Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24
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FALSE: every interior local modulus minimum forces constancy

Statement

Every holomorphic function on a complex domain whose modulus has an interior local minimum is constant.

Facts & Assumptions

Given: The open unit disc D and the identity function f(z)=z, which is holomorphic and has derivative 1 (Linearity, product, reciprocal, and quotient rules for complex derivatives). Complex modulus is nonnegative and vanishes only at 0 (Conjugation is an involutive real-field automorphism, zz‾=∣z∣2, and modulus is definite, multiplicative, and subadditive).

[L1]

A nowhere-zero holomorphic function on a complex domain cannot have an interior local modulus minimum unless it is constant (Minimum modulus principle for a nowhere-zero holomorphic function).

Refutation

technique · direct
1.1

The identity function f(z)=z is holomorphic on the nonempty complex domain D.

given
2.1

Its modulus satisfies ∣f(z)∣=∣z∣≥0=∣f(0)∣, so it has a global, and hence local, minimum at the interior point 0.

step 1.1algebra
3.1

The map is not constant, since f(0)=0 and f(1/2)=1/2. The valid theorem [L1] does not apply because f vanishes at the minimizer, so nonvanishing is essential.

step 1.1step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources