LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 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.

Koszul Differential Coordinate Formula

For $I = (i_{1} < \dots < i_{p})$ , $d (e_{I} \otimes m) = \sum_{j = 1}^{p} (- 1)^{j - 1} e_{I ∖ i_{j}} \otimes x_{i_{j}} m$ .

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Complex Of A Sequence With Coefficients, Exterior Algebra Basis Monomials.

technique · direct

1.1givenalgebra

Apply the graded Leibniz rule to the ordered wedge and use $d (e_{i_{j}}) = x_{i_{j}}$ .

2.1step 1.1algebra∎

Passing the differential through $j - 1$ degree-one factors gives $(- 1)^{j - 1}$ , yielding the formula.

Depends on