Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck 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: Every finite-dimensional endomorphism has Jordan form over its base field

Statement

False claim. Every endomorphism of a finite-dimensional vector space has Jordan canonical form over its base field.

Facts & Assumptions

Given: The real quarter-turn R=(0−110).

[L1]

Jordan form over F exists exactly when the characteristic polynomial splits over F (Jordan form over the base field exists exactly when the characteristic polynomial splits).

Refutation

technique · counterexample
1.1algebra

The characteristic polynomial of R is x2+1, which has no real root and does not split over R.

2.1step 1.1L1∎

By [L1], R has no Jordan form over its real base field, so the claim is false; scalar extension to C is essential in this example.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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