ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

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.

Ising Clifford versus Fibonacci dense qubit

Example

In a Majorana–Ising realization satisfying Ising–Majorana exchange interface, braid-only gates on a fixed parity sector normalize the logical Pauli group and form a finite projective Clifford subgroup. In the three- $τ$ , total- $τ$ Fibonacci sector, the two braid generators instead generate a subgroup dense in $P U (2)$ . The models therefore have different projective gate power even though both use unitary braid matrices.

Facts & Assumptions

Given: The fixed Ising and Fibonacci sectors and the countable-choice assumption for the Fibonacci density result.

[F1]

Even-Ising braid operations in the stated Majorana interface normalize the logical Pauli group and have finite projective image. Even-Majorana Ising exchanges normalize logical Paulis

[F2]

Three-Fibonacci-anyon braid generators are dense in $P U (2)$ under ${A C}_{ω}$ . Fibonacci single-qubit projective density The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F3]

The Majorana–Ising interface explicitly assumes the pair-channel identification and common relative phase for every adjacent pair. Ising–Majorana exchange interface

Verification

1.1F1F3

Under the stated interface [F3], the Ising conclusion is [F1]: signed permutations of the Majorana generators normalize the finite logical Pauli group, so the projective image is finite.

2.1F1F2∎

By [F2], the subgroup generated by the two Fibonacci braid generators is dense in $P U (2)$ . In contrast, [F1] gives a finite Ising projective image, which cannot be dense in the infinite projective unitary group. Thus these models have different projective gate power under their stated hypotheses.

Remarks

The Ising comparison is conditional on the explicit Majorana–Ising interface. The three-anyon Ising supplier supplies the relative phase in its stated sector but does not establish the all- $m$ channel identification.

The diagram fixes only the crossing convention; none of the proofs relies on the drawing. Time runs from top to bottom, the positive generator is the exchange whose channel matrix is $R$ , and the continuous strand is drawn over the broken strand.

The drawn braid is a convention aid for the adjacent generator on strands $1, 2$ ; an identical local crossing on strands $2, 3$ gives $σ_{2}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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

Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)
John Preskill, Lecture Notes for Physics 219: Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)