DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

Regular and critical points, regular and critical values, and level sets

Definition

Let $m, n \geq 1$ , let $U \subseteq R^{m}$ be open, and let $f : U \to R^{n}$ be $C^{1}$ .

A point $a \in U$ is a regular point of $f$ when $f$ is a submersion at $a$ , and a critical point otherwise (Submersions and immersions between Euclidean open sets).
A value $c \in R^{n}$ is a regular value when every $a \in f^{- 1} ({c})$ is a regular point. A value that is not regular is a critical value. In particular every value outside $f [U]$ is regular by vacuous truth.
The level set or fibre over $c$ is $f^{- 1} (c) : = {x \in U : f (x) = c}$ .

Regularity is a condition on the derivative at points of the fibre, not a claim that the fibre is nonempty.

Depends on

Submersions and immersions between Euclidean open sets

Used by

A regular level set is locally a Cᵏ graph of dimension m-n Corollary
A critical value can have a smooth level set Counterexample
The cone x²+y²=z² has a rank drop at its apex Counterexample
The cusp y²=x³ has a rank drop at the origin Counterexample
The tangent space to a regular level set Definition
The orthogonal group is a regular level set of dimension n(n-1)/2 Example
FALSE: a critical value must have a singular level set False statement
FALSE: every level set of a smooth map is locally a graph False statement
A constant-rank level set is locally a coordinate slice Theorem
Lagrange multipliers for a regular vector-valued level-set constraint Theorem

Dependency tree · two levels

4 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

J. M. Lee, Introduction to Smooth Manifolds, Section 8 (standard reference, not scraped)
L. W. Tu, An Introduction to Manifolds, Section 11.2 (standard reference, not scraped)