Alphabeta MathHow 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.
Under the grand theorem's Choice hypothesis, plane-domain simple connectivity means any grand-equivalent clause
Before this page, the track deliberately said homologically simply connected whenever only the cycle/index criterion had been proved. After For a plane domain, the complement, homology, primitive, logarithm, conjugate, conformal, homotopy, and contractibility conditions are equivalent, and under its Axiom-of-Choice hypothesis, that caution is no longer needed for plane domains: all of the analytic, homological, homotopic, conformal, and contractibility clauses on the theorem are equivalent.
Accordingly, whenever this track invokes that theorem or a corollary derived from it, a simply connected plane domain means a complex domain satisfying any, and hence every, clause of the grand theorem under the same Axiom-of-Choice hypothesis.
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Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
15 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
- L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 4, §4.2 (standard reference, not scraped)