Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30‡ rests on unproved material
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.

‡ Rests on 1 statement not proved in this library. Every dependency marked ‡ below is recorded with a citation but is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

FALSE: the diagram lemmas in an abelian category follow from the module case by the embedding theorem

Statement

The diagram lemmas for an arbitrary abelian category can be proved on this page simply by reducing to the already-published module case via Freyd-Mitchell.

Facts & Assumptions

Given: The embedding-theorem route just described.

[L1]

The connecting morphism is constructed intrinsically on this page (The connecting morphism exists and is unique).

Refutation

1.1

The proposed reduction already fails at scope: Freyd-Mitchell gives a fully faithful exact functor from every small abelian category to a module category ‡ records the smallness condition on Freyd-Mitchell, so the route is not a theorem about arbitrary abelian categories.

given
2.1

Even inside that smaller scope, The library does not use Freyd-Mitchell to prove the diagram lemmas records that this library does not take the embedding-theorem route, and [L1] supplies the intrinsic construction it uses instead. Therefore the statement is false as a description of the page's proof method.

L1step 1.1∎

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