LemmaStatement: AI-adaptedProof: 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.

Vector bundle transition functions satisfy the cocycle identities

Statement

For vector bundle charts on a rank- $r$ bundle, the transition functions satisfy

$g_{α α} (p) = I_{r}, g_{γ α} (p) = g_{γ β} (p) g_{β α} (p)$

on every overlap where the expressions are defined.

Facts & Assumptions

Given: Three vector bundle charts $(U_{α}, Φ_{α})$ , $(U_{β}, Φ_{β})$ , and $(U_{γ}, Φ_{γ})$ on one rank- $r$ bundle.

[L1]

In a vector bundle chart overlap, $Φ_{β} \circ Φ_{α}^{- 1} (p, v) = (p, g_{β α} (p) v)$ with $g_{β α} (p) \in G L (r, R)$ (Vector bundle charts and transition functions).

Proof

technique · direct

1.1

On $U_{α}$ , the map $Φ_{α} \circ Φ_{α}^{- 1}$ is the identity, so [L1] gives $(p, g_{α α} (p) v) = (p, v)$ for every $v \in R^{r}$ . Hence $g_{α α} (p) = I_{r}$ .

L1given

2.1

On a triple overlap, $Φ_{γ} \circ Φ_{α}^{- 1} = (Φ_{γ} \circ Φ_{β}^{- 1}) \circ (Φ_{β} \circ Φ_{α}^{- 1})$ . Applying [L1] to $(p, v)$ gives $(p, g_{γ α} (p) v) = (p, g_{γ β} (p) g_{β α} (p) v)$ for every $v$ , so $g_{γ α} (p) = g_{γ β} (p) g_{β α} (p)$ .

L1step 1.1algebra∎

Depends on

Vector bundle charts and transition functions

Used by

Construction of a vector bundle from a smooth cocycle Theorem

Dependency tree · two levels

2 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)