DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Morse functions and excellent Morse functions

Definition

Let $M$ be a smooth manifold and let $f : M \to R$ be smooth.

The function $f$ is a Morse function when every critical point of $f$ is nondegenerate (Critical points and critical values of a smooth function, Nondegenerate critical points, nullity, index, and coindex).
The function $f$ is an excellent Morse function when it is Morse and any two distinct critical points have distinct critical values.

Thus excellence is stronger than the Morse condition: it excludes repeated critical values, not repeated indices.

Depends on

Dependency tree · 2 levels

Sources