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.
- 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.
Local finiteness, finiteness, and finite dimensionality of a simplicial complex
Definition
Let be an abstract simplicial complex.
The complex is finite if has only finitely many simplices.
It is locally finite if each vertex lies in only finitely many simplices of .
It is finite dimensional if there is an integer such that every simplex of has dimension at most .
These are distinct conditions: finite implies locally finite and finite dimensional, but local finiteness and finite dimensionality do not imply finiteness, and finite dimensionality does not imply local finiteness.