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.

A spectral window giving condition number below two

Statement

For a connected graph with positive degrees and normalized Laplacian as defined above, suppose $0 \leq η < 1 / 3$ and $∣ 1 - λ_{i} ∣ \leq η$ for $2 \leq i \leq n$ . Then $κ (G) \leq (1 + η) / (1 - η) < 2$ .

Facts & Assumptions

[F1]

The normalized Laplacian has ordered nontrivial eigenvalues $λ_{2}, \dots, λ_{n}$ and $κ (G) = λ_{n} / λ_{2}$ (Normalized graph Laplacian and its nontrivial spectrum).

Proof

Given: The objects and hypotheses in the statement.

Goal: Establish all stated claims.

1.1givenF1algebra

The assumed window gives $λ_{2} \geq 1 - η > 0$ and $λ_{n} \leq 1 + η$ . Dividing nonnegative quantities by the positive lower bound yields $κ (G) \leq (1 + η) / (1 - η)$ .

2.1givenstep 1.1algebra∎

Since $1 - η > 0$ , the inequality $(1 + η) / (1 - η) < 2$ is equivalent to $1 + η < 2 - 2 η$ , or $3 η < 1$ , which is assumed. Combining the bounds proves the claim.

Depends on

Normalized graph Laplacian and its nontrivial spectrum

Used by · 1

Uniform spectral control transfers past connectivity

Dependency tree · 2 levels

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