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 euclidean metric and its musical maps

Example

On Euclidean $R^{n}$ , $v^{♭} = \sum_{i} v^{i} d x^{i}$ , $α^{♯} = \sum_{i} α_{i} \partial_{i}$ , and $grad f = \sum_{i} (\partial_{i} f) \partial_{i}$ .

Facts & Assumptions

Given: $g = \sum_{i} d x^{i} \otimes d x^{i}$ , $v = \sum_{i} v^{i} \partial_{i}$ , and $α = \sum_{i} α_{i} d x^{i}$ .

[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]

The musical maps are smooth inverse bundle isomorphisms: $♭ : T M \to T^{*} M$ and $♯ : T^{*} M \to T M$ are smooth inverse bundle isomorphisms.

[F3]

The gradient is characterized by inner products: The gradient is the unique smooth vector field $Y$ satisfying $g (Y, X) = X f$ for every smooth vector field $X$ .

[F4]

Musical isomorphisms: The musical maps are defined by $v^{♭} = g (v, \cdot)$ and by $g (α^{♯}, w) = α (w)$ for every tangent vector $w$ .

Verification

technique · direct

1.1F1F2F4given

The matrix of $g$ is $I_{n}$ , which is smooth, symmetric, and has $v^{T} I_{n} v = \sum_{i} (v^{i})^{2} > 0$ for $v \neq 0$ . For every basis vector $\partial_{j}$ , F4 gives $v^{♭} (\partial_{j}) = g (v, \partial_{j}) = v^{j}$ , hence $v^{♭} = \sum_{i} v^{i} d x^{i}$ . If $α^{♯} = \sum_{i} w^{i} \partial_{i}$ , then F4 gives $w^{j} = g (α^{♯}, \partial_{j}) = α (\partial_{j}) = α_{j}$ , hence $α^{♯} = \sum_{i} α_{i} \partial_{i}$ . Substitution in either order returns the original coefficients, as also required by F2.

2.1F3step 1.1∎

Since $d f = \sum_{i} (\partial_{i} f) d x^{i}$ , the inner-product characterization gives $grad f = (d f)^{♯} = \sum_{i} (\partial_{i} f) \partial_{i}$ . In particular $f (x) = \frac{1}{2} \sum_{i} (x^{i})^{2}$ has $d f = \sum_{i} x^{i} d x^{i}$ and $grad f = \sum_{i} x^{i} \partial_{i}$ .

Source locator

Lee, Example 13.1, p.328; musical isomorphisms and gradient, p.342. The quadratic-function instance is calculated above.

Depends on

Used by

Distance hessian in euclidean space Example

Dependency tree · two levels

8 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)
Ved Datar, Lectures on Riemannian Geometry, September 2025 (standard reference, not scraped)