CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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 partition (4,2,1) is not self-conjugate

Statement refuted

The partition

$(4, 2, 1)$

is self-conjugate.

Facts & Assumptions

Given: the partition $λ = (4, 2, 1)$ .

[F1]

Conjugation transposes the Ferrers diagram, so the conjugate partition has column lengths equal to the row lengths of the transposed diagram (Ferrers and Young diagrams, conjugate partitions, self-conjugacy, and the Durfee square).

Counterexample

technique · direct

1.1

The Ferrers diagram of $λ$ has column lengths $3, 2, 1, 1$ , so [F1] gives $λ^{'} = (3, 2, 1, 1)$ .

2.1

Since $(3, 2, 1, 1) \neq (4, 2, 1)$ , the partition $λ$ is not self-conjugate. Therefore the displayed claim is false.

step 1.1∎

Depends on

Ferrers and Young diagrams, conjugate partitions, self-conjugacy, and the Durfee square

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.