Alphabeta Math
False statementConstruction: 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.

FALSE: extension classes automatically form a set

Statement

FALSE: for fixed objects in an arbitrary abelian category, merely defining equivalence of extensions proves that the extension classes form a set.

Facts & Assumptions

Given: An arbitrary abelian category and fixed objects M,N, with no local-smallness, essential-smallness, or enough-resolution hypothesis imposed.

Refutation

technique · direct
1.1givenalgebra

The axioms stated here do not bound the possible middle objects E by a set. Thus the definition initially supplies only a class of extensions and an equivalence relation on it. Taking a quotient of a class by an equivalence relation does not, by itself, prove that the quotient is a set.

2.1step 1.1algebra∎

The cited corollary obtains a set only after a resolution-based classification identifies the classes with a set-sized Ext⁡1(M,N). That extra smallness discharge is not a consequence of the equivalence-relation definition alone, which refutes the claim as now stated.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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