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Regular spectral expansion implies synchronization — Theorem 1.4

PDF p. 1: Abstract; Theorem 1.4; Section 5 proof; PDF p. 7: Abstract; Theorem 1.4; Section 5 proof; PDF p. 28: Abstract; Theorem 1.4; Section 5 proof; PDF p. 29: Abstract; Theorem 1.4; Section 5 proof; PDF p. 30: Abstract; Theorem 1.4; Section 5 proof

Statement

Assume countable choice. Every $d$ -regular $(n, d, α)$ -expander with $d > 0$ and $α \leq 0.0816$ is globally synchronizing. Graphs may have loops, counted once in degrees. The explicit positive-degree condition separates the zero-degree boundary of the unrestricted extraction; the full positive-degree regular theorem and endpoint $0.0816$ are preserved.

Facts & Assumptions

[F1]

The graph and expander conventions are Adjacency spectral expander — Definition 1.3 and Adjacency, degrees, graph Laplacian, and spectral notation. The critical-PSD conclusion under exactly these hypotheses is All critical PSD equilibria synchronize on positive regular expanders.

[F2]

Under The Axiom of Countable Choice (AC_ω), Positive-semidefinite-equilibrium criterion for global synchrony implies Globally synchronizing graph — Definition 1.1 when all PSD equilibria synchronize.

[F3]

Normalized phase volume is the complete probability measure of Normalized phase volume and almost-everywhere assertions: for the quotient map $q$ and $D = (- π, π]^{V}$ , a set $B$ is phase-measurable exactly when $q^{- 1} (B) \cap D$ is Lebesgue measurable, and $μ (B) = (2 π)^{- n} λ_{n} (q^{- 1} (B) \cap D)$ .

[F4]

Under The Axiom of Countable Choice (AC_ω), every affine hyperplane ${x \in R^{n} : ⟨ u, x ⟩ = c}$ with $n \geq 1$ , $u \neq 0$ and $c \in R$ , and every subset of it, is Lebesgue measurable with measure zero (Every affine hyperplane of ℝⁿ, and hence every proper linear subspace, is Lebesgue null, claim 1).

Original claim and scope correction

The exact original extracted claim is retained as attribution:

Every $d$ -regular $(n, d, α)$ -expander with $α \leq 0.0816$ is globally synchronizing.

Proof

Given: The graph and positive-degree expansion hypotheses, and the unrestricted extracted formulation for the boundary example.

Goal: Prove global synchrony under the stated hypotheses and verify the zero-degree boundary example.

1.1F1given

Apply [F1] to the graph in the Statement. Positivity of $d$ is explicit, and the spectral expansion bound is exactly the required one, including every $α \leq 0.0816$ . Thus every equilibrium with positive-semidefinite Hessian is fully synchronized.

1.2F1F2givenalgebra

Take the edgeless graph on $n = 2$ vertices, $d = 0$ and $α = 0.0816$ . It is $0$ -regular, and $A - (d / n) J = 0$ , so the displayed expansion inequality is $0 \leq 0$ . In the homogeneous Kuramoto equation of [F2], $A = 0$ makes each rotating-frame derivative zero. Hence the rotating-frame trajectory from every initial phase is constant and converges to a fully synchronized state exactly when its initial phase is fully synchronized.

1.3F2F3F4algebra

Let $S$ be the fully synchronized states in the two-vertex phase torus. In the unique representative square $D = (- π, π]^{2}$ of [F3], two coordinates are equal modulo $2 π$ exactly when their representatives are equal: their difference lies in $(- 2 π, 2 π)$ , whose only multiple of $2 π$ is zero. Thus $q^{- 1} (S) \cap D = {(a, a) : - π < a \leq π}$ . This is a subset of the affine hyperplane $H = {(a, b) \in R^{2} : a - b = 0}$ , with nonzero normal $u = (1, - 1)$ and $c = 0$ . Applying [F4] in dimension two, under the stated countable-choice assumption, makes this representative set Lebesgue measurable with $λ_{2} (q^{- 1} (S) \cap D) = 0$ .

2.1F2step 1.1

The global-synchrony criterion [F2] applies under countable choice and yields convergence to a fully synchronized state from phase-volume almost every initial phase. This is the asserted global synchrony.

3.1F2F3step 1.2step 1.3algebra∎

By [F3] and step 1.3, $S$ is phase-measurable and $μ (S) = (2 π)^{- 2} λ_{2} (q^{- 1} (S) \cap D) = 0$ . Its complement has phase volume one. By step 1.2 this complement is exactly the set of initial phases whose rotating-frame trajectories fail to converge to a fully synchronized state. Hence global synchrony fails by [F2]. This verifies a counterexample to the unrestricted frozen extraction; it makes no claim against the positive-degree theorem.

Source notes

The original p. 7 theorem states no explicit $d > 0$ condition. Its preceding p. 6 convention chooses $d^{*} = d$ for regular graphs; calling it an “expected average degree” alone does not exclude zero. Section 2.1 on p. 11 subsequently assumes connectedness while deriving phase estimates. The edgeless example is outside that local standing assumption. Final assessment must resolve the intended standing convention before attributing an original-article defect. The counterexample above concerns the actual unrestricted frozen predicate.

The proof follows the original Section 5 two-stage amplification and uses the same exact parameters and cap comparisons. Its scalar suppliers already provide rational interval certificates rather than floating-point table readings. The new regular PSD mass bootstrap is a direct composition of the regular Hessian test and constant-vector projection; it avoids importing a local-minimum-only version of Remark 2.7. No claim of PSD implying local minimality is made.

Depends on

Used by · 1

Ramanujan and typical regular graphs synchronize for degree at least 600

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 — Abdalla, Bandeira, Kassabov, Souza, Strogatz, Townsend (2026) · PDF p. 1, Abstract; Theorem 1.4; Section 5 proof; PDF p. 7, Abstract; Theorem 1.4; Section 5 proof; PDF p. 28, Abstract; Theorem 1.4; Section 5 proof; PDF p. 29, Abstract; Theorem 1.4; Section 5 proof; PDF p. 30, Abstract; Theorem 1.4; Section 5 proof