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.

Nondegenerate critical points, nullity, index, and coindex

Definition

Let $f : M \to R$ be smooth and let $p$ be a critical point of $f$ . The Hessian ${Hess}_{p} (f)$ is a symmetric bilinear form on the finite-dimensional real vector space $T_{p} M$ (The intrinsic Hessian of a smooth function at a critical point).

The nullity of $p$ for $f$ is ${null}_{p} (f) : = \dim \ker ({Hess}_{p} (f)) .$
The critical point $p$ is nondegenerate when ${null}_{p} (f) = 0$ .
The index of $p$ for $f$ is the largest dimension of a subspace of $T_{p} M$ on which ${Hess}_{p} (f)$ is negative definite.
The coindex of $p$ for $f$ is the largest dimension of a subspace of $T_{p} M$ on which ${Hess}_{p} (f)$ is positive definite.

The positivity and negativity conventions are those of Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form.

Depends on

Dependency tree · 2 levels

Sources