Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29
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.

A non-split short exact sequence of abelian groups

Statement refuted

Every short exact sequence of abelian groups splits.

Facts & Assumptions

Given: The short exact sequence 0→Z→×2Z→qZ/2→0.

[L1]

The category Ab is abelian (Abelian groups form an abelian category).

[L2]

A short exact sequence splits exactly when the quotient map has a section (Split short exact sequence in an abelian category, Splitting lemma in an abelian category).

Counterexample

technique · direct
1.1L1givenalgebra

The sequence is short exact in Ab by the usual kernel-image computation.

1.2L2assume-hypalgebra

If q had a section s, then x:=s(1+2Z) would be an odd integer with 2x=s(0)=0, impossible in Z. So no section exists.

2.1L2step 1.1step 1.2∎

By [L2], the short exact sequence is nonsplit. This refutes the statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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