LemmaStatement: 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.

The annihilator ideal of a distribution is frame-independent

Statement

For a constant-rank distribution $D$ , the local ideal generated by any local frame of $D^{\circ}$ is independent of that frame and consists exactly of forms generated by one-forms vanishing on $D$ .

Facts & Assumptions

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

[F1]

The preceding result states that A differential ideal is a graded wedge ideal $I \subseteq Ω^{*} (M)$ satisfying $d I \subseteq I$ . (A differential ideal in the algebra of forms).

Proof

technique · direct

1.1

At a point, complete a local annihilator frame $θ^{1}, \dots, θ^{r}$ to a coframe. A form vanishes whenever all inputs lie in $D$ exactly when every wedge monomial contains some $θ^{a}$ .

F1given

2.1

Hence those forms constitute the ideal generated by the frame, a description independent of the frame chosen.

step 1.1∎

Depends on

Used by

The Pfaffian Frobenius criterion Theorem

Dependency tree · two levels

10 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)