CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

The exponential function omits 0 and infinity as a meromorphic map on the plane

Statement refuted

A nonconstant meromorphic function on $C$ omits at most one sphere value.

Facts & Assumptions

Given: The exponential function regarded as a meromorphic map $e^{z} : C \to \hat{C}$ .

[L1]

The exponential is entire (The complex exponential is entire and its complex derivative is itself).

[L2]

Its image is $C ∖ {0}$ (The complex exponential maps $C$ onto $C ∖ {0}$ ).

Counterexample

technique · direct

1.1

Fact [L1] makes $e^{z}$ meromorphic on $C$ with no poles, and [L2] identifies its image as $C ∖ {0}$ .

L1L2given

2.1

As a sphere-valued map, the omitted values are therefore $0$ and $\infty$ . Since $e^{z}$ is nonconstant, this refutes the claim that at most one sphere value can be omitted.

step 1.1∎

Depends on

Used by

FALSE: a nonconstant meromorphic function on the plane omits at most one sphere value False statement

Dependency tree · two levels

16 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

Aleksander Simonic, The Ahlfors lemma and Picard's theorems (standard reference, not scraped)