RemarkRemark: AI-adaptedProof: AI-generated

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Measure-zero convergence to strict saddles

PDF p. 3: Section 1, citation to [21, Lemma A.1]; PDF p. 4: Section 1, citation to [21, Lemma A.1]

Statement

Assume countable choice. Let $G = (V, E)$ be a finite undirected graph with nonempty vertex set, allowing self-loops, and let $A$ be its symmetric adjacency matrix. On the phase torus $T_{V} = R^{V} / (2 π Z)^{V}$ , consider the homogeneous Kuramoto energy $E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (θ_{x} - θ_{y}))$ and its downward gradient flow $Φ_{t}$ for the flat metric. A strict saddle is a critical point of $E_{G}$ whose Hessian in angular coordinates has a negative eigenvalue. Then the set of initial phases $θ$ for which $Φ_{t} (θ)$ converges, as $t \to \infty$ , to some strict saddle has normalized phase volume zero. The union is over all strict saddles, which need not be isolated.

Facts & Assumptions

[F1]

The phase torus has angular charts whose transitions are locally translations and a flat metric with Euclidean coefficients (Finite phase tori, local lifts and the flat metric). The displayed energy is a well-defined finite sum on this torus (Homogeneous Kuramoto energy).

[F2]

Sine and cosine are periodic in angular lifts, with derivatives $\cos$ and $- \sin$ respectively; finite sums may be differentiated termwise (Phasors, trigonometric identities and periodic derivatives).

[F3]

Under countable choice, the downward gradient of a smooth potential on this torus has a complete flow (The Axiom of Countable Choice (AC_ω), Gradient trajectories and complete compact phase dynamics). The rotating-frame homogeneous Kuramoto equation is the downward gradient equation for $E_{G}$ (Rotating-frame gradient formulation).

[F4]

A strict saddle of this energy is a critical point whose angular Hessian has a negative eigenvalue; this condition is independent of angular chart (Strict saddle critical point).

[F5]

Under countable choice, for a $C^{2}$ potential on the finite phase torus with complete downward gradient flow, the union of the basins of all critical points with a negative Hessian eigenvalue is phase-null (A null basin for all strict saddles of a smooth compact gradient flow). Phase-null means measurable with normalized phase volume zero (Normalized phase volume and almost-everywhere assertions).

Proof

Given: Countable choice, the graph, energy and downward gradient flow in the Statement.

Goal: Show that the set of initial phases converging to any strict saddle is phase-null.

1.1givenF1F2F3

In every angular chart the energy is a finite sum of cosine functions of linear differences of coordinates. By the differentiation rules and periodic derivatives in [F2], repeated differentiation gives derivatives of every order; the translation transitions in [F1] make these local expressions compatible. Thus $E_{G}$ is smooth, and in particular $C^{2}$ , on $T_{V}$ . By [F3], its downward gradient flow is complete.

2.1F4F5step 1.1∎

By [F4], the strict saddles in the Statement are exactly the critical points with a negative angular Hessian eigenvalue to which [F5] applies. Applying [F5] to the $C^{2}$ energy and complete flow from step 1.1 shows that the set of phases converging to any one of these points, taken as a union over all of them, is phase-null. By the phase-volume definition in [F5], it has normalized phase volume zero. The conclusion allows non-isolated strict saddles.

Source notes

The article's assertion is attributed to Expander graphs are globally synchronizing, PDF pp. 3–4. The argument above uses the stated mathematical suppliers, rather than the article as proof.

Depends on

Used by · 1

Positive-semidefinite-equilibrium criterion for global synchrony

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.

Sources

Expander graphs are globally synchronizing · PDF pp. 3–4, Section 1, citation to [21, Lemma A.1]