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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Historical regular-graph synchronization question and subsequent negative-resolution claims

PDF p. 39: Section 7, first paragraph

Historical question

For every fixed d≥3, is a uniformly random d-regular graph globally synchronizing with high probability as the number of vertices grows?

Here graphs follow Adjacency, degrees, graph Laplacian, and spectral notation, the uniform labelled simple model follows Uniform random regular graph convention, and high probability means probability tending to one through feasible orders as in Asymptotic quantifiers and uniform parameter ranges. Global synchronization has exactly the almost-every-initial-state rotating-frame meaning in Globally synchronizing graph — Definition 1.1.

Source history

The question and positive expectation are preserved from the original article, Section 7, PDF p. 39, first paragraph. That passage associates d≥3 with the regular-graph connectivity threshold and describes d≥66 as a limitation of its starting estimates. These are historical source reports, not a current claim that the question is open.

Colbrook's manuscript dated 11 September 2026, Theorem 1 (p. 2), claims that, for uniformly random labelled simple cubic graphs on even n, the probability that every local minimum of the homogeneous Kuramoto energy is synchronized tends to zero. Its stronger assertion supplies, with probability tending to one, a nonsynchronized critical point with positive edge cosines and Hessian positive definite on the mean-zero subspace. The repository labels this a negative resolution and reports independent agent checks; its submission record explicitly distinguishes those checks from external human peer review and formal certification. Missing supplementary programs are disclosed there and are not claimed to have been run here.

Carvallo, Groisman and Mitsche's arXiv:2610.09135v1 (6 October 2026), Theorem 1.2 (p. 2), claims directly that random simple cubic graphs are not globally synchronizing with probability tending to one. The version-specific record inspected here is v2, withdrawn on 10 October 2026. Its stated reason is the prior GitHub solution and lack of added value. Withdrawal for that stated reason is neither a disproof nor an independent verification of either argument; the retained v1 must not be presented as an unwithdrawn latest paper.

Applicability and verification limits

For the dynamical comparison assume countable choice, as in The Axiom of Countable Choice (AC_ω) and Globally synchronizing graph — Definition 1.1. The cubic manuscript's graph law, even-order limit, phase torus and unit edge couplings agree with the cubic specialization of the historical question. Its unordered-edge energy equals EG in Homogeneous Kuramoto energy: in a simple graph each edge appears twice in that definition's ordered-pair sum, cancelling the factor 1/2. The energy is smooth, being a finite sum of smooth cosine differences, and invariant under common rotation because those differences are unchanged. By Rotating-frame gradient formulation, its negative gradient is exactly the homogeneous rotating-frame dynamics.

The manuscript's stronger assertion gives a nonsynchronized critical point q with Hessian positive on 1⊥, rather than merely an arbitrary local minimum. Common rotation preserves every phase difference, so no rotation of q is synchronized. Thus the smooth rotation-invariant energy and q meet the hypotheses of A positive transverse Hessian gives an open nonsynchronous basin: the nonsynchronous rotation orbit attracts a nonempty open set of positive phase volume. This obstructs the almost-every-initial-state convergence in Globally synchronizing graph — Definition 1.1. Consequently, if the asserted stronger existence conclusion is established, it addresses the intended dynamical question; the weaker bare local-minimum wording alone should not be silently substituted for it.

The manuscript's Sections 2–4 and the actual statements/model assumptions of its Bordenave and Johnson inputs were inspected. This status note does not certify a complete independent proof, reconstruct those external theorems, or establish the existence claim as a logical supplier. Independent scientific acceptance of the negative-resolution arguments remains unresolved here. The accurate present description is a historical question with explicit, applicable negative-resolution claims: neither continued worldwide openness nor independently established resolution is asserted by this note. A cubic negative resolution, if established, answers the universal d≥3 expectation negatively; it gives no conclusion for each remaining degree.

Source notes

The statement-bearing source records support attribution and status reporting; their negative-resolution claims are not adopted as theorem premises. The conditional dynamical implication uses the declared mathematical suppliers. The original question remains visible in full. This remark supplies no proof of the conjecture or its negation and is not an open-problem proof supplier. The original article is attribution only; no error in it is asserted from a later claimed resolution.

Depends on

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Matthew J. Colbrook, Random cubic graphs admit nonsynchronized stable Kuramoto equilibria (11 September 2026) · Theorem 1, p. 2; Lemma 2 and Sections 3–4, pp. 2–6; packaging note p. 1

Theorem 1 claims that for uniformly random labelled simple cubic graphs on even n at least four, the probability that every local minimum of the unit-coupling Kuramoto energy is synchronized tends to zero. More strongly, with probability tending to one there is a nonsynchronized critical point with uniformly positive edge cosines and Hessian positive definite on the mean-zero subspace.

Hypotheses: Uniform labelled simple 3-regular graph on {1,...,n}, even n at least four, n tending to infinity through even orders; phases in (R/(2 pi Z))^n; energy summed once over unordered edges, with unit attractive couplings. The stronger conclusion concerns the Hessian transverse to common rotations.

Reading limits: The supplied eight-page manuscript text was read for the theorem, energy convention, construction and scope; page 2 was visually checked. Sections 2–4 and their cited interfaces were inspected for this status comparison, without a complete independent proof acceptance. Supplementary programs were not executed; no external peer review or formal certification is asserted.

MD-06 repository status and submission record · Resolution dated 11 September 2026; linked submission record

The repository labels MD-06 solved with a negative resolution dated 11 September 2026, reports two independent agent reviews, and expressly distinguishes these from external human peer review and formal proof-assistant certification.

Hypotheses: A report of the repository’s status and review claims for the uniform labelled simple cubic graph problem, not an adoption of its proof or review verdicts.

Reading limits: The repository README and its linked submission record were read via the live pages for this assignment. Their agent-review claims are attributed reports only; no review verdict or program execution is inherited.

Carvallo, Groisman and Mitsche, Random 3-regular graphs are not globally synchronizing, v1 · Definition 1.1 and Theorem 1.2, pp. 1–2

Definition 1.1 uses convergence to synchrony from a uniform phase state with probability one. Theorem 1.2 claims that the probability that a uniformly random simple 3-regular graph on 2n labelled vertices is not globally synchronizing tends to one.

Hypotheses: Uniform simple cubic graph on an even number 2n of labelled vertices, n tending to infinity; homogeneous Kuramoto dynamics with unit couplings on the phase torus, in a rotating frame. This describes the retained v1 claim of a subsequently withdrawn paper.

Reading limits: The supplied v1 PDF extracted pages 1–3 were read for the model, definition, theorem and overview; the page 2 theorem was visually checked. The rest of its proof was not checked.

arXiv:2610.09135v2 withdrawal record · Submission history and Comments, 10 October 2026

The record lists v1 submitted on 6 October 2026 and v2 withdrawn on 10 October 2026; the authors give an existing GitHub solution and lack of added value as their reason. Version 2 has no PDF.

Hypotheses: Version-specific arXiv status and author-provided withdrawal explanation, not a mathematical error finding or a correctness certificate.

Reading limits: The live version-specific arXiv record was read for submission dates, withdrawn status, lack of a v2 PDF and stated reason. No correctness guarantee or identified mathematical error follows from this record.

Bordenave, A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts · v4, p. 2, uniform simple graph model and Theorem 1

For fixed integer d at least three, a uniform simple d-regular graph has max(mu_2,abs(mu_n)) below 2 sqrt(d-1)+epsilon with probability tending to one for each epsilon>0, through orders with nd even (Theorem 1).

Hypotheses: The distribution is uniform on simple d-regular graphs on {1,...,n}; d is fixed, epsilon>0 is fixed, nd is even, and the graph space is nonempty. Only the interface cited by Colbrook is compared, not imported here as an accepted theorem premise.

Reading limits: The supplied v4 extracted pages 1–2 were read for the uniform simple model and Theorem 1, in comparison with Colbrook Section 4. The proof was not reconstructed; this is status bibliography only.

Johnson, Exchangeable pairs, switchings, and random regular graphs · v5, p. 3, Section 2 and Proposition 1(a); p. 12, Theorem 11

Proposition 1(a) bounds the probability that a fixed subgraph H of K_n with minimum degree at least two and e edges is present by c_1(d-1)^e/n^e. Theorem 11 bounds total variation between cycle counts through r and independent Poisson variables of means (d-1)^k/(2k) by c_6 sqrt(r)(d-1)^(3r/2-1)/n.

Hypotheses: Uniform simple d-regular graph; parity and feasibility as in Section 2. Proposition 1(a) requires d<=n^(1/3) and e<=2 n^(1/10). Theorem 11 states n>=1 and integers r,d>=3 whenever the model exists. Colbrook uses d=3 and fixed subgraph sizes and fixed cycle lengths before the graph order grows. Interface comparison only, not a theorem import.

Reading limits: The supplied v5 extracted page 3 (Section 2, Proposition 1(a)) and page 12 (Theorem 11) were read for model and precise hypotheses, in comparison with Colbrook Section 4. The proof was not reconstructed; this is status bibliography only.

Colbrook MD-06 submission record: attribution, review boundaries and missing files · Results and boundaries; Original package and missing files

The record attributes the manuscript to Matthew J. Colbrook and reports agent checks separately from external human peer review or formal certification. It discloses absent supplementary programs and declines to adopt claims that those files were supplied or their tests ran; they are not premises of the asymptotic proof.

Hypotheses: A report of the submission package and its separately described review evidence; no reconstruction of missing files, novelty claim, or independent acceptance is implied.

Reading limits: The supplied submission-record Markdown and the linked live record were read for attribution, review boundaries and missing-file disclosures. Review reports and programs were not independently executed or accepted.

Sources