Alphabeta Math
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 Is Well Defined And Squares To Zero

Statement

The derivation defining d annihilates the exterior relations, so it descends to ⋀Rn⊗M, and its square is zero. Thus K(x;M) is a chain complex.

Facts & Assumptions

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, Koszul Differential Coordinate Formula, Koszul Differential Square Pairwise Cancellation.

Proof

technique · direct
1.1

The graded derivation sends ei∧ei to xiei−eixi=0 and therefore respects the alternating quotient. Its coordinate action is the stated deletion formula.

givenalgebra
2.1

The pairwise cancellation calculation proves d2=0, so the graded modules and this differential satisfy the chain-complex axioms.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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