CorollaryStatement: Literature-sourcedProof: Literature-sourcedjudge 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.

The Morse index detects local extrema and saddles

Statement

Let $f : M \to R$ be smooth and let $p$ be a nondegenerate critical point of index $λ$ on an $n$ -manifold.

If $λ = 0$ , then $p$ is a strict local minimum of $f$ .
If $λ = n$ , then $p$ is a strict local maximum of $f$ .
If $0 < λ < n$ , then $p$ is a saddle point of $f$ .

When $n = 0$ , the first two clauses coincide.

Facts & Assumptions

Given: A smooth function $f : M \to R$ and a nondegenerate critical point $p$ of index $λ$ .

[L1]

Morse coordinates put $f - f (p)$ into the signed quadratic normal form with exactly $λ$ negative squares (Morse lemma).

Proof

technique · normal form reading

1.1

By [L1], choose local coordinates centered at $p$ in which $f - f (p) = - \sum_{i = 1}^{λ} (x^{i})^{2} + \sum_{i = λ + 1}^{n} (x^{i})^{2}$ .

L1givenconstruct

2.1

If $λ = 0$ , the first sum is empty, so $f - f (p) = \sum_{i = 1}^{n} (x^{i})^{2}$ , which is strictly positive for every nearby point other than $p$ ; hence $p$ is a strict local minimum. If $n = 0$ , this same formula is $f - f (p) = 0$ , so the local minimum and maximum clauses coincide.

step 1.1algebra

2.2

If $λ = n$ , the second sum is empty, so $f - f (p) = - \sum_{i = 1}^{n} (x^{i})^{2}$ , which is strictly negative away from $p$ . Hence $p$ is a strict local maximum.

step 1.1algebra

2.3

If $0 < λ < n$ , then along the $x^{1}$ -axis one has $f - f (p) = - (x^{1})^{2} < 0$ for nearby nonzero points, while along the $x^{n}$ -axis one has $f - f (p) = (x^{n})^{2} > 0$ for nearby nonzero points. Therefore every neighbourhood of $p$ contains points where $f < f (p)$ and points where $f > f (p)$ , so $p$ is a saddle.

step 1.1algebra

3.1

The three cases above exhaust the possible values of $λ$ .

step 2.1step 2.2step 2.3∎

Depends on

Morse lemma

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Michele Audin and Mihai Damian, Morse Theory and Floer Homology (standard reference, not scraped)
Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (standard reference, not scraped)