False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

FALSE: complex sine and cosine are bounded on the complex plane

Statement

False claim. The functions $\sin : C \to C$ and $\cos : C \to C$ are bounded.

Facts & Assumptions

Given: The complex sine and cosine functions.

[L1]

Neither $\sin : C \to C$ nor $\cos : C \to C$ is bounded (Complex sine and cosine are unbounded on the complex plane).

Refutation

technique · direct

1.1L1

The sourced proposition [L1] directly contradicts the asserted global boundedness of both functions.

2.1step 1.1∎

Thus the claim that both functions are bounded on $C$ is false.

Depends on

Complex sine and cosine are unbounded on the complex plane

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

E. Stein and R. Shakarchi, Complex Analysis, Ch. 1, §2.3 (standard reference, not scraped)