.
  - The bounds of Lemma 3.1 against the exact Irwin–Hall density (m ≤ 80) and CDF (m ≤ 60).
  - E Y = 4/27 for Theorem 1.6(a).
  - Strip minimal cuts in windows, cross-checked by brute force and max-flow (evidence).
  - Monte Carlo chain profiles (evidence).
  - All five scripts were rerun on 2026-09-29 with byte-identical output.
- **Audit (lead)** (`lead/`):
  - Lemma 5.2 for 1 ≤ x < 3000 by a constrained dual BFS: 0 violations.
  - Lemma 5.1 exhaustively on the windows [−2,2] and [−3,3] (24 and 100 bonds): all have the stated form.
  - No pendant edges for |x| ≤ 10^6.
  - The Step-3 inequality for 528 ≤ a ≤ 10^6 (not needed).
  - P(K_0 ≤ 1) = 47/90 exactly, and P(E) ≥ 0.105965 (Cantelli; exact rationals with downward rounding).
  - Monte Carlo (4000 samples, |k| ≤ 12): P(|G| = 4) = 0.543, i.e. 0.54 ± 0.01 (one standard error ≈ 0.008). The
    numerical convolution of referee 2 (below) gives P(|G| = 4) ≈ 0.5555; the Monte Carlo value is about 1.6
    standard errors low.
- **Independent verifiers** (`referee/`, written from the claim before seeing the lead's audit code):
  - exact Irwin–Hall comparison for m ≤ 150;
  - a transfer-matrix check of the ground-state structure on windows of L(m), enumerating middle spins directly;
  - a constrained dual BFS for the strip length bound (x < 1200), plus the unconstrained version that shows why the
    constraint is essential;
  - Theorem D bounds for the original (multigraph) construction.
- **Second referee** (`referee2/`, written before reading the finder's or the lead's scripts; all four scripts rerun
  on 2026-09-29 with byte-identical output):
  - Lemma 3.1(b),(c) against the exact Irwin–Hall law (max ratio 0.999062 for m ≤ 80; 0 violations of (c) for
    m ≤ 40); the constants of Theorem 1.6(a), Remark 4.1 and Proposition 6.1.
  - Theorem 1.6(b): P(K_0 ≤ 1) = 47/90 by two independent exact methods (Dirichlet integrals, polynomial
    convolution); P(E) ≥ 0.105965 in exact rationals with downward rounding.
  - Theorem 1.5: Lemma 5.1 exhaustively on the windows [−2,2], [−3,3], [−4,4] (24, 100 and 374 bonds, all simple
    BOTTOM→squares→TOP dual paths); Lemma 5.2 for 1 ≤ x ≤ 3000 (0 violations; 2946 if walks may pass through
    TOP/BOTTOM); the Step-3 and Step-6 inequalities.
  - Lemma 2.1 by brute force, a finite-window analogue of Proposition 2.3 (84 windows) and the combinatorial part of
    Lemma 2.2: 0 failures.
  - Numerical (not rigorous) evaluation of Theorem 1.6(b): P(|G| = 2) ∈ [0.4444, 0.4446], so P(|G| = 4) ≈ 0.5555.

## Independent adversarial audit
Two independent verifications (2026-09-29):

| Item | Verifier 1 | Verifier 2 |
|---|---|---|
| Source fidelity (OWR p. 635, EJP §4) | CONFIRMED (fetched anonymously; byte-identical to the saved PDF) | CONFIRMED |
| Answers the question as intended | CONFIRMED (not a misprint or literal-reading exploit) | CONFIRMED |
| Theorem A (thickened line) | CONFIRMED by hand | CONFIRMED by hand |
| Theorems B, C, D and the MaxFlow example | CONFIRMED by hand | not fully checked (asked for independent verification) |
| Novelty | no prior answer found | no prior answer found |
| Classification | PAPER_CANDIDATE | HF_CORRECTION |

Required fixes, all applied:
1. Split the corpus record. Problem 1 is answered (no). Problem 2 is open-ended, with partial answers only.
2. Theorem B, Step 5:
   - define "touches column x" precisely;
   - state that a bond is a simple dual path from BOTTOM to TOP that never re-enters the outer faces;
   - drop the vacuous pendant-edge sentence.
3. Theorem D:
   - use Cantelli;
   - state Var(min of n uniforms) = n/((n+1)²(n+2)).

   In addition, the audit found that the finder's gadget 0 (three paths of length 1) was a multigraph. It was
   replaced by K_{2,3}, and the constants were recomputed: 47/90 and 0.105 instead of 1/6 and 0.1339.
4. Label the quasi-transitive remark as a sketch.
5. State the Kolmogorov–Rogozin inequality precisely, and restrict the "every continuous law" remark to
   Σ m_k^{−1/2} < ∞.
6. Frame the paper as a short note:
   - the mechanism is elementary, and ferromagnetic laws are easy (Remark 4.1);
   - novelty is claimed only as "not found in the literature";
   - the arXiv search was rerun before release.
7. Scope statements:
   - connectedness and local finiteness, with the trivial disconnected counterexample;
   - cite the EJP §4 form alongside the OWR report;
   - list the open cases.
8. Cite White–Fisher (PRL 96 (2006) 137204) as related heuristic work.

The recurrent examples asked for by verifier 2 were verified by verifier 1 and again in this audit. The audit
re-derived every proof and checked the new length bound and the corrected Theorem D by computer.

The two verifiers saw the pre-audit version (Theorem D with a multigraph gadget 0, the finder's weaker length bound
and the older Step 5). The revised parts were checked by the audit programs and by a second referee.

### Second referee (2026-09-29)
An independent adversarial referee checked the audited version line by line and wrote its own code before reading
the finder's or the lead's scripts (`reproducibility/referee2/`). It also reran all eleven earlier release scripts
and obtained byte-identical outputs. Verdicts:

| Item | Verdict |
|---|---|
| Statement fidelity (OWR 11/2020, p. 635; EJP 2019, §4) | Correct; Problem 1 quoted verbatim, both EJP alternatives addressed |
| Proofs | Correct; every proof checked line by line, no mathematical error and no gap |
| Computations | Correct, except that the Monte Carlo value "≈ 0.54" was imprecise (true value ≈ 0.5555) |
| Novelty | Not found in the literature; the White–Fisher citation misdescribed their model |
| Presentation / house style | Good; some citations inaccurate and one sentence overstated what the verifiers checked |
| Release package | Complete and consistent; to be regenerated after the fixes |
| Fatal? | No |

All eight required fixes were applied:
1. White–Fisher described correctly (mechanism paragraph, Scope paragraph, this report and the working notes): a
   three-dimensional ferromagnet with random-sign couplings across one plane, half-space coupling of order √(L²) = L,
   four ground states, heuristic, couplings not identically distributed. The paper now says that this square-root
   mechanism is the one behind the threshold Σ m_k^{−1/2} of Theorem 1.3.
2. OpenAlex citers of the EJP paper: "two records, the arXiv and journal versions of Arguin–Hanson (ECP 2020)"
   (paper and this report).
3. Arguin–Damron qualified with "for continuous coupling laws with support ℝ" (Introduction, Open question 2, this
   report); Open question 2 states that the half-plane with, e.g., U(−1,1) couplings is not covered.
4. Monte Carlo value corrected: 0.54 ± 0.01 (4000 samples) together with ≈ 0.5555 from a numerical convolution
   (Verification item 2, this report, README row for `lead/check_theorem_D.py`).
5. Wehr (J. Stat. Phys. 87 (1997) 439–447, doi:10.1007/BF02181495) and Wehr–Woo (Ann. Probab. 26 (1998) 358–367,
   doi:10.1214/aop/1022855423) cited after Theorem 1.5 and in the Scope paragraph, with the minimal-cut/geodesic
   correspondence and the explanation that Theorem 1.5 is consistent with them because the strip is not
   quasi-transitive (nor invariant under horizontal translations).
6. Verification item 3 made accurate: the two verifiers checked the pre-audit version; the revised parts (gadget 0 =
   K_{2,3} with 47/90 and 0.105, Lemmas 5.1–5.3 as now stated) were rechecked by the audit programs and by the second
   referee, whose independent checks are listed (bond counts 24/100/374, Lemma 5.2 for x ≤ 3000, P(E) ≥ 0.105965,
   47/90 by two methods, P(|G| = 4) ≈ 0.5555).
7. Linear growth versus absolute continuity: Table 1 caption ("rows 2–6 have linear growth; row 1 holds for every
   absolutely continuous law, and not every such law has linear growth") and Corollary 1.4 ("in particular for every
   absolutely continuous law of linear growth").
8. Release regenerated: tectonic rebuild with no overfull boxes and all 13 pages inspected; main.tex, references.bib
   and paper.pdf copied to release/; referee 2's scripts added to `reproducibility/referee2/` with README rows;
   source.zip, the three zenodo/ copies and the hashes in ZENODO_METADATA.md regenerated.

Optional suggestions also applied: Newman–Stein (arXiv:2510.27507) cited in Open question 3; the component R of
Step 6 renamed Q and x_1 < x_3 renamed x_1 < x_2; the EJP section numbers flagged as those of arXiv v2; the sentence
that the proofs do not depend on the computations except for the constant 0.105 added to the Verification paragraph.
The pre-revision source is kept as `paper/main_v1_prereferee_2026-09-29.tex`.

## Relation to the literature, novelty and scope
- **Searches (September 2026).**
  - Finder: arXiv API, OpenAlex, Crossref, zbMATH and one web search.
  - Verifiers: OpenAlex, Semantic Scholar, the arXiv API (rate-limited for one verifier) and one web search each.
  - Audit (2026-09-29): three arXiv API abstract queries ("number of ground states" with spin glass; ground states,
    spin glass and recurrence or transience; ground states, Edwards–Anderson or spin glass, and "general graphs" or
    "locally finite"), an OpenAlex re-check of the citers of the EJP paper, and an OpenAlex keyword search.
  - Second referee (2026-09-29): arXiv API (ground states with spin glass and graphs; ground states with
    Edwards–Anderson; domain walls in ladders, strips or slabs; author Bäumler), OpenAlex (citers of the EJP paper;
    "number of ground states" spin glass), zbMATH, Crossref DOI checks, one web search, and the arXiv PDF of
    White–Fisher.
  - Nothing answers Problem 1 or constructs a graph with i.i.d. couplings of linear growth and finitely many but more
    than two ground states.
- **Related work.**
  - Arguin–Damron (AIHP 50 (2014) 28–62), for continuous coupling laws with support ℝ: |G(J)| is 2 or ∞ on the
    half-plane; it is a.s. constant on ℤ^d, with unknown value. The half-plane with, e.g., U(−1,1) couplings is not
    covered.
  - Wehr (J. Stat. Phys. 87 (1997) 439–447): for random ferromagnets on ℤ^d, d ≥ 2, |G| ∈ {2, ∞}. Wehr–Woo
    (Ann. Probab. 26 (1998) 358–367): |G| = 2 on the half-plane. Ferromagnetic ground states correspond to minimal
    cuts, in the plane to first-passage-percolation geodesics of the dual. Theorem 1.5 (a ferromagnetic example) is
    consistent with them because the strip Γ is not quasi-transitive (nor invariant under horizontal translations).
  - OpenAlex lists two records citing the EJP paper: the arXiv and journal versions of Arguin–Hanson (ECP 25 (2020)),
    on disorder chaos on ℤ^d.
  - Itoi (JPSJ 90 (2021) 033002) concerns a different uniqueness question.
  - Newman–Stein (arXiv:2510.27507, 2025) prove that on ℤ² the periodic-boundary-condition metastate is supported on
    one pair of spin-reversed ground states; |G(J)| on ℤ² remains open.
  - White and Fisher (PRL 96 (2006) 137204) describe, non-rigorously, a three-dimensional ferromagnet with random-sign
    couplings across one plane; the coupling between the two half-spaces is of order √(L²) = L, and they describe four
    ground states. Their couplings are not identically distributed (hence not i.i.d.). The same square-root mechanism
    is behind the threshold Σ m_k^{−1/2} of Theorem 1.3. This is a related heuristic, not an answer.
- **Novelty.** Not found in the literature. The mechanism (a unique cheapest finite domain wall on a two-ended graph
  with growing cross-section) is elementary, and experts may consider it natural. This negative search is not a
  proof of priority.
- **Scope.**
  - The counterexample of Corollary 1.4 has unbounded degrees. Bäumler's tree theorem covers all locally finite trees,
    so such graphs are within the scope of the question.
  - With bounded degree, the note has a counterexample only for a ferromagnetic law (Theorem 1.5).
  - Open: Problem 1 for bounded-degree graphs with a symmetric law; Problem 1 for quasi-transitive graphs (including
    ℤ^d); whether |G(J)| > 2 is possible on a recurrent graph with a symmetric law (a negative answer would give
    |G(J)| = 2 on ℤ²).
- **Recommended corpus status.**
  - Problem 1: solved (answered negatively; this note).
  - Problem 2: open-ended. Partial answers: transient with |G| = 2 (Corollary 1.4); recurrent with |G| = 4
    (Theorems 1.5, 1.6); MaxFlow > 0 on a recurrent graph with |G| = 2 (Proposition 6.1).
  - Recommendation: split the record, or label the merged record 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/