Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-05‡ rests on unproved material (inherited)
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 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and 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: universality removes the need for supplied resolution data

Statement

False. Because derived functors are universal delta functors, one never needs to supply projective or injective resolution data in their definition.

Facts & Assumptions

Given: The derived-functor construction and its universality theorem.

[L1]

Left and right derived objects are defined relative to supplied projective or injective resolution data (Supplied projective resolution data, Supplied injective resolution data).

[L2]

The previous page proves well-definedness only after comparing different supplied data by natural isomorphism (Two supplied projective resolution data define naturally isomorphic left derived functors, Two supplied injective resolution data define naturally isomorphic right derived functors).

[L3]

Universality is a later comparison principle for already constructed delta functors (Derived functors are universal delta functors, A morphism between universal delta functors is determined in degree zero).

Refutation

technique · direct
1.1

The construction of the derived objects starts with the chosen resolution data in [L1]. Without those data there is no deleted resolution to which the functor can be applied.

L1givenalgebra
2.1

Item [L2] shows that even the basic well-definedness claim is a separate comparison theorem about different supplied data. Only after that construction work is in place does [L3] compare the resulting delta functors abstractly. Therefore universality does not remove the need for supplied resolution data at the definition stage.

L2L3step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 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