Alphabeta MathHow 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.
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- 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 minimal polynomial divides the characteristic polynomial,
Statement
For every endomorphism of a finite-dimensional vector space,
Facts & Assumptions
Given: A finite-dimensional endomorphism with minimal polynomial and characteristic polynomial (The basis-independent characteristic polynomial of an endomorphism of a finite-dimensional space, including in dimension zero).
Cayley–Hamilton states (Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, ).
A polynomial annihilates if and only if it is divisible by (The annihilator ideal is nonzero and has a unique monic generator; if and only if ).
Proof
By [L1], annihilates .
Therefore [L2] gives . In dimension zero both polynomials are , and the same argument applies.
Depends on
The annihilator ideal is nonzero and has a unique monic generator; if and only if
Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial,
The basis-independent characteristic polynomial of an endomorphism of a finite-dimensional space, including in dimension zero