Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: 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.

The identity on F2 has no cyclic vector

Statement refuted

Every endomorphism of a finite-dimensional vector space has a cyclic vector.

Facts & Assumptions

Given: The identity endomorphism I of F2.

[L1]

An endomorphism has a cyclic vector exactly when its minimal and characteristic polynomials are equal (A cyclic vector exists exactly when the minimal and characteristic polynomials agree).

Counterexample

technique · counterexample
1.1algebra

For any v∈F2, every power Ikv equals v, so Z(v;I)=Fv has dimension at most one and cannot equal F2.

2.1L1algebra∎

Equivalently, μI=x−1 while χI=(x−1)2, so [L1] also rules out a cyclic vector. Thus the identity on F2 refutes the universal claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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