ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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 tautological line bundle over real projective space

Example

The tautological line bundle over ${R P}^{n}$ is

$γ^{1} : = {([x], v) \in {R P}^{n} \times R^{n + 1} : v \in R x} \to {R P}^{n},$

with projection to the first factor.

Facts & Assumptions

Given: The standard affine cover $U_{i} = {[x] : x_{i} \neq 0}$ of ${R P}^{n}$ with its usual affine coordinates.

[L1]

A smooth cocycle on a countable cover constructs a smooth vector bundle (Construction of a vector bundle from a smooth cocycle).

Verification

technique · direct

1.1

On $U_{i}$ , every line $[x]$ has a unique representative with $i$ -th coordinate $1$ . Let $s_{i} ([x])$ be that representative. Then $s_{i} ([x])$ spans the fibre of $γ^{1}$ over $[x]$ , so $s_{i}$ is a local frame of the tautological line bundle on $U_{i}$ .

givenconstruct

2.1

On $U_{i} \cap U_{j}$ , the two generators satisfy $s_{j} = \frac{x_{i}}{x_{j}} s_{i}$ , because both are rescalings of the same representative vector $x$ . Therefore a vector with $s_{i}$ -coordinate $a$ has $s_{j}$ -coordinate $(x_{j} / x_{i}) a$ , so the chart transition $g_{j i}$ is the nonzero smooth scalar $x_{j} / x_{i}$ . By [L1], these transition functions define a smooth line bundle, namely $γ^{1}$ .

L1step 1.1givenalgebra∎

Depends on

Used by

Pullback of the tautological line bundle along the antipodal cover Example

Dependency tree · two levels

9 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Rob van der Vorst, Introduction to differentiable manifolds (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)