Alphabeta Math
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 trivial line bundle and its sections as functions

Example

For a smooth manifold M, the trivial line bundle M×R→M has smooth sections exactly of the form

p↦(p,f(p))

for smooth functions f:M→R.

Facts & Assumptions

Given: The trivial line bundle pr⁡1:M×R→M.

[L1]

A smooth section is a smooth map whose composition with the bundle projection is the identity (Smooth sections, local sections, and support).

Verification

technique · direct
1.1

If s:M→M×R is a section, then pr⁡1(s(p))=p, so s(p)=(p,f(p)) for a unique scalar function f:M→R.

L1given
2.1

The map s is smooth exactly when its second component f is smooth. Conversely every smooth f gives a smooth section p↦(p,f(p)). Thus sections of the trivial line bundle are the same as smooth functions on M.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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