ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31

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 $2 \times 2$ Jordan block splits into two eigenvalues separated by a square root

Example

At $λ = 0$ , the defective family

$A_{ε} = (\begin{array}{cc} 0 & 1 \\ ε & 0 \end{array})$

has characteristic polynomial $z^{2} - ε$ , so its eigenvalues are $\pm \sqrt{ε}$ .

Facts & Assumptions

Given: The perturbed Jordan block above.

[L1]

A defective Jordan block can split at square-root scale (A defective Jordan block can split under perturbation at square-root scale).

Verification

technique · direct

1.1

Computing the determinant of $z I - A_{ε}$ gives $z^{2} - ε$ , so the roots are $\pm \sqrt{ε}$ .

algebra

2.1

The separation between the two eigenvalues is therefore $2 \sqrt{ε}$ , not a quantity linear in $ε$ . This is exactly the square-root splitting described in [L1].

L1step 1.1∎

Depends on

A defective Jordan block can split under perturbation at square-root scale

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Benjamin Texier, Basic matrix perturbation theory (standard reference, not scraped)

A 2×22×2 Jordan block splits into two eigenvalues separated by a square root