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

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The volume coordinate expression changes sign under a reflection

Statement refuted

False claim: the coordinate expression of a top-degree form is unchanged by a reflection.

Facts & Assumptions

Given: The reflection $R : R^{2} \to R^{2}$ , $R (x, y) = (x, - y)$ , and the form $ω = d x \land d y$ .

[L1]

Pullback preserves wedge products (Pullback of forms is smooth functorial and preserves wedges).

Counterexample

technique · direct

1.1

The reflection satisfies $R^{*} d x = d x$ and $R^{*} d y = d (- y) = - d y$ .

givenalgebra

2.1

By [L1], $R^{*} ω = R^{*} (d x \land d y) = R^{*} d x \land R^{*} d y = d x \land (- d y) = - d x \land d y .$

L1step 1.1algebra

3.1

Therefore the top-degree form changes sign under the reflection, so the claimed invariance is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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)