Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: 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.

Empty And Unit Koszul Boundaries

Example

For nonzero M, compare K(∅;M), K(0;M), and K(1;M); the first is M, the second has H0=H1=M, and the third is acyclic.

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Empty Koszul Complex Is The Coefficient Module, One Element Koszul Homology, Regular Sequence On A Module.

Proof

technique · direct
1.1

The empty complex is M; for 0 the two-term differential is zero, and for 1 it is an isomorphism.

givenalgebra
2.1

Thus H0=H1=M in the zero case and the unit case is acyclic; the unit fails the proper-quotient regularity convention.

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