TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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 wedge product is associative and graded commutative

Statement

For alternating covectors $α \in {Alt}^{k} (V)$ , $β \in {Alt}^{ℓ} (V)$ , and $γ \in {Alt}^{m} (V)$ ,

$(α \land β) \land γ = α \land (β \land γ)$

and

$α \land β = (- 1)^{k ℓ} β \land α .$

Facts & Assumptions

Given: Alternating covectors $α, β, γ$ of degrees $k, ℓ, m$ .

[F1]

The wedge product is the normalized alternation of the tensor product, equivalently the signed shuffle sum (The wedge product of alternating covectors).

[L1]

The wedge product is alternating and bilinear (The wedge product is alternating and bilinear).

Proof

technique · direct

1.1F1L1givenalgebra

Using [F1] twice and bilinearity from [L1], both $(α \land β) \land γ$ and $α \land (β \land γ)$ are the full alternation of the multilinear tensor $α \otimes β \otimes γ$ with the same normalization factor. Hence they are equal.

1.2F1givenalgebra

In the shuffle formula of [F1], swapping the $k$ inputs destined for $α$ with the $ℓ$ inputs destined for $β$ contributes the sign of the block permutation, namely $(- 1)^{k ℓ}$ . Therefore every term of $α \land β$ matches the corresponding term of $(- 1)^{k ℓ} β \land α$ .

The wedge product is associative and graded commutative

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources