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 $F : M \to N$ be smooth and let $ω$ be a differential form on $N$ . If $d ω = 0$ , then $d (F^{*} ω) = 0$ ; if $ω = d η$ 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

1.1F1given

Naturality gives $d (F^{*} ω) = F^{*} (d ω)$ , so a closed form pulls back to a closed form.

2.1F1∎

If $ω = d η$ , the same equality applied to $η$ reads $F^{*} ω = d (F^{*} η)$ , proving exactness preservation.

Depends on

The exterior derivative commutes with pullback

Used by

Compact Kähler Hodge decomposition from elliptic harmonic forms Theorem

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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)