CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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.

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Holomorphic functions of several variables have no isolated singularities

Statement

Let $m \geq 2$ , let $Ω \subseteq C^{m}$ be a domain, and let $a \in Ω$ . A holomorphic function on $Ω ∖ {a}$ cannot have a genuine isolated singularity at $a$ : it always extends holomorphically across $a$ .

Facts & Assumptions

Given: A domain $Ω \subseteq C^{m}$ with $m \geq 2$ , a point $a \in Ω$ , and a holomorphic function on $Ω ∖ {a}$ .

[L1]

In complex dimension at least two, a holomorphic function on a punctured domain extends uniquely across the puncture (An isolated puncture is removable in complex dimension at least two).

Proof

technique · direct

1.1

Apply [L1] to the punctured domain $Ω ∖ {a}$ . It produces a holomorphic extension across $a$ .

2.1

So the deleted point cannot support a nonremovable isolated singularity.

step 1.1∎

Depends on

An isolated puncture is removable in complex dimension at least two

Used by

FALSE: isolated singularities in several variables can be poles or essential False statement

Dependency tree · two levels

6 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

J. Lebl, Tasty Bits of Several Complex Variables, §1.6 (standard reference, not scraped)