CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

A locally bounded meromorphic quotient has no genuine pole

Statement

Let $U \subseteq C^{m}$ be a domain, let $g, h : U \to C$ be holomorphic with $g \equiv̸ 0$ , and suppose the quotient $h / g$ is locally bounded on $U ∖ Z (g)$ near every point of $Z (g)$ . Then $h / g$ extends holomorphically to all of $U$ .

Facts & Assumptions

Given: The domain $U$ , holomorphic functions $g$ and $h$ with $g \equiv̸ 0$ , and local boundedness of $h / g$ near $Z (g)$ .

[L1]

On the open set where $g \neq 0$ , the quotient of holomorphic functions is holomorphic (Sums, products and nonvanishing quotients of holomorphic functions are holomorphic).

[L2]

A locally bounded holomorphic function on $U ∖ Z (g)$ extends uniquely across $Z (g)$ (Riemann extension across a holomorphic hypersurface zero set).

Proof

technique · direct

1.1

By [L1], the quotient $h / g$ is holomorphic on $U ∖ Z (g)$ . The local boundedness hypothesis is exactly the extra condition required by [L2].

givenL1

2.1

Applying [L2] to the holomorphic function $h / g$ on $U ∖ Z (g)$ gives the required holomorphic extension to all of $U$ .

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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

Jiří Lebl, Tasty Bits of Several Complex Variables, Section 1.6 (standard reference, not scraped)
Jaap Korevaar and Jan Wiegerinck, Several Complex Variables, Section 4.7 (standard reference, not scraped)