False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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 fibrewise quotient of a vector bundle by arbitrary varying subspaces is a vector bundle

Statement

The fibrewise quotient of a vector bundle by arbitrary varying subspaces is always a smooth vector bundle.

Facts & Assumptions

Given: The displayed claim.

[L1]

A quotient bundle theorem requires a smooth vector subbundle, in particular constant fibre dimension and smooth local frames (A vector bundle quotient by a subbundle is a smooth vector bundle, Vector subbundles).

Refutation

technique · direct

1.1

In the trivial line bundle $R \times R \to R$ , let $S_{x} = {0}$ for $x \neq 0$ and let $S_{0} = R$ . Then the quotient fibre is one-dimensional for $x \neq 0$ and zero-dimensional at $x = 0$ .

L1givenconstruct

2.1

A smooth vector bundle has locally constant fibre dimension, so this family of quotients cannot be a vector bundle. The missing hypothesis is exactly that the subspaces form a smooth subbundle as in [L1].

L1step 1.1algebra∎

Depends on

Used by

A rank-jumping kernel is not a vector subbundle Counterexample

Dependency tree · two levels

7 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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)