ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

The hyperbolic upper half space metric

Example

For $n \geq 1$ , on ${(x_{1}, \dots, x_{n - 1}, y) : y > 0}$ the metric $g = y^{- 2} (\sum_{i < n} d x_{i}^{2} + d y^{2})$ has density $y^{- n} ∣ d x_{1} \dots d x_{n - 1} d y ∣$ .

Facts & Assumptions

Given: The upper half-space with its Euclidean open-subset smooth structure.

[F1]

Coordinate criterion for a riemannian metric: A tensor $g = \sum_{i, j} g_{i j} d x^{i} \otimes d x^{j}$ is Riemannian exactly when its coordinate matrix $G = (g_{i j})$ has smooth entries and is symmetric positive definite. Under $J = \partial x / \partial y$ it transforms by $G_{y} = J^{T} G_{x} J$ .

[F2]

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 prop-coordinate-criterion-for-a-riemannian-metric, so its determinant is positive and smooth. The density frames and their absolute-Jacobian law are def-density-bundle-and-smooth-density. 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 lem-the-riemannian-volume-density-is-coordinate-independent.

Verification

technique · direct

1.1F1given

For $y > 0$ , $y^{- 2}$ is smooth and positive. Hence $G = y^{- 2} I_{n}$ is a smooth symmetric positive-definite matrix, since $v^{T} G v = y^{- 2} \sum_{i} v_{i}^{2} > 0$ when $v \neq 0$ . It defines a Riemannian metric.

2.1F2step 1.1∎

Its determinant is $y^{- 2 n}$ and the positive square root is $y^{- n}$ . The density definition therefore gives the asserted formula. At $(0, \dots, 0, 2)$ the matrix is $\frac{1}{4} I_{n}$ and the density coefficient is $2^{- n}$ .

Source locator

Lee, p.328, coordinate positive-definiteness criterion; Proposition 15.31, p.390, volume coefficient; pp.430–431, Riemannian density. The upper-half-space coefficients are computed above; no curvature or completeness statement is asserted.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)