ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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 connection one form on a trivial line bundle

Example

For any supplied smooth one-form $a$ on $M$ , $\nabla (u e) = (d u + a u) e$ is a connection on $M \times R$ , where $e$ is the constant unit frame. Its connection form is $a$ . In particular $a = x d y$ on $R^{2}$ gives $\nabla_{\partial_{x}} (u e) = \partial_{x} u e$ and $\nabla_{\partial_{y}} (u e) = (\partial_{y} u + x u) e$ .

Facts & Assumptions

Given: A smooth one-form $a$ and the specified product line frame.

[F1]

A smooth one-form in a global line frame determines a connection (Local connection forms glue exactly when they obey the transformation law).

Verification

1.1

Apply [F1] with the one-by-one matrix $a$ . The formula is real-linear, and $d (f u) + a (f u) = d f u + f (d u + a u)$ explicitly verifies the one-form Leibniz identity. Applying it to $u = 1$ gives $\nabla e = a e$ , hence exactly the asserted coefficient.

F1given

2.1

For $a = x d y$ , evaluate on $\partial_{x}, \partial_{y}$ to obtain the displayed two derivatives. For example $u = y$ gives $\nabla_{\partial_{y}} (y e) = (1 + x y) e$ , which equals $2 e$ at $(1, 1)$ . At $x = 0$ this coefficient form vanishes but the derivative of $y$ still contributes $e$ . With $a = 0$ the connection is $d$ ; with $u = 0$ it is zero. These formulas apply to an empty or zero-dimensional base using the unique empty or zero one-form, and require no choices.

step 1.1∎

Depends on

Local connection forms glue exactly when they obey the transformation law

Used by

Nothing in the library uses this result yet.

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

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