Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13
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.

If 0∈S, localisation collapses: S−1R is the zero ring

Statement refuted

Localising a nonzero ring always produces a nonzero ring.

Facts & Assumptions

Given: A nonzero commutative ring R and a multiplicative subset S containing 0.

[F1]

A fraction r/s is zero exactly when ur=0 for some u∈S (Equality, vanishing, and the kernel of the localisation map).

Counterexample

technique · direct
1.1

Let R be any nonzero commutative ring and let S be multiplicative with 0∈S, for example S=R. For every r/s∈S−1R, choose u=0. Then ur=0, so [F1] gives r/s=0.

F1
2.1

Thus S−1R has one element and is the zero ring, refuting the statement even though R itself is nonzero.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 7 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources