Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16‡ 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.

Special adjoint functor theorem, data-supplied functor form

Statement

Under either hypothesis branch of Special adjoint functor theorem, objectwise form with explicit intersection smallness or preservation data, suppose an initial object of (C↓U) is supplied for every C∈C. Then these data determine a left adjoint to U.

Facts & Assumptions

Given: The SAFT hypotheses and a supplied initial object in every comma category.

[L1]

The objectwise SAFT gives an initial object of each fixed comma category under either explicit intersection branch (Special adjoint functor theorem, objectwise form with explicit intersection smallness or preservation data).

[L2]

A supplied family of initial comma objects determines a left adjoint and its adjunction uniquely (A left adjoint exists exactly when chosen initial objects are supplied in every comma category).

Proof

technique · direct
1.1

Apply [L2] to the supplied initial comma objects. They assemble into a functor F:C→A and an adjunction F⊣U.

L1L2
2.1

The supplied family is essential data: [L1] is objectwise and does not itself choose one initial object over a proper class. With the family supplied, [L2] completes the construction.

step 1.1L1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 34 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources