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: zero derivative on an open set forces constancy

Statement

False claim: if a differentiable map has zero derivative at every point of an open Euclidean domain, then it is constant.

Facts & Assumptions

Given: No connectedness hypothesis is imposed in the false claim.

[L1]

There is an open set U and a function on it whose total derivative is zero at every point of U, but which is not constant on U (Zero derivative need not give constancy on a disconnected open set).

[L2]

If U is nonempty, open, and connected, then Df=0 on U if and only if f is constant on U (A differentiable map on a connected open Euclidean set has zero derivative exactly when it is constant).

Refutation

technique · direct
1.1

The construction in [L1] satisfies the derivative hypothesis and violates the conclusion, so it refutes the claim.

L1
2.1

The comparison with [L2] identifies connectedness as the missing hypothesis: adding it restores exactly the stated equivalence.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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