Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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
1.1

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

algebra
2.1

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.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 19 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources