Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21
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: every open C1 map has invertible derivative

Statement

False claim: every open C1 map between open subsets of Euclidean space has invertible derivative at every point.

Facts & Assumptions

Given: No assumptions beyond the false claim.

[L1]

The map x↦x3 is a smooth open bijection of R with derivative zero at the origin, but its inverse is not differentiable there (x↦x3 is a C1 bijection whose inverse is not differentiable at zero).

[L2]

A C1 map with everywhere-invertible derivative is an open map (A C1 map with everywhere-invertible derivative is open).

Refutation

technique · direct
1.1

The cube map in [L1] is open and C1, but its derivative at zero is the zero linear map, which is not invertible.

L1algebra
2.1

Therefore openness does not imply derivative invertibility. The valid result [L2] is only the forward sufficient implication from derivative invertibility to openness.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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