ExampleConstruction: 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.

Determinant as the pairing of top exterior powers

Example

On $R^{n}$ with standard basis $e_{1}, \dots, e_{n}$ and dual basis $e^{1}, \dots, e^{n}$ , the pairing

⟨ e^{1} \land \dots \land e^{n}, v_{1} \land \dots \land v_{n} ⟩

is exactly $\det [v_{1} \dots v_{n}]$ .

Facts & Assumptions

Given: Vectors $v_{1}, \dots, v_{n} \in R^{n}$ .

[L1]

The exterior-power pairing on decomposable elements is the determinant of the evaluation matrix (Exterior-power duality pairing).

[L2]

The top exterior power is one-dimensional (The top exterior power is one-dimensional).

Verification

technique · direct

1.1

The matrix with entries $e^{i} (v_{j})$ is exactly the coordinate matrix $[v_{1} \dots v_{n}]$ of the ordered $n$ -tuple $(v_{1}, \dots, v_{n})$ .

given

2.1

Applying [L1] to $α_{i} = e^{i}$ gives $⟨ e^{1} \land \dots \land e^{n}, v_{1} \land \dots \land v_{n} ⟩ = \det (e^{i} (v_{j})) = \det [v_{1} \dots v_{n}] .$

L1step 1.1algebra

3.1

By [L2], this scalar determines the decomposable top wedge relative to the standard volume form.

L2step 2.1

4.1

Thus the determinant is the top-degree pairing against $e^{1} \land \dots \land e^{n}$ .

step 2.1step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

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