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.
The global Cauchy equivalences give primitives, zero periods, and holomorphic logarithms, which in turn give holomorphic roots
Remark
The genuinely new work on this page is the bridge between planar homotopy, planar complement topology, and the older homological criterion. The analytic equivalences themselves were already proved on Equivalent characterisations of a homologically simply connected domain:
- every cycle has zero period against every holomorphic function;
- every holomorphic function has a primitive;
- every nowhere-zero holomorphic function has a holomorphic logarithm.
The one-way root consequence is already present too: once the logarithm exists, A nonvanishing holomorphic function on such a domain has holomorphic roots of every positive order gives holomorphic roots of every positive order. The grand theorem below cites those results instead of restating their proofs.
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Used by
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Sources
- L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 4, §4.4 (standard reference, not scraped)
- J. Lebl, Guide to Cultivating Complex Analysis, Ch. 4, §4.3 (standard reference, not scraped)