Contradiction threshold for amplification

PDF p. 23: Equation (4.2)

Assume countable choice. Let $G=(V,E)$ be a finite connected undirected graph, possibly with loops, with $n=\mathrm{\mid }V\mathrm{\mid }>0$ and symmetric binary adjacency matrix $A$. Connected means that every two vertices are joined by a finite edge path. Let $d^{\ast }\ge 0$ and $\alpha \le 1\mathrm{/}5$, and suppose $G$ is an $(n,d^{\ast },\alpha )$-expander: $\mathrm{\parallel }A-(d^{\ast }\mathrm{/}n)J\mathrm{\parallel }\le \alpha d^{\ast }$, where $J$ is the all-ones matrix and the norm is the Euclidean operator norm.

Let $\theta$ be a nonsynchronized stable state: the phases are not all equal modulo $2\pi$, and $\theta$ is a local minimum on the phase torus $(\mathbb{R}\mathrm{/}2\pi \mathbb{Z}{)}^V$ of ${\mathcal{E}}_G(\theta )=\frac{1}{2}{\sum }_{x,y\in V}{A}_{xy}(1-\cos ({\theta }_x-{\theta }_y))$. Choose representatives ${\theta }_x\in (-\pi ,\pi ]$ after a common rotation for which the first order parameter ${\rho }_1=n^{-1}{\sum }_x{e}^{i{\theta }_x}$ is real and nonnegative. For $0\le \psi \le \pi$, put ${\mathcal{C}}_{\psi }=\{x\in V:\mathrm{\mid }{\theta }_x\mathrm{\mid }\ge \psi \}$. Let $M\ge 0$ be an integer and let $\pi \mathrm{/}2={\beta }_0>\cdots >{\beta }_M\ge 0$ and $c_0<\cdots <c_M$ be real numbers with $c_0=\mathrm{\mid }{\mathcal{C}}_{\pi \mathrm{/}2}\mathrm{\mid }$ and $\mathrm{\mid }{\mathcal{C}}_{{\beta }_s}\mathrm{\mid }\ge c_s$ for $0\le s\le M$; these are an amplification sequence.

For $t\in (-\pi ,\pi ]$, define $H(t)={\sin }^2\mathrm{\mid }t\mathrm{\mid }$ when $\mathrm{\mid }t\mathrm{\mid }\le \pi \mathrm{/}2$ and $H(t)=1$ when $\pi \mathrm{/}2\le \mathrm{\mid }t\mathrm{\mid }\le \pi$. Then $c_M{\sin }^2{\beta }_M\le 5{\alpha }^{2}n\mathrm{/}2$. Consequently the additional inequality $c_M{\sin }^2{\beta }_M>5{\alpha }^{2}n\mathrm{/}2$ is incompatible with these hypotheses.

Adjacency, degrees, graph Laplacian, and spectral notation

Adjacency spectral expander — Definition 1.3

Equilibria, stable states, synchronized states, and spurious minima

Daido order parameters

Truncated sine-square function

Angle and mass sequences in the amplification argument

Phase tail sets for amplification

Finite phase tori, local lifts and the flat metric

Phasors, trigonometric identities and periodic derivatives

The Axiom of Countable Choice (AC_ω)

Equation (4.2) on PDF page 23 of the cited article states the contradiction threshold. The article citation is attribution; the proof above uses the two declared mathematical results.