LemmaStatement: AI-generatedProof: 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.

Critical PSD phases: kernel inequality and half-circle rigidity

Statement

Let $G$ have $n > 0$ vertices and symmetric binary adjacency matrix $A$ , with possible loops. Let $d > 0$ , $0 \leq a < 1 / 32$ and $∥ A - (d / n) J ∥ \leq a d$ , where $J$ is the all-ones matrix. 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. Choose phase representatives in $(- π, π]$ after rotation so that $ρ = n^{- 1} \sum_{x} e^{i θ_{x}}$ is real and nonnegative. These are the hypotheses of the critical-PSD quadratic-test and mass lemma.

The off-diagonal graph has vertex set $V (G)$ and an edge ${x, y}$ precisely when $x \neq y$ and $A_{x y} = 1$ ; connectedness means that every two vertices are joined by a finite path of such edges, with a path of length zero allowed. A configuration is synchronized when all phases are equal modulo $2 π$ . For $u, v \in (- π, π]$ , put $K (u, v) = \sin (∣ u ∣ - \min (∣ v ∣, π / 2))$ . Then $\sum_{x} A_{x y} K (θ_{x}, θ_{y}) \geq 0$ for every $y$ . If the off-diagonal graph is connected and every $∣ θ_{x} ∣ < π / 2$ , all representatives equal zero under the normalization $ρ \geq 0$ . Thus a normalized nonsynchronized critical PSD state on a connected off-diagonal graph has $∣ {x : ∣ θ_{x} ∣ \geq π / 2} ∣ > 0$ .

Statement references

Critical PSD phases: quadratic tests and small-alpha mass bound

Phasors, trigonometric identities and periodic derivatives

Connected graphs and connected components defined by the existence of vertex paths

Facts & Assumptions

[F1]

Under the stated graph, energy and phase hypotheses, the critical-PSD quadratic-test and mass lemma gives $r_{y} = \sum_{x} A_{x y} \cos (θ_{x} - θ_{y}) \geq 0$ and $\sum_{x} A_{x y} e^{i θ_{x}} = r_{y} e^{i θ_{y}}$ (Critical PSD phases: quadratic tests and small-alpha mass bound). No local-minimum hypothesis is assumed.

[F2]

Phases are angles modulo $2 π$ , and $e^{i t} = \cos t + i \sin t$ for real $t$ , with $\exp (0) = 1$ (Phasors, trigonometric identities and periodic derivatives). In particular $\sin 0 = 0$ , $\cos 0 = 1$ , and real and imaginary parts commute with finite sums.

[F3]

Sine and cosine satisfy their addition formulas and the parity identities $\sin (- t) = - \sin t$ , $\cos (- t) = \cos t$ (The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine).

[F4]

Sine is strictly increasing on $[- π / 2, π / 2]$ and strictly decreasing on $[π / 2, 3 π / 2]$ ; cosine is strictly decreasing on $[0, π]$ (Signs, monotonicity intervals, and ranges of sine and cosine). The quarter-turn values include $\sin (π / 2) = 1$ , $\cos (π / 2) = 0$ and $\sin π = 0$ , and $\sin (t + π / 2) = \cos t$ (Quarter-turn values and shifts by pi/2 and pi). Sine is nonnegative on $[0, π]$ (Angular order and elementary inverse-trigonometric bounds). Together with [F2] and [F3], these facts imply $\sin t > 0$ for $0 < t < π$ , $\sin t < 0$ for $- π / 2 < t < 0$ , and $\cos t \leq 0$ for $π / 2 \leq ∣ t ∣ \leq π$ .

[F5]

A nonempty simple graph is connected when every pair of vertices can be joined by a path; the one-vertex graph permits the path of length zero (Connected graphs and connected components defined by the existence of vertex paths). This definition applies to the off-diagonal graph specified in the Statement.

Proof

Given: The graph, critical PSD configuration, representatives and normalization in the Statement.

Goal: Prove the kernel inequality and half-circle rigidity, and then its stated contrapositive.

1.1F1F2F3F4algebra

Taking real and imaginary parts of the phasor identity in [F1], both directly and after multiplication by $e^{- i θ_{y}}$ , gives $\sum_{x} A_{x y} \cos θ_{x} = r_{y} \cos θ_{y}$ and $\sum_{x} A_{x y} \sin (θ_{x} - θ_{y}) = 0$ . For $∣ θ_{y} ∣ \leq π / 2$ , put $b = ∣ θ_{y} ∣$ . Parity gives $\cos ∣ θ_{x} ∣ = \cos θ_{x}$ , and nonnegativity of sine on $[0, π]$ gives $\sin ∣ θ_{x} ∣ \geq ∣ \sin θ_{x} ∣$ . If $θ_{y} \geq 0$ , the addition formulas yield $K (θ_{x}, θ_{y}) - \sin (θ_{x} - θ_{y}) = (\sin ∣ θ_{x} ∣ - \sin θ_{x}) \cos b \geq 0$ . If $θ_{y} \leq 0$ , they yield $K (θ_{x}, θ_{y}) + \sin (θ_{x} - θ_{y}) = (\sin ∣ θ_{x} ∣ + \sin θ_{x}) \cos b \geq 0$ . Here $\cos b \geq 0$ by [F4]. Since $A_{x y} \geq 0$ , summing and using the zero sine sum proves the kernel inequality in this region.

2.1F1F3F4step 1.1algebra

If $∣ θ_{y} ∣ \geq π / 2$ , the quarter-turn and parity identities give $K (θ_{x}, θ_{y}) = \sin (∣ θ_{x} ∣ - π / 2) = - \cos θ_{x}$ . Therefore $\sum_{x} A_{x y} K (θ_{x}, θ_{y}) = - r_{y} \cos θ_{y} \geq 0$ , since $r_{y} \geq 0$ and $\cos θ_{y} \leq 0$ . This includes the boundary $∣ θ_{y} ∣ = π / 2$ and completes the kernel inequality.

3.1F4F5step 1.1step 2.1algebra

Suppose the off-diagonal graph is connected and every absolute phase is below $π / 2$ . The finite nonempty vertex set has a vertex with maximal absolute phase $c < π / 2$ . At any vertex $y$ attaining $c$ , the kernel inequality reads $\sum_{x} A_{x y} \sin (∣ θ_{x} ∣ - c) \geq 0$ . Each argument lies in $[- c, 0]$ , so each term is nonpositive, and an off-diagonal neighbor of smaller absolute phase would contribute a strictly negative term. Consequently every neighbor also attains $c$ . Along a path from $y$ to any vertex this property propagates one edge at a time; connectedness therefore gives $∣ θ_{x} ∣ = c$ for every vertex. Loops contribute zero and do not affect the argument.

4.1F2F4F5step 1.1step 3.1algebra∎

If $c > 0$ and both classes $V_{-} = {x : θ_{x} = - c}$ and $V_{+} = {x : θ_{x} = c}$ are nonempty, a path between them has an edge crossing the classes. Hence $q = \sum_{y \in V_{-}, x \in V_{+}} A_{x y} > 0$ . Sum the zero sine identities of step 1.1 over $y \in V_{-}$ : terms within $V_{-}$ vanish and every term from $V_{+}$ equals $A_{x y} \sin (2 c)$ , giving $0 = q \sin (2 c)$ . This contradicts $q > 0$ and $\sin (2 c) > 0$ for $0 < 2 c < π$ . Thus all representatives have one common value $t$ with $∣ t ∣ < π / 2$ , including $t = 0$ if $c = 0$ . Now $ρ = e^{i t}$ is real, so $\sin t = 0$ ; strict monotonicity of sine on $[- π / 2, π / 2]$ and $\sin 0 = 0$ force $t = 0$ . This proves the rigidity assertion. A nonsynchronized normalized state on a connected off-diagonal graph therefore cannot have every absolute phase below $π / 2$ , which is exactly the asserted positive cardinality.

Depends on

Used by · 2

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.