CounterexampleConstruction: 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.

An inward field without a negative-time flow in the half-line

Statement refuted

On $[0, \infty)$ , $X = \partial_{x}$ is inward at $0$ but has no negative-time flow through $0$ staying in the half-line.

Facts & Assumptions

Given: The half-line $M = [0, \infty)$ with boundary point $0$ and the constant smooth vector field $X = \partial_{x}$ .

[L1]

In a boundary chart with half-space coordinate increasing into the manifold, a vector is inward exactly when its last coordinate is positive (Inward, outward, and boundary-tangent vectors).

Counterexample

technique · direct

1.1

In the identity boundary chart on $M$ , the vector $X_{0} = \partial_{x}$ has coordinate $+ 1$ , so it is inward at $0$ by [L1].

givenL1algebra

2.1

Any integral curve through $0$ must satisfy $γ^{'} (t) = 1$ and $γ (0) = 0$ , hence $γ (t) = t$ . It lies in $M$ for $t \geq 0$ but not for any $t < 0$ . Therefore no flow through $0$ can be defined for negative time while remaining in the half-line.

givenstep 1.1algebra∎

Depends on

Inward, outward, and boundary-tangent vectors

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)