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 Append One Element Mapping Cone Identification

Statement

If y∈R, then K(x,y;M) is chain-isomorphic to the mapping cone of multiplication by y on K(x;M).

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Complex Concatenation Tensor Isomorphism, The mapping cone of a chain map.

Proof

technique · direct
1.1

Split ⋀p(Rn⊕Re) as ⋀pRn⊕e∧⋀p−1Rn. Send the second summand to the shifted cone summand.

givenalgebra
2.1

The coordinate formula gives d(e∧z)=yz−e∧dz, exactly the cone differential; hence the splitting is a chain isomorphism.

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

7 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