Alphabeta Math
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.

Koszul Differential Square Pairwise Cancellation

Statement

The two terms of d2(eI⊗m) obtained by deleting ia and ib in opposite orders have equal coefficient and opposite sign; hence d2=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
1.1

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 xiaxib.

givenalgebra
2.1

Every summand of d2 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.

step 1.1algebra∎

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