Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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: any sequence of functors with long exact sequences is a delta functor

Statement

False. Any family of functors that sends every short exact sequence to a long exact sequence is automatically a delta functor.

Facts & Assumptions

Given: The homology delta functor on complexes and one short exact sequence of complexes whose connecting map is nonzero.

[L1]

A delta functor requires naturality of the connecting maps, not only exactness of the long sequence (Homological delta functor, Cohomological delta functor).

[L2]

Homology of complexes is a genuine homological delta functor (Homology of complexes satisfies the delta-functor naturality and exactness laws).

[L3]

There exists a short exact sequence of complexes with a nonzero connecting map (A degreewise split sequence with nonzero connecting map).

Refutation

technique · direct
1.1

Start from the homology delta functor of [L2]. Keep all functors Hn and all connecting maps unchanged except on one chosen short exact sequence with nonzero connector from [L3], where replace the connecting map by its negative. Each long exact sequence stays exact, because negating one map does not change its image or kernel.

L2L3givenconstruct
2.1

By construction, the modified family still has long exact sequences, but on an isomorphism between the altered sequence and an unaltered copy the connecting square no longer commutes: one side uses ∂ and the other uses −∂, which are different because ∂≠0. Thus [L1] fails, so the modified family is not a delta functor.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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