ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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.

The eigenvalues of a principal 2x2 submatrix interlace those of a 3x3 symmetric matrix

Example

For

$A = (\begin{array}{ccc} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{array}), B = (\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}),$

the eigenvalues of $A$ are $3, 3, 1$ and the eigenvalues of the principal submatrix $B$ are $3, 1$ , so they interlace.

Facts & Assumptions

Given: The matrices $A$ and $B$ above, where $B$ is the compression of $A$ to the coordinate hyperplane $span (e_{1}, e_{2})$ .

[L1]

Orthogonal compression to a hyperplane gives interlacing eigenvalues for a self-adjoint operator (The eigenvalues of the orthogonal compression of a self-adjoint endomorphism to a hyperplane interlace those of the original endomorphism).

Verification

technique · direct

1.1

The characteristic polynomial of $A$ is $(x - 3)^{2} (x - 1)$ , so its ordered eigenvalues are $3, 3, 1$ . The characteristic polynomial of $B$ is $(x - 3) (x - 1)$ , so its ordered eigenvalues are $3, 1$ .

algebra

2.1

These satisfy $3 = λ_{1} (A) \geq μ_{1} (B) = 3 \geq λ_{2} (A) = 3$ and $3 = λ_{2} (A) \geq μ_{2} (B) = 1 \geq λ_{3} (A) = 1$ , exactly the pattern asserted by [L1].

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Christoph Helmberg et al., An interlacing property of the signless Laplacian of threshold graphs (standard reference, not scraped)