Bond Cohomology of Icosahedral Quasicrystals: A Fibonacci Approximant Approach

We compute the first cohomology group of the bond graph of the icosahedral quasicrystal via an explicit Fibonacci approximant sequence. The bond graph has vertices at atomic positions and edges connecting nearest-neighbor pairs at the two icosahedral bond lengths  and . For the -th Fibonacci approximant  with  atoms and icosahedral coordination shell (), we prove , giving  exponentially fast (rate ).
The limiting rank  follows from two facts: (1) Euler’s formula for graphs, and (2) the geometric identity , which holds because the cut-and-project construction from  contributes exactly two bond orientations  for each of the  independent projection directions. This gives a direct elementary proof of the result  first established by Forrest, Hunton and Kellendonk (2002) via -theory and groupoid -algebras.
The cohomology group  carries the -representation of the icosahedral group  — the unique five-dimensional irreducible representation — which coincides with the phason strain tensor space of quasicrystal physics. The Fibonacci approximant hierarchy exhibits the symmetry breaking chain , connecting this paper to earlier work on FCC lattice cohomology (Paper 1 of this series, zenodo.18793518). A universal conjecture is proposed: for any cut-and-project quasicrystal from  with full coordination shell (), the bond cohomology satisfies .