ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

A multiplicity-two vertex contributes two path states

Example

Assume a finite semisimple fusion category $C$ has a representative set $Irr (C)$ containing labels $x, a, y$ with $N_{x a}^{y} = 2$ . Fix representative labels and basis indices at every other vertex of a path, and suppose the remaining vertex is $x, a \to y$ . Choose a basis $i_{1}, i_{2}$ of the split space $Hom (y, x \otimes a)$ . The two complete path labels differ only by this vertex index.

Facts & Assumptions

Given: A fixed surrounding fusion path with one vertex whose fusion multiplicity is $N_{x a}^{y} = 2$ .

[F1]

Fixed-boundary anyonic fusion-path space: each vertex has its own split-space basis index $μ_{j}$ .

[F2]

Fusion paths give a basis of the fixed-boundary path space: distinct admissible paths and distinct vertex basis-index tuples give distinct basis vectors of the fixed-boundary path space.

Verification

Given: The two-dimensional split space and the fixed labels at the other vertices.

1.1F1F2givenalgebra

The fusion rule gives $N_{x a}^{y} = \dim Hom (x \otimes a, y) = 2$ . Apply the $n = 1$ case of [F2] with boundary $x$ , site charge $a$ , and endpoint $y$ : it gives $\dim Hom (y, x \otimes a) = N_{x a}^{y} = 2$ . Therefore the chosen basis of this split space consists of exactly $i_{1}, i_{2}$ .

2.1F1F2step 1.1algebra∎

Inserting $i_{1}$ and $i_{2}$ into the same surrounding path produces two different vertex-index tuples. By [F2] both composites are members of a path basis and are linearly independent. The local two-dimensional multiplicity contributes exactly two basis vectors; identifying the indices would lose one dimension.

Remarks

This is a conditional multiplicity-indexing calculation. It neither constructs a fusion category with this fusion rule nor uses a multiplicity-free Ising or Fibonacci specialization.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

David Penneys, Unitary Fusion Categories, Math 8800 lecture notes (standard reference, not scraped)
Shu-Heng Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries (standard reference, not scraped)