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 complete weighted network distinguishes a loop cap from spectral control

Statement

On $V = {1, 2}$ take $A = 2 J$ , $d_{0} = 4$ , $α = 1 / 1000$ , $c^{-} = - 1 / 100$ and $c^{+} = 1 / 100$ , where $J$ is the $2 \times 2$ all-ones matrix. Degrees and Laplacians have the network convention Finite symmetric networks and loop removal, and matrix norms and order have the vertex-space convention Finite vertex Hilbert space in the article convention. Both $∥ A - (d_{0} / 2) J ∥ \leq α d_{0}$ and $c^{-} d_{0} I ⪯ L - d_{0} I + (d_{0} / 2) J ⪯ c^{+} d_{0} I$ hold, and $c^{-} > - 1$ , $0 < α < 1 / 5$ , $c^{-} < 0 < c^{+}$ . Yet $d_{\max} = 4 > (1 + c^{+}) d_{0} + 1 - d_{0} / 2 = 76 / 25$ . The loop-dependent bound of Loop-dependent weighted degree envelope remains valid. This example concerns this degree bound only; it makes no assertion that a synchronization theorem fails.

Facts & Assumptions

[F1]

Degrees are row sums and the network Laplacian is $D - A$ (Finite symmetric networks and loop removal); finite matrix action and norms use Finite vertex Hilbert space in the article convention.

[F2]

Common positive rescaling preserves the normalized centered-matrix inequalities (Covariance under a common positive rescaling of weights).

[F3]

The upper degree envelope with arbitrary loops has $ℓ_{\max}$ in place of a fixed unit cap (Loop-dependent weighted degree envelope).

Proof

Given: The data in the Statement.

Goal: Establish the stated conclusions.

1.1F1F2algebra

Begin with $A = J$ on two vertices and $d_{0} = 2$ . Both row degrees are $2$ , so $L = 2 I - J$ and both centered matrices are zero. Apply [F2] with $λ = 2$ : the asserted $A = 2 J$ , $d_{0} = 4$ have degrees $4$ , $L = 4 I - 2 J$ , and again both residuals are zero. A zero norm satisfies the positive adjacency bound, and $- (1 / 25) I ⪯ 0 ⪯ (1 / 25) I$ follows by testing quadratic forms. All strict sign inequalities in the Statement follow immediately from the chosen rational numbers.

2.1F1F3step 1.1algebra∎

The proposed unit-cap expression is $(101 / 100) 4 + 1 - 2 = 76 / 25 < 4$ . In contrast, here $ℓ_{\max} = 2$ , so [F3] gives $d_{\max} \leq (101 / 100) 4 - 2 + 2 = 101 / 25$ , which is true. Thus the failure comes exactly from substituting a unit loop cap for an arbitrary nonnegative loop, despite exact centered spectral residuals.

Depends on

Used by · 1

Claimed extension to nonnegative weighted graphs

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.