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

Statement

False claim: in complex dimension at least two, an isolated singularity can still be removable, a pole, or essential just as in one variable.

Facts & Assumptions

Given: Complex dimension m≥2.

[L1]

In one variable every isolated singularity is exactly one of the removable, pole, or essential cases (Every isolated singularity is removable, a pole, or essential).

[L2]

In several variables with m≥2, an isolated deleted point is always removable (Holomorphic functions of several variables have no isolated singularities).

Refutation

technique · direct
1.1

The one-variable trichotomy of [L1] distinguishes three genuinely different behaviors at a puncture.

L1
2.1

In several variables with m≥2, [L2] collapses that trichotomy at an isolated point: the only possible behavior is removability. So poles and essential singularities do not occur at isolated deleted points.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources