DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Riemannian volume density

The Riemannian volume density is $μ_{g} = \sqrt{\det G_{x}} ∣ d x^{1} \dots d x^{n} ∣$ in coordinates.

The matrix is that of Coordinate criterion for a riemannian metric, so its determinant is positive and smooth. The density frames and their absolute-Jacobian law are Density bundle and smooth density fields. In dimension zero take the empty determinant to be one, giving weight one at every point, independently of orientation. The compatibility of these local formulas is proved in The riemannian volume density is coordinate independent ↗.

Lee, Propositions 15.29–15.33 and Corollary 15.34, pp.389–391; density construction and integration pp.428–433.

Depends on