False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-08-29

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.

FALSE: an orientation determines an inner product

Statement

An orientation of a finite-dimensional real vector space determines an inner product on that space.

Facts & Assumptions

Given: The orientation of $R^{2}$ represented by $(e_{1}, e_{2})$ , the standard inner product $g$ , and the scaled inner product $2 g$ .

[L1]

The oriented unit volume form is attached to the pair of an orientation and an inner product (The oriented unit volume form).

[L2]

The Gram pairing on the exterior powers is built from the inner product (The Gram formula gives a well-defined positive-definite inner product on exterior powers, and $∥ v_{1} \land \dots \land v_{k} ∥^{2}$ is the Gram determinant).

Refutation

technique · direct

1.1

The two inner products $g$ and $2 g$ are different: $⟨ e_{1}, e_{1} ⟩_{g} = 1$ but $⟨ e_{1}, e_{1} ⟩_{2 g} = 2$ . However, orientation classes of ordered bases are defined purely by determinants of change-of-basis maps, with no metric input, so both metrics sit over the same orientation class of $(e_{1}, e_{2})$ .

givenalgebra

1.2

By [L1], the unit volume form for $(orientation, g)$ is $e_{1} \land e_{2}$ , while for $(orientation, 2 g)$ it is $(e_{1} / \sqrt{2}) \land (e_{2} / \sqrt{2}) = \frac{1}{2} e_{1} \land e_{2} \neq e_{1} \land e_{2}$ .

L1algebra

2.1

By [L2], the two metrics also induce different Gram pairings on the exterior powers: norms are rescaled, so the metric data is genuinely different even though the orientation is the same.

L2step 1.1

3.1

Steps 1.1, 1.2 and 2.1 exhibit two different inner products over one fixed orientation, refuting the claim.

step 1.1step 1.2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Reyer Sjamaar, Manifolds and Differential Forms, §8.3 (standard reference, not scraped)