Fibonacci Anyons

Anyon theory with fusion τ×τ=1+τ and dense braid representations on appropriate fusion spaces. The main concepts are golden ratio, F moves, braid matrices. Each must be defined in the context of a concrete physical model, mathematical representation, or experimental task. Terminology alone does not establish the behavior of the system.

Fibonacci anyons contain the vacuum object and one nontrivial anyon τ satisfying τ×τ=1+τ. The allowed fusion trees form Hilbert spaces whose dimensions obey Fibonacci recurrences, while the F move transforms between different fusion associations and the R move describes elementary exchanges. Braiding matrices become noncommuting operators on encoded fusion spaces; their unitarity and consistency follow from pentagon and hexagon relations. Global topological charge constraints determine which encoded states are physically allowed.

A precise simulator needs a fixed total charge, an ordering of punctures and consistent F and R gauge conventions. Build three- and four-anyon spaces, compute braid generators, verify braid relations and compile a simple logical gate while tracking leakage. An attractive braid plot without consistent phases and fixed total charge is not a physically meaningful encoded operation.