Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31
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: every quasi-isomorphism is a chain homotopy equivalence

Statement

Every quasi-isomorphism is a chain homotopy equivalence.

Facts & Assumptions

Given: The acyclic noncontractible complex 0→Z→2Z→ mod 2Z/2→0, and the zero map from it to the zero complex.

[A1]

The statement refuted is: every quasi-isomorphism is a chain homotopy equivalence.

[L1]

A quasi-isomorphism is a chain map inducing isomorphisms on all homology objects (Quasi-isomorphism).

[L2]

Every chain homotopy equivalence is a quasi-isomorphism (A chain homotopy equivalence is a quasi-isomorphism).

[L3]

A complex is zero in the homotopy category exactly when it is contractible (Zero homology does not make an object zero in the homotopy category).

Refutation

technique · direct
1.1L1givenalgebra

Both the source complex and the zero complex have zero homology in every degree, so the zero map between them is a quasi-isomorphism by [L1].

2.1A1L2L3step 1.1algebra∎

If that map were a chain homotopy equivalence, its source would be isomorphic to the zero object in the homotopy category. By [L3], the source would then be contractible, contrary to the explicit nonsplit example. Hence [A1] is false. This does not contradict [L2], which gives only the forward implication.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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