CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01

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.

A vector field with no pullback under a noninjective map

Statement refuted

False claim: for every smooth map $F : M \to N$ and vector field $Y$ on $N$ , there is a vector field $X$ on $M$ satisfying $d F_{p} (X_{p}) = Y_{F (p)}$ for every $p \in M$ .

Facts & Assumptions

Given: The constant map $F : R \to R^{2}$ , $F (t) = (0, 0)$ , and the constant vector field $Y = \partial / \partial x$ on $R^{2}$ .

[L1]

A general mixed tensor field does not have a pullback by every smooth map (A general mixed tensor field does not have a pullback by every smooth map).

Counterexample

technique · direct

1.1

The map $F$ is noninjective and has differential $d F_{t} = 0$ for every $t \in R$ .

given

2.1

A vector field $X$ as in the false claim would satisfy $d F_{t} (X_{t}) = Y_{F (t)}$ at each point. Step 1.1 makes that impossible, because $d F_{t} (X_{t}) = 0$ for every $X_{t} \in T_{t} R$ , whereas $Y_{F (t)} \neq 0$ .

givenstep 1.1algebra

3.1

Hence $Y$ has no pullback along $F$ , giving the announced counterexample and agreeing with [L1].

L1step 2.1∎

Depends on

A general mixed tensor field does not have a pullback by every smooth map

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Will J. Merry, Differential Geometry (standard reference, not scraped)