CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

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A coordinate tuple is not an intrinsic tangent vector

Statement refuted

The coordinate tuple of a tangent vector is itself an intrinsic tangent vector.

Facts & Assumptions

Given: The manifold $R$ at $0$ with coordinates $x (t) = t$ and $y (t) = 2 t$ .

[L1]

Tangent bases change by the Jacobian of the coordinate transition (Change-of-coordinate formula for tangent bases).

Counterexample

technique · direct

1.1

Let $v = \partial_{x} ∣_{0}$ . In the $x$ -chart its coordinate tuple is $1$ , while [L1] gives $v = 2 \partial_{y} ∣_{0}$ , so in the $y$ -chart its coordinate tuple is $2$ .

L1given

2.1

The same intrinsic tangent vector therefore has different coordinate tuples in different charts.

step 1.1

3.1

Hence the tuple itself is chart dependent and not the intrinsic tangent vector.

step 1.1step 2.1∎

Depends on

Change-of-coordinate formula for tangent bases

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

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