ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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 nearfield identity block requires full rank for subunit spectral error

Example

Let $τ = {1, 2}$ have coordinates $0$ and $1$ , choose leaf threshold $b = 2$ , and use the block $τ \times τ$ . Its distance from itself is zero, so it is inadmissible and terminates as a dense near block. For $I_{2}$ , no rank-one approximation has operator-norm error below one.

Facts & Assumptions

Given: The cluster $τ$ , its two coordinates, the threshold $b = 2$ , and the identity matrix $I_{2}$ .

[F1]

Finite cluster trees and geometric admissibility: a pair is admissible only at positive distance, and the block recursion stops when both clusters are leaves.

[F2]

H-matrix block representation: near block leaves are stored as full dense matrices.

[F3]

The best rank-at-most-k approximation in operator norm is the rank-k truncation of a singular value decomposition: every rank-at-most-one matrix has operator-norm error at least the second singular value of the identity matrix.

[F4]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: the operator norm is the maximum vector stretch over unit vectors.

Verification

1.1givenF1F2algebra

Since $dist (τ, τ) = 0$ , $(τ, τ)$ is not admissible. The cluster has size $b$ and is a leaf, so its block pair is terminal and near by [F1]; [F2] therefore stores this block densely.

2.1givenF3F4algebra∎

The singular values of $I_{2}$ are $1, 1$ . Applying [F3] with rank cap one shows $∥ I_{2} - B ∥_{2} \geq 1$ for every rank-at-most-one $B$ , where [F4] identifies this norm as the maximum unit-vector stretch. Thus replacing this dense near block by a rank-at-most-one matrix cannot meet any spectral error tolerance strictly below one.

Depends on

Used by

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Dependency tree · two levels

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Sources

Steffen Börm, Lars Grasedyck, and Wolfgang Hackbusch, Hierarchical Matrices, MiS Lecture Note 21 (2003; revised June 2006), §§1.5–1.6 (standard reference, not scraped)