False statementConstruction: Literature-sourcedVerification: 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.

The exterior derivative depends on a Riemannian metric

Statement

The assertion that constructing the exterior derivative requires a Riemannian metric is false.

Facts & Assumptions

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

[F1]

The preceding result states that For $ω \in Ω^{k} (M)$ , define the candidate $d ω$ on smooth vector fields by $d ω (X_{0}, \dots, X_{k}) = \sum_{i} (- 1)^{i} X_{i} ω (X_{0}, {\hat{X}}_{i}, \dots, X_{k}) + \sum_{i < j} (- 1)^{i + j} ω ([X_{i}, X_{j}], X_{0}, {\hat{X}}_{i}, {\hat{X}}_{j}, \dots, X_{k}) .$ The next lemma proves that this candidate is a form. (The exterior derivative by the invariant vector-field formula).

Refutation

technique · direct

1.1F1given

The invariant formula uses only the form, vector fields, their action on functions, and their Lie brackets.

2.1step 1.1∎

No metric, connection, or inner product occurs in that construction, so the asserted dependence is false.

Depends on

The exterior derivative by the invariant vector-field formula

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

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