False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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 tangent space at a boundary point has dimension n-1

Statement

False. $T_{p} M$ has dimension $n$ at a boundary point; only $T_{p} \partial M$ has dimension $n - 1$ .

Facts & Assumptions

Given: An $n$ -dimensional smooth manifold $M$ with boundary and a point $p \in \partial M$ .

[L1]

Boundary-germ derivations form an $n$ -dimensional tangent space at every point (Tangent and cotangent bundles extend over a boundary).

[L2]

For $n \geq 1$ , $T_{p} \partial M$ is the boundary-tangent hyperplane in $T_{p} M$ (The boundary tangent space is the boundary-tangent hyperplane).

Refutation

technique · direct

1.1

By [L1], the $n$ coordinate derivations form a basis of the full tangent space $T_{p} M$ .

givenL1

2.1

By [L2], only the last-coordinate-zero span is $T_{p} \partial M$ , an $(n - 1)$ -dimensional hyperplane when $n \geq 1$ . Thus the false statement confuses the boundary tangent space with the full tangent space.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

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)