LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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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.

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Koszul Differential Square Pairwise Cancellation

Statement

The two terms of $d^{2} (e_{I} \otimes m)$ obtained by deleting $i_{a}$ and $i_{b}$ in opposite orders have equal coefficient and opposite sign; hence $d^{2} = 0$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Differential Coordinate Formula, Exterior Multiplication Koszul Sign Rule.

Proof

technique · direct

1.1givenalgebra

Fix two deleted positions $a < b$ . Deleting $a$ then $b$ has sign $(- 1)^{a - 1} (- 1)^{b - 2}$ , while the reverse order has sign $(- 1)^{b - 1} (- 1)^{a - 1}$ ; these differ by $- 1$ and have the same scalar $x_{i_{a}} x_{i_{b}}$ .

2.1step 1.1algebra∎

Every summand of $d^{2}$ has a unique unordered pair of deleted indices, so the pairs cancel, including in characteristic $2$ where $- 1 = 1$ and the two equal terms add to $0$ .

Depends on

Used by

Dependency tree · two levels

5 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

The Stacks Project, Koszul complexes and regular sequences (standard reference, not scraped)