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 is triangularisable over its base field

Statement

False claim. Every endomorphism of a finite-dimensional vector space is triangularisable over its base field.

Facts & Assumptions

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

[L1]

An endomorphism is triangularisable over F exactly when its characteristic polynomial splits over F (T is triangularisable iff its minimal polynomial splits iff its characteristic polynomial splits).

Refutation

technique · counterexample
1.1

Direct determinant computation gives χR=x2+1. It has no root in R, so it does not split over the base field.

algebra
2.1

Fact [L1] therefore says that R is not triangularisable over R, refuting the universal claim.

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: 24 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