False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

FALSE: Geometric multiplicity alone determines Jordan block sizes

Statement

False claim. The geometric multiplicity of an eigenvalue determines all Jordan block sizes at that eigenvalue.

Facts & Assumptions

Given: For any $λ \in F$ , $A = J_{3} (λ) \oplus J_{1} (λ), B = J_{2} (λ) \oplus J_{2} (λ) .$

[L1]

The geometric multiplicity is the number of Jordan blocks for the eigenvalue (For split operators, Jordan blocks read off eigenspace multiplicities and both canonical polynomials).

[L2]

Ranks of shifted powers determine every Jordan block size (Ranks of shifted powers determine Jordan form up to block order).

Refutation

technique · counterexample

1.1L1algebra

Each matrix has exactly two $λ$ -blocks, so [L1] gives geometric multiplicity two for both.

1.2L2algebra

Yet $(A - λ I)^{2}$ has rank one, contributed by $J_{3} (0)^{2}$ , while $(B - λ I)^{2} = 0$ ; [L2] therefore distinguishes the block multisets ${3, 1}$ and ${2, 2}$ .

2.1step 1.1step 1.2∎

Equal geometric multiplicity has not determined the sizes, so the claim is false.

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

S. Treil, Linear Algebra Done Wrong, Chapter 9, Sections 4.3-4.4 (standard reference, not scraped)