DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Closed and exact differential forms

Definition

For the complex De rham cochain complex, put $Z^{k} (M) = \ker (d : Ω^{k} (M) \to Ω^{k + 1} (M))$ and $B^{k} (M) = im (d : Ω^{k - 1} (M) \to Ω^{k} (M))$ . A form is closed if it belongs to $Z^{k}$ and exact if it belongs to $B^{k}$ .

If $ω = d η$ , then $d ω = d^{2} η = 0$ by The exterior derivative squares to zero, so $B^{k} \subseteq Z^{k}$ . In particular $B^{0} = 0$ , since $Ω^{- 1} = 0$ . The zero form is both closed and exact in every degree.

Source locator

Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 17, pp.441–443, Proposition 17.2 and Corollary 17.3; local quotient calculations below.

Closed and exact differential forms

Definition

Source locator

Depends on

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Sources