False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-24

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: some rational number squares to 2

Statement

False claim: there exists $x \in Q$ with $x^{2} = 2$ .

Facts & Assumptions

Given: The rationals as quotients of integer pairs (The rationals as equivalence classes of pairs of integers).

[A1]

Every rational can be written $a / b$ in lowest terms, with $a, b$ not both even (cancel common factors of $2$ ; the process terminates by well-ordering of $N$ ).

[A2]

Parity in $Z$ : if $k^{2}$ is even then $k$ is even (an odd $k = 2 j + 1$ has odd square $4 j^{2} + 4 j + 1$ ).

Refutation

technique · contradiction

1.1assume-contraA1

Assume, for contradiction, that some $x \in Q$ has $x^{2} = 2$ ; write $x = a / b$ in lowest terms, $a, b$ not both even.

2.1step 1.1algebra

Clearing denominators: $a^{2} = 2 b^{2}$ .

3.1step 2.1A2

$a^{2}$ is even, hence $a$ is even: $a = 2 c$ .

4.1step 3.1A2algebra

Substituting: $4 c^{2} = 2 b^{2}$ , so $b^{2} = 2 c^{2}$ is even, hence $b$ is even.

5.1step 4.1step 1.1discharge-contradiction∎

Both $a$ and $b$ are even, contradicting the lowest-terms choice of step 1.1; the assumption fails: no rational squares to $2$ .

FALSE: some rational number squares to 2

Statement

Facts & Assumptions

Refutation

Depends on

Dependency tree · 2 levels

Sources