Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ 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.

Ext balance isomorphisms satisfy change-of-resolution cocycle laws

Statement

If βP,I denotes the balance map attached to supplied projective data P and injective data I, then βP,I is the identity when the comparison data are unchanged, and comparison through any intermediate resolution gives the same map as direct comparison.

Facts & Assumptions

Given: Two choices of resolution data and, where needed, a third intermediate choice.

Proof

technique · direct
1.1

The direct comparison and the composite through the intermediate data are morphisms of the same Ext delta functors and both restrict to the identity on degree-zero Hom⁡.

givenconstruct
2.1

The Ext balance isomorphism is independent of resolution comparison data identifies these morphisms. The same argument with identical data gives the identity law, and The Ext balance isomorphism is natural in both variables makes the laws natural.

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

7 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