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 C1 bijection has a C1 inverse

Statement

False claim: every C1 bijection between open subsets of Euclidean space has a C1 inverse.

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).

Refutation

technique · direct
1.1

The map in [L1] satisfies the C1 bijection hypothesis of the false claim.

L1
2.1

Its inverse fails even to be differentiable at zero by [L1], and therefore cannot be C1. Hence the claim is false.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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