Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01‡ 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: the connecting morphism is defined by choosing one lift with no independence proof

Statement

The connecting morphism is defined by choosing one lift of one cycle representative, with no independence proof required.

Facts & Assumptions

Given: A short exact sequence of module complexes.

[A1]

The statement refuted is: the connecting morphism is defined by choosing one lift of one cycle representative, with no independence proof required.

[L1]

The module formula proves independence of the chosen lift and of the chosen cycle representative (Elementwise formula for the connecting map in module categories).

[L2]

The categorical construction leaves no residual choice at all once the universal-property data are fixed (The connecting morphism depends on no choices).

Refutation

technique · direct
1.1

The claim in [A1] ignores exactly the two choice-independence checks named in [L1] and the universal-property uniqueness recorded in [L2]. A single lift can at best produce one candidate value; it does not define a map on homology classes.

A1L1L2givenalgebra
2.1

Since the actual construction either proves independence of all allowed choices or avoids those choices altogether, [A1] contradicts the established definition of the connecting morphism. Therefore [A1] is false.

A1L1L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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