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.

Synchronization landscape preserved by vertex gluing

PDF p. 7: Remark 1.7, gluing assertion

Statement

Assume countable choice. Let $H$ and $H^{'}$ be finite undirected graphs with nonempty vertex sets, with homogeneous Kuramoto energies $E_{H}$ and $E_{H^{'}}$ . Identify one vertex of $H$ with one vertex of $H^{'}$ and otherwise keep their vertex sets disjoint, obtaining $G$ . Suppose each component energy has no spurious local minima and every critical point that is not a local minimum is a strict saddle. Then $E_{G}$ has no spurious local minima and every critical point that is not a local minimum is a strict saddle.

A phase state assigns an angle modulo $2 π$ to each vertex. A local minimum is a state whose energy is no greater than that of every state in some neighborhood; it is spurious if its vertex phases are not all equal. A critical point has all first angular partial derivatives zero, and it is a strict saddle if its Hessian has a strictly negative eigenvalue. Here the saddle condition is stated explicitly for all non-minimum critical points.

Statement references

Adjacency, degrees, graph Laplacian, and spectral notation

Finite phase tori, local lifts and the flat metric

Homogeneous Kuramoto energy

Equilibria, stable states, synchronized states, and spurious minima

Strict saddle critical point

The Axiom of Countable Choice (AC_ω)

Facts & Assumptions

[F1]

For a finite graph, $E (θ) = \frac{1}{2} \sum_{x, y} A_{x, y} (1 - \cos (θ_{x} - θ_{y}))$ . Its adjacency matrix is symmetric and binary, and the phase space is the finite phase torus (Adjacency, degrees, graph Laplacian, and spectral notation, Finite phase tori, local lifts and the flat metric, Homogeneous Kuramoto energy).

[F2]

For real $t$ , $e^{i t} = \cos t + i \sin t$ and $∣ e^{i t} ∣ = 1$ ; sine and cosine have the stated periodic derivatives and finite sums may be differentiated termwise (Phasors, trigonometric identities and periodic derivatives).

[F3]

For two finite connected networks meeting at exactly one vertex, the sum of two $C^{2}$ phase-difference energies inherits synchronized local minima and a negative Hessian direction at every non-minimum critical point, provided each component has those properties (One-vertex composition of phase-difference landscapes).

[F4]

A self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).

[F5]

A symmetric nonnegative adjacency matrix is a finite network. Its loop-free support graph has the same vertices and an edge ${x, y}$ exactly when $x \neq y$ and $A_{x y} > 0$ (Finite symmetric networks and loop removal). Connected components are induced by reachability via paths, equivalently walks; a graph is connected when it is nonempty and every two vertices are reachable. A one-vertex graph is connected (Connected graphs and connected components defined by the existence of vertex paths). Connectedness of a network here means connectedness of this support graph.

[A1]

The graphs and their energies satisfy the hypotheses of the Statement.

Proof

Given: The graphs, identified vertices, and energy hypotheses in the Statement.

Goal: Establish both asserted landscape properties for the glued energy.

1.1A1F1F2F5

For either graph, consider its loop-free support graph from [F5]. Every energy summand is nonnegative: by [F2], $\cos t$ is the real part of a complex number of modulus one, so $\cos t \leq 1$ . If the support graph were disconnected, fix a vertex $v$ and let $C$ be its connected component. By [F5], $C$ is nonempty and proper. No support edge joins $C$ to its complement: such an edge would extend a walk from $v$ to its other endpoint, contradicting the definition of $C$ . Assign phase $0$ on $C$ and phase $π$ on its complement. The endpoints of every nonloop edge lie on the same side, and every loop has identical endpoints. Thus every edge has equal endpoint phases and [F1] gives energy zero. This state is a global, hence local, minimum but is not synchronized, contradicting [A1]. Thus both support graphs are connected, which is the connectedness required by [F3]; loops affect neither this support graph nor the energy.

1.2A1F1F2algebra

By [F1] and [F2], each component energy is a real $C^{2}$ function of phase differences. Its Hessian in angular coordinates is real symmetric: differentiating an edge term twice gives symmetric mixed coefficients. A negative eigenvalue supplies a negative quadratic-form direction. Thus [A1] gives the component hypotheses of [F3], including a negative Hessian direction at every non-minimum critical point.

2.1F1F3step 1.1step 1.2

The vertex sets of the two component copies meet only at the identified vertex. Their edge sets are disjoint as edge sets except possibly for a loop at that vertex; such a loop contributes zero to the energy. Consequently the glued energy is the sum of the two restricted component energies. Applying [F3] with steps 1.1 and 1.2, every local minimum of $E_{G}$ is synchronized, and every non-minimum critical point has a negative Hessian direction.

3.1F1F2F4step 1.2step 2.1algebra∎

The glued energy is also a finite sum of the edge terms in [F1], so its angular Hessian is real symmetric as in step 1.2. By [F4], it has an orthonormal eigenbasis. If its quadratic form is negative on a vector, the eigenbasis expansion expresses that value as a weighted sum of eigenvalues with nonnegative squared coefficients; at least one eigenvalue must therefore be negative. The non-minimum critical points in step 2.1 are strict saddles, and its synchronized minima are precisely the non-spurious minima required.

Source notes

The gluing assertion is attributed to Expander graphs are globally synchronizing, PDF p. 7, Remark 1.7. The proof above uses the declared mathematical suppliers; the article attributes the assertion to Canale and Monzon.

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.

Sources

Expander graphs are globally synchronizing · PDF p. 7, Remark 1.7, gluing assertion