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Regular critical PSD phases: a direct mass bootstrap at 0.0816

Statement

Let $a = 0.0816$ , and let $G$ be a finite nonempty undirected graph with symmetric binary adjacency $A$ (loops allowed), every row sum equal to $d > 0$ , and $∥ A - (d / n) J ∥ \leq a d$ . Let $θ$ be a critical point of $E = \frac{1}{2} \sum_{x, y} A_{x y} (1 - \cos (θ_{x} - θ_{y}))$ with positive-semidefinite angular Hessian. Rotate and choose representatives in $(- π, π]$ so $ρ = n^{- 1} \sum_{x} e^{i θ_{x}} \geq 0$ is real. Write $X = ρ^{2}$ , $r = ∣ n^{- 1} \sum_{x} e^{2 i θ_{x}} ∣$ , $H (t) = \sin^{2} t$ for $∣ t ∣ \leq π / 2$ and $H (t) = 1$ otherwise, and $M = n^{- 1} \sum_{x} H (θ_{x})$ . Then $X > 4 / 5$ , $M < a^{2} / 4 < 0.00832$ , and for each $y$ , $(A e^{i θ})_{y} = r_{y} e^{i θ_{y}}$ with $r_{y} \geq 0$ . In particular $∣ {x : ∣ θ_{x} ∣ \geq ξ} ∣ \sin^{2} ξ < 0.00832 n$ for $0 < ξ < π / 2$ .

Facts & Assumptions

[F1]

Use the graph, phase, algebraic and analytic interfaces of Homogeneous Kuramoto energy, Adjacency, degrees, graph Laplacian, and spectral notation, Finite phase tori, local lifts and the flat metric, Phasors, trigonometric identities and periodic derivatives, Angular order and elementary inverse-trigonometric bounds. Every configuration assumed here is a critical point with positive-semidefinite Hessian; local minimality is never inferred from that condition.

[F2]

In the open angular boxes of Finite phase tori, local lifts and the flat metric, the finite cosine sum defining the energy has continuous second derivatives. A critical point has zero coordinate gradient and its angular Hessian is the matrix of second partial derivatives (The Hessian matrix and critical points of a scalar field). Positive semidefiniteness means $h^{T} B h \geq 0$ for every real coordinate vector $h$ , where $B$ is that Hessian (Positive definite, negative definite, semidefinite, and indefinite quadratic forms). Changes between these angular lifts are locally translations, so the gradient and Hessian conventions agree on overlaps.

[F3]

Vector norms and adjoints are those of the finite-dimensional complex vertex space Finite vertex Hilbert space in the article convention; matrix norms are its induced operator norms (The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖). Scaling a nonzero vector to a unit vector gives $∥ T v ∥ \leq ∥ T ∥ ∥ v ∥$ , and the inequality also holds for $v = 0$ . Cauchy–Schwarz gives $∣ v^{*} T v ∣ \leq ∥ v ∥ ∥ T v ∥ \leq ∥ T ∥ ∥ v ∥^{2}$ (Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors). Its linear-first convention has the same norm and absolute pairing as the vertex convention.

[F4]

A real symmetric quadratic form nonnegative on real vectors is nonnegative on complex vectors: for $z = u + i v$ , $z^{*} B z = u^{T} B u + v^{T} B v$ . For real symmetric weights $b_{x y}$ , its Laplacian has quadratic form $\frac{1}{2} \sum_{x, y} b_{x y} ∣ z_{x} - z_{y} ∣^{2}$ , including signed weights (Real symmetric quadratic forms tested on phasors).

[F5]

For the finite symmetric binary adjacency in this statement, including loops, Gradient and Hessian formulas gives the angular gradient components $\sum_{x} A_{x y} \sin (θ_{y} - θ_{x})$ and the Hessian $B = diag (\sum_{x} b_{x y}) - b$ , where $b_{x y} = A_{x y} \cos (θ_{x} - θ_{y})$ . Diagonal loop terms cancel from the Hessian, and the formulas are independent of angular lifts.

Proof

Given: The hypotheses of the Statement.

Goal: Establish its conclusions directly for critical PSD phases.

1.1F1F2F4F5algebra

Apply [F5] to the energy in an open angular box. Its hypotheses hold because the vertex set is finite and nonempty and $A$ is symmetric and binary, with loops permitted. By [F2], criticality makes the gradient vanish; reversing the sine difference gives $\sum_{x} A_{x y} \sin (θ_{x} - θ_{y}) = 0$ . The Hessian is the real symmetric matrix $B = diag (\sum_{x} b_{x y}) - b$ with $b_{x y} = A_{x y} \cos (θ_{x} - θ_{y})$ . By [F4], $z^{*} B z = \frac{1}{2} \sum_{x, y} A_{x y} \cos (θ_{x} - θ_{y}) ∣ z_{x} - z_{y} ∣^{2} \geq 0$ for every complex $z$ . Its real coordinate-vector test shows $\sum_{x \neq y} A_{x y} \cos (θ_{x} - θ_{y}) \geq 0$ . Adding $A_{y y} \geq 0$ proves $r_{y} = \sum_{x} A_{x y} \cos (θ_{x} - θ_{y}) \geq 0$ , and criticality gives the phasor identity.

2.1F1F3step 1.1algebra

Set $f_{x} = e^{i θ_{x}}$ , $g_{x} = e^{2 i θ_{x}}$ and $Δ = A - (d / n) J$ . Both vectors have squared norm $n$ , so [F3] and the norm bound give $f^{*} A f \leq d n X + a d n$ , $g^{*} A g \geq d n r^{2} - a d n$ , while $1^{T} A 1 = d n$ by regularity. The PSD test $z = f$ in step 1.1, using $∣ f_{x} - f_{y} ∣^{2} = 2 (1 - \cos (θ_{x} - θ_{y}))$ , gives $\sum A_{x y} \cos (θ_{x} - θ_{y}) \geq \sum A_{x y} \cos^{2} (θ_{x} - θ_{y})$ . Using $\cos^{2} t = (1 + \cos 2 t) / 2$ yields $X \geq (1 + r^{2}) / 2 - 3 a / 2 \geq 1 / 2 - 3 a / 2 = 0.3776 > 1 / 3$ .

3.1F1F3step 1.1step 2.1algebra

Since $Δ 1 = 0$ and $∥ f - ρ 1 ∥^{2} = n (1 - X)$ , [F3] gives $∥ W ∥^{2} \leq a^{2} d^{2} n (1 - X)$ for $W = Δ f = Δ (f - ρ 1)$ . By step 1.1, $e^{- i θ_{y}} W_{y} = r_{y} - d ρ e^{- i θ_{y}}$ . Its imaginary part is $d ρ \sin θ_{y}$ . For $∣ θ_{y} ∣ \geq π / 2$ , its real part is at least $- d ρ \cos θ_{y} \geq 0$ , so the sum of its squared real and imaginary parts is at least $d^{2} X$ . Thus $∣ W_{y} ∣^{2} \geq d^{2} X H (θ_{y})$ for every $y$ , giving $M \leq a^{2} (1 - X) / X$ . With $X > 1 / 3$ from step 2.1, this is $M < 2 a^{2}$ . Put $ρ_{2} = n^{- 1} \sum_{x} e^{2 i θ_{x}}$ , so $r = ∣ ρ_{2} ∣$ . Since $Re ρ_{2} = 1 - (2 / n) \sum \sin^{2} θ_{x} \geq 1 - 2 M$ , we have $r > 1 - 4 a^{2} = 0.97336576 > 0.97$ . Inserting this into step 2.1 gives $X > (1 + {0.97}^{2}) / 2 - 0.1224 = 0.84805 > 4 / 5$ . Therefore $M < a^{2} / 4 = 0.00166464 < 0.00832$ .

4.1F1step 3.1algebra∎

On the tail $∣ θ_{x} ∣ \geq ξ$ , $H (θ_{x}) \geq \sin^{2} ξ$ directly from the piecewise definition and sine monotonicity on $[0, π / 2]$ (equivalently, cosine monotonicity there and $\sin t = \cos (π / 2 - t)$ ). Summing gives $∣ {x : ∣ θ_{x} ∣ \geq ξ} ∣ \sin^{2} ξ \leq n M < 0.00832 n$ . Every argument above uses criticality and PSD rather than a local-minimum hypothesis.

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