CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

A forward-only mapping that is not a correct many-one reduction

Statement refuted

A gadget reduction is correct as soon as every yes-instance maps to a yes-instance.

Facts & Assumptions

Given: The constant map sending every $3$ -CNF formula to the CLIQUE instance $(K_{1}, 1)$ .

[F1]

The graph $K_{1}$ has one vertex, so $(K_{1}, 1)$ is a yes-instance of CLIQUE, by Clique, independent set, and vertex cover decision problems.

[L1]

A polynomial-time many-one reduction requires an iff between source and target membership, by Polynomial-time many-one reductions.

[L2]

The false slogan being refuted is recorded in A gadget reduction is correct as soon as every yes-instance maps to a yes-instance.

Counterexample

technique · direct

1.1

The constant map in the given data is total and polynomial-time, and every satisfiable $3$ -CNF formula is sent to the yes-instance $(K_{1}, 1)$ . So it preserves yes-instances in exactly the way promised by [L2].

F1L2givenconstruct

2.1

Consider the unsatisfiable formula $(x \lor x \lor x) \land (\neg x \lor \neg x \lor \neg x)$ . It is still mapped to $(K_{1}, 1)$ , which is a yes-instance by [F1]. Thus the reverse implication in [L1] fails, so the map is not a correct many-one reduction.

F1L1step 1.1

3.1

This explicit map therefore refutes the slogan [L2].

L2step 2.1∎

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

Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)