False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: every holomorphic function on a punctured several-variable domain is unbounded near the puncture

Statement

False claim: if $m \geq 2$ and $f$ is holomorphic on a punctured neighborhood of $0 \in C^{m}$ , then $f$ must be unbounded near $0$ .

Facts & Assumptions

Given: The bounded coordinate function $f (z) = z_{1}$ on $Δ_{1}^{m} ∖ {0}$ with $m \geq 2$ .

[L1]

A holomorphic function on a punctured several-variable domain extends holomorphically across the missing point (An isolated puncture is removable in complex dimension at least two).

Refutation

technique · direct

1.1givenalgebra

The function $f (z) = z_{1}$ is holomorphic on $Δ_{1}^{m} ∖ {0}$ and satisfies $∣ f (z) ∣ \leq 1$ there, so it is bounded near the puncture.

2.1step 1.1L1∎

Step 1.1 already contradicts the displayed claim, and [L1] explains why no singularity is hiding here: the function extends holomorphically across $0$ as the same coordinate function.

Depends on

An isolated puncture is removable in complex dimension at least two

Used by

Nothing in the library uses this result yet.

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)