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 acyclic complex is contractible

Statement

Every acyclic chain complex is contractible.

Facts & Assumptions

Given: The three-term complex $0 \to Z \overset{2}{\to} Z \overset{m o d 2}{\to} Z / 2 \to 0,$ placed in degrees $2, 1, 0$ .

[A1]

The statement refuted is: every acyclic chain complex is contractible.

[L1]

Contractible complexes are acyclic (A contractible complex is acyclic).

[L2]

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).

[L3]

Contractibility means the identity map is null-homotopic (A contractible complex).

Refutation

technique · direct

1.1givenalgebra

The displayed complex is acyclic because the image of multiplication by $2$ is the kernel of reduction modulo $2$ , and the map $Z \to Z / 2$ is surjective.

2.1A1L1L2L3step 1.1algebra∎

If the complex were contractible, then by [L3] its identity would be null-homotopic. In degree $0$ that would give a section $Z / 2 \to Z$ of the quotient map $Z \to Z / 2$ , which is impossible. Hence the complex is not contractible, so [A1] is false; [L1] and [L2] explain the genuine implication and its homotopy-category meaning.

Depends on

Used by

fs-the-derived-category-is-the-same-category-as-the-homotopy-category.md False statement

Dependency tree · two levels

9 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

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)