Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: the holomorphic inverse function theorem is global

Statement

False claim: the holomorphic inverse function theorem makes a map with everywhere-invertible complex Jacobian globally invertible.

Facts & Assumptions

Given: The false claim above.

[L1]

The several-variable holomorphic inverse function theorem is local: it produces biholomorphic neighbourhoods around each point (The holomorphic inverse function theorem in several complex variables).

[L2]

The exponential counterexample has invertible complex Jacobian everywhere and is not injective (The map (z1,z2)↦(ez1,z2) has invertible complex Jacobian everywhere and is not injective).

Refutation

technique · direct
1.1

Fact [L1] says that an invertible complex Jacobian gives a local biholomorphism near each point, not a global inverse on the whole domain.

L1
2.1

Fact [L2] realizes exactly that gap: the map (z1,z2)↦(ez1,z2) satisfies the local hypothesis everywhere, yet it is not injective and therefore has no global inverse. So the claim is false.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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