Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-sol)
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.

Polynomial rings are Cohen--Macaulay

Example

Assume the Axiom of Choice. For every field k and every integer n≥0, the polynomial ring k[X1,…,Xn] is a Cohen--Macaulay ring. At the homogeneous maximal ideal m=(X1,…,Xn), the variables form a regular system of parameters of the local ring.

Facts & Assumptions

Given: The Axiom of Choice; a field is a zero-dimensional Cohen--Macaulay ring.

Verification

technique · direct
1.1given

Under the stated Choice premise, iterating Polynomial extensions of Cohen--Macaulay rings shows that k[X1,…,Xn] is globally Cohen--Macaulay.

2.1step 1.1algebra∎

In the localization at m, multiplication by Xi remains injective after quotienting by the preceding variables, and the terminal quotient is k. Thus the displayed variables are a regular parameter system of length n.

Depends on

Used by

Nothing in the library uses this result yet.

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