ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

Brauer and partition loops contribute their parameters

Example

For a Brauer cup $U : 0 \to 2$ and cap $A : 2 \to 0$ , $A \circ U = δ 1_{0}$ . For the partition algebra $P_{1} (δ)$ , let $d = {{1}, {1^{'}}}$ . Then $d^{2} = δ d$ : the glued middle singleton blocks form one removed component, while the two outer singleton blocks remain.

Facts & Assumptions

Given: The cup, cap, and singleton partition diagrams with the stacking convention in Typed Brauer and partition diagram categories, and optional substitutions $δ = m$ or $δ = - 2 n$ .

[F1]

Typed Brauer and partition diagram categories: each closed component removed in stacking contributes one factor $δ$ .

Verification

1.1givenF1construct

Gluing the two vertices of $U$ to the two vertices of $A$ closes one matched component and leaves no outer vertex, so the result is one loop times the empty identity diagram, namely $δ 1_{0}$ .

2.1F1step 1.1algebra∎

In $d \circ d$ , the two middle singleton blocks are identified with each other and with no outer vertex, so they form exactly one removed class. The outer vertices remain two singleton blocks, giving $δ d$ . Substituting $δ = m$ or $δ = - 2 n$ gives those same loop scalars in the respective classical branches.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

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Sources

Lehrer and Zhang, The Brauer Category and Invariant Theory, §2.1 (standard reference, not scraped)
Halverson and Ram, Partition Algebras, §2 (standard reference, not scraped)