False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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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.

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

Statement

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

Facts & Assumptions

Given: A source language $A$ having at least one no-instance and a target language $B$ having at least one yes-instance $y_{B}$ .

[L1]

A gadget reduction is not complete until both directions of correctness have been proved, by Reduction gadgets and interface invariants.

[L2]

A polynomial-time many-one reduction requires $x \in A$ if and only if $f (x) \in B$ , by Polynomial-time many-one reductions.

Refutation

technique · direct

1.1

Define the constant map $f (x) : = y_{B}$ for every source instance $x$ . This map is total and polynomial-time, and it certainly sends every yes-instance of $A$ to a yes-instance of $B$ .

L2givenconstruct

2.1

Let $x_{0} \notin A$ . Then $f (x_{0}) = y_{B} \in B$ , so the reverse implication in [L2] fails: $f (x_{0}) \in B$ does not force $x_{0} \in A$ . Therefore yes-preservation alone does not make $f$ a correct many-one reduction.

L1L2step 1.1

3.1

The statement is false.

step 2.1∎

Depends on

Used by

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

Dependency tree · two levels

4 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)