CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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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The function e^(1/z) shows that essential singularities lie outside the argument principle

Statement refuted

Refuted claim: the argument principle still applies unchanged when the enclosed singularity is essential.

Facts & Assumptions

Given: The punctured unit disc and the function $f (z) = e^{1 / z} - 1$ .

[L1]

The argument principle requires a finite zero-minus-pole count for a meromorphic function (The argument principle for an admissible null-homologous cycle).

Counterexample

technique · direct

1.1

The equation $e^{1 / z} - 1 = 0$ is equivalent to $1 / z = 2 π i k$ for some nonzero integer $k$ , so the zeros are $z_{k} = \frac{1}{2 π i k} (k \in Z ∖ {0}) .$ These are infinitely many distinct points, and they accumulate at $0$ .

givenalgebra

2.1

Thus every small circle around $0$ encloses infinitely many zeros of $e^{1 / z} - 1$ , while $0$ is an essential singularity rather than a pole. The finite meromorphic count required by [L1] is unavailable, so the naive extension of the argument principle to essential singularities is false.

step 1.1L1∎

Depends on

The argument principle for an admissible null-homologous cycle

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, Guide to Cultivating Complex Analysis, §5.4 (standard reference, not scraped)