LemmaStatement: 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.

Kernel stability condition — Lemma 2.8

PDF p. 16: Lemma 2.8, equation (2.14); PDF p. 17: Lemma 2.8, proof

Statement

Assume countable choice. Let $G = (V, E)$ be a finite undirected graph with nonempty vertex set, self-loops allowed, and symmetric binary adjacency matrix $A$ . A phase state assigns a real angle modulo $2 π$ to each vertex. Let $θ$ be stable, meaning a local minimum on the phase torus of $E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (θ_{x} - θ_{y}))$ , and choose representatives $θ_{x} \in (- π, π]$ . Put $n = ∣ V ∣$ and $ρ_{1} = n^{- 1} \sum_{x} e^{i θ_{x}}$ , and suppose $ρ_{1} \in [0, 1]$ . Define $K (α, β) = \sin (∣ α ∣ - \min {∣ β ∣, π / 2})$ . Then, for every $y \in V$ ,

$\sum_{x \in V} A_{x, y} K (θ_{x}, θ_{y}) \geq 0.$

Facts & Assumptions

[F1]

The adjacency entries are nonnegative, and the phase torus, energy and stable-state convention are as in the Statement (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).

[F2]

The first Daido order parameter and kernel are the functions displayed in the Statement (Daido order parameters, Kernel for stability amplification).

[F3]

A real-valued $C^{2}$ function on an open Euclidean set has zero gradient at a local minimum (Local minima and nonnegative second variation).

[F4]

Under the stated graph, stability and representative hypotheses, if $0 \leq θ_{y} \leq π / 2$ then $K (θ_{x}, θ_{y}) \geq \sin (θ_{x} - θ_{y})$ for every $x$ ; if $- π / 2 \leq θ_{y} \leq 0$ then $K (θ_{x}, θ_{y}) \geq - \sin (θ_{x} - θ_{y})$ for every $x$ . If $∣ θ_{y} ∣ \geq π / 2$ , the entire kernel sum is nonnegative (Pointwise comparisons used in kernel stability).

[F5]

Under countable choice, an equilibrium is a critical point of the energy, and at an equilibrium $\sum_{x} A_{x, y} \sin (θ_{x} - θ_{y}) = 0$ for every $y$ (The Axiom of Countable Choice (AC_ω), Equilibria, stable states, synchronized states, and spurious minima, Real and complex equilibrium identities).

[F6]

Sine and cosine are continuously differentiable, with $\sin^{'} = \cos$ and $\cos^{'} = - \sin$ ; finite sums may be differentiated termwise (Phasors, trigonometric identities and periodic derivatives).

Proof

Given: The graph, stable state, representatives, kernel and countable choice in the Statement.

Goal: Prove the displayed inequality for every vertex $y$ .

1.1F1F3F5F6

Choose a local angular chart around $θ$ as in [F1]. In this chart the energy is the finite cosine sum in the Statement, so it is a real-valued $C^{2}$ function on an open Euclidean box by [F6]. The local minimum $θ$ is an interior local minimum of this coordinate function. By [F3] its gradient vanishes; the flat angular coordinates in [F1] identify this with vanishing of the energy gradient. Thus $θ$ is an equilibrium by [F5], and [F5] gives $\sum_{x} A_{x, y} \sin (θ_{x} - θ_{y}) = 0$ for every $y$ .

2.1F1F4step 1.1algebra

Fix $y \in V$ . If $0 \leq θ_{y} \leq π / 2$ , multiply the first pointwise comparison in [F4] by the nonnegative $A_{x, y}$ and sum over $x$ . The right-hand side is zero by step 1.1, so the kernel sum is nonnegative.

3.1F1F4step 1.1step 2.1algebra∎

If $- π / 2 \leq θ_{y} \leq 0$ , the second comparison in [F4], nonnegativity of the adjacency entries and step 1.1 similarly bound the kernel sum below by zero. If $∣ θ_{y} ∣ \geq π / 2$ , its nonnegativity follows directly from [F4]. These cases together with step 2.1 cover $(- π, π]$ , including the endpoints, proving the claim. The stated condition on $ρ_{1}$ is retained from the article but is not needed for this deduction.

Source notes

Lemma 2.8, equation (2.14), of Expander graphs are globally synchronizing, PDF pages 16–17, attributes the claim. The deduction above uses the stated suppliers.

Depends on

Used by · 3

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. 16–17, Lemma 2.8, equation (2.14) and proof