Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.
Finite multisets as finitely supported multiplicity functions
Definition
Let be a set. A finite multiset on is a function with finite support . The multiplicity is the number of occurrences of , and the size is . Two finite multisets are equal exactly when their multiplicity functions agree at every element of . We write for the multiset whose multiplicity at is the number of indices with ; denotes the empty multiset.
Used by
- Aggregation-based message-passing graph neural networks Definition
- Finite-round one-dimensional Weisfeiler–Leman color refinement Definition
- A six-cycle and two disjoint triangles are regular but one-WL-indistinguishable Example
- An irrational GIN self-weight separates two bounded color neighborhoods Example
- Mean and max aggregation can lose multiset multiplicity Example
- Countable bounded multisets admit injective sum encodings Lemma
- One-dimensional Weisfeiler–Leman refinement is invariant under graph isomorphism Lemma
- Injective message passing matches the one-dimensional Weisfeiler–Leman test Theorem
- Message-passing GNNs cannot distinguish beyond one-dimensional Weisfeiler–Leman Theorem
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka, How Powerful are Graph Neural Networks? (standard reference, not scraped)
- Christopher Morris et al., Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks (standard reference, not scraped)
- Pan Li and Jure Leskovec, Graph Representation Learning (standard reference, not scraped)