DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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 signed one-density on a real vector space

Definition

For an $n$ -dimensional real vector space $V$ , a one-density is a real-valued function $δ$ on ordered bases such that $δ (v A) = ∣ \det A ∣ δ (v) (A \in G L (n, R)) .$ For $n \geq 1$ extend its value by zero to dependent $n$ -tuples. For $n = 0$ it is an arbitrary real scalar on the empty basis (the empty determinant is one). Write $D (V)$ for these densities. Positive means strictly positive on every basis; nonnegative includes zero. Negative scalar multiples remain densities; positivity is extra structure on their one-dimensional real space.

Depends on

The top exterior power is one-dimensional

Used by

The density line and its positive cone Proposition

Dependency tree · two levels

3 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

Lee (16.18) and Proposition 16.35, pp.428–429 (standard reference, not scraped)