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.

Rational bootstrap through the regular endpoint

Statement

Assume countable choice. Let $G = (V, E)$ be a finite connected undirected graph, with $n = ∣ V ∣ > 0$ , symmetric binary adjacency matrix $A$ , and self-loops allowed and counted once in each row sum. Suppose every row sum is $d \geq 0$ , let $α \geq 0$ , and assume $∥ Δ ∥ \leq α d$ , where $Δ = A - (d / n) J$ and $J$ is the all-ones matrix; the norm is the Euclidean operator norm. Let $θ$ be a stable state, meaning a local minimum on the phase torus of $E_{G} (θ) = \frac{1}{2} \sum_{x, y} A_{x y} (1 - \cos (θ_{x} - θ_{y}))$ . Choose representatives in $(- π, π]$ after common rotation so that $ρ_{1} \geq 0$ is real, where $ρ_{k} = n^{- 1} \sum_{x} e^{i k θ_{x}}$ . Put $X = ρ_{1}^{2}$ , $r = ∣ ρ_{2} ∣$ , and $M = n^{- 1} \sum_{x} H (θ_{x})$ , where $H (t) = \sin^{2} ∣ t ∣$ for $∣ t ∣ \leq π / 2$ and $H (t) = 1$ for $π / 2 \leq ∣ t ∣ \leq π$ .

If $0 \leq α \leq 4917 / 20000 = 0.24585$ , then $X \geq 451 / 1000$ .

Facts & Assumptions

[F1]

Regular Lemma 2.1 gives $X \geq (1 + r^{2}) / 2 - 3 α / 2$ (First and second order parameter inequality — Lemma 2.1).

[F2]

For $d > 0$ and $X > 0$ , one has $r \geq 1 - 2 α^{2} (1 - X) / X$ . For $d = 0$ , $X = 1$ (Refined H mass bound in a regular graph).

[F10]

Definition or convention used in the Statement: Adjacency, degrees, graph Laplacian, and spectral notation.

[F11]

Definition or convention used in the Statement: Adjacency spectral expander — Definition 1.3.

[F12]

Definition or convention used in the Statement: Equilibria, stable states, synchronized states, and spurious minima.

[F13]

Definition or convention used in the Statement: Daido order parameters.

[F14]

Definition or convention used in the Statement: Bounds, saturation, and rotation of order parameters.

[F15]

Definition or convention used in the Statement: Truncated sine-square function.

[F16]

Definition or convention used in the Statement: Centered graph matrices and reference degree.

[F17]

Definition or convention used in the Statement: Loewner order, Hermitian inner product, and operator norm.

[F18]

Definition or convention used in the Statement: Phasors, trigonometric identities and periodic derivatives.

[F19]

Definition or convention used in the Statement: The Axiom of Countable Choice (AC_ω).

Proof

Given: The graph, stable normalized state, parameters and notation in the Statement.

Goal: Establish all the asserted estimates.

1.1F1F2F14givenalgebra

The zero-degree case follows from [F2]. Otherwise $α \leq 0.24585 < 123 / 500$ and $α^{2} < (123 / 500)^{2} < 61 / 1000$ . Dropping $r^{2}$ in [F1] gives $X \geq 1 / 2 - 3 α / 2 > 131 / 1000$ . If $X \geq s / 1000$ for $s \geq 131$ , then $0 \leq 1 - X$ and $1 - 2 α^{2} (1 - X) / X \geq 1 - (61 / 500) (1000 - s) / s = (561 s - 61000) / (500 s) > 0$ . Hence [F1] and [F2] may be squared to give $X \geq 131 / 1000 + \frac{1}{2} ((561 s - 61000) / (500 s))^{2}$ .

2.1F1F2step 1.1algebra∎

For successive integer pairs $(s, t) = (131, 149), (149, 176), (176, 222), (222, 294), (294, 380), (380, 451)$ , the integer differences $(561 s - 61000)^{2} - 500 s^{2} (t - 131)$ are respectively $1576081, 10740421, 14597696, 20939764, 40994356, 54752400$ , all positive. Thus the last lower bound of step 1.1 exceeds $t / 1000$ for each pair. Start with $s = 131$ and apply these six inequalities consecutively to get $X \geq 451 / 1000$ , including the exact endpoint $α = 4917 / 20000$ .

Depends on

Used by · 2

Dependency tree · 2 levels

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