PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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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.

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Boundary-defining functions exist locally and detect inward vectors

Statement

Every boundary point has a boundary-defining function; for any such $ρ$ , a vector $v$ is inward exactly when $d ρ_{p} (v) > 0$ .

Facts & Assumptions

Given: A boundary point $p$ of a smooth manifold $M$ , a boundary chart at $p$ , and, for the sign assertion, a boundary-defining function $ρ$ near $p$ and a vector $v \in T_{p} M$ .

[L1]

A boundary-defining function is nonnegative, vanishes exactly on the boundary, and has nonzero differential there (Boundary-defining functions).

[L2]

Inward, outward, and boundary-tangent vectors have respectively positive, negative, and zero last coordinate in a boundary chart (Inward, outward, and boundary-tangent vectors).

Proof

technique · direct

1.1

In the chosen boundary chart take $ρ = x^{n}$ . It is nonnegative, vanishes precisely on the face, and has nonzero differential, so it is a boundary-defining function by [L1].

givenL1construct

2.1

For any boundary-defining function $ρ$ , its restriction to the face is zero, so $d ρ_{p}$ vanishes on the tangent hyperplane. Its derivative in the positive normal coordinate is nonnegative because $ρ \geq 0$ on the half-space and $ρ (p) = 0$ ; by [L1] it is nonzero, hence positive. Therefore $d ρ_{p} (v)$ has the sign of the last coordinate of $v$ , and [L2] gives $d ρ_{p} (v) > 0$ exactly for inward $v$ .

givenL1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)