ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

The unsupported sock swap

Statement

In the second Fraenkel system with the full group of permutations preserving each atom pair setwise, a pair outside a proposed choice function's finite support admits the permutation that swaps exactly its two atoms and fixes every other atom. It fixes every input pair but changes the chosen output.

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

The second Fraenkel model has countable pairs without a choice function fixes the paired atoms and group.

Proof

1.1

Write $P_{n} = {a_{n}, b_{n}}$ . If $E$ is a finite atom support, choose the least $n$ with $P_{n} \cap E = \emptyset$ . The permutation $π$ interchanging $a_{n}, b_{n}$ and fixing all other atoms is allowed, fixes $E$ , and satisfies $π P_{k} = P_{k}$ for every $k$ .

2.1

If $c$ were supported by $E$ and $c (k) \in P_{k}$ , then $π c = c$ , so graph evaluation gives $c (n) = π (c (n))$ . But $π$ has no fixed point in $P_{n}$ , contradiction.

F1step 1.1∎

Depends on

The second Fraenkel model has countable pairs without a choice function

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Jech, The Axiom of Choice, §4.4 (standard reference, not scraped)