Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Pullback carries closed forms to closed forms and exact forms to exact forms

Statement

Let F:M→N be smooth and let ω be a differential form on N. If dω=0, then d(F∗ω)=0; if ω=dη for a differential form η on N, then F∗ω=d(F∗η).

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

Exterior differentiation commutes with pullback by a smooth map (The exterior derivative commutes with pullback).

Proof

technique · direct
1.1

Naturality gives d(F∗ω)=F∗(dω), so a closed form pulls back to a closed form.

F1given
2.1

If ω=dη, the same equality applied to η reads F∗ω=d(F∗η), proving exactness preservation.

F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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