Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17
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.

False statement: every module has a composition series

Facts & Assumptions

Given: The hypotheses and objects in the false claim.

[L1]

A module with a composition series is both Noetherian and Artinian. The zero module has the empty composition series. (A module has a composition series if and only if it is Noetherian and Artinian, the converse using dependent choice).

[L2]

The regular Z-module Z is Noetherian but not Artinian. (Z is Noetherian but not Artinian as a module over itself).

Refutation

technique · direct
1.1L1L2givenalgebra

The module Z over itself is Noetherian but not Artinian, so the equivalence theorem rules out a composition series.

2.1step 1.1L2givenalgebra∎

The failure is exhibited by an explicit chain rather than only by the equivalence: the submodules Z>2Z>4Z>⋯>2kZ>⋯ are strictly descending and never stabilize, so Z is not Artinian. The false claim is therefore refuted by a single module, not merely restricted in scope.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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