Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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FALSE: every holomorphic function on a punctured several-variable domain is unbounded near the puncture

Statement

False claim: if m≥2 and f is holomorphic on a punctured neighborhood of 0∈Cm, then f must be unbounded near 0.

Facts & Assumptions

Given: The bounded coordinate function f(z)=z1 on Δ1m∖{0} with m≥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
1.1

The function f(z)=z1 is holomorphic on Δ1m∖{0} and satisfies ∣f(z)∣≤1 there, so it is bounded near the puncture.

givenalgebra
2.1

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.

step 1.1L1∎

Depends on

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