A fractal that looks like a butterfly hides an order that is not visible at first glance. In the quantum world of one-dimensional quasicrystals described by the Komoto model, the energy spectrum forms a whimsical shape: the Komoto butterfly. For a long time, it could not be fully colored, because each gap in the spectrum needed an integer, a topological index, but the standard tools failed due to the discontinuous potential. Chern numbers, familiar from smooth systems, are simply not defined here.

The genealogy of quasiperiodicity is the key the authors propose. They build an infinite directed tree, where each vertex corresponds to a spectral band or gap in a periodic approximation of the quasiperiodic operator. This tree grows from the continued fraction digits of the irrational number that defines the quasiperiodicity. By tracing paths in this tree, one can unambiguously assign an index to each gap: for each vertex, a matrix Q is computed, and its determinant yields an integer. The ambiguity that previously allowed a shift by the denominator of the approximation disappears.

The result is striking: the phase diagram is fully colored. Each gap in the spectrum now has a unique integer index, and the order of these indices follows the tree structure. Unlike the Hofstadter butterfly, the complement of the Komoto butterfly consists of a single connected component, so the indices are not tied to isolated regions. Red phases can terminate without a continuous closing of the gap, a behavior that defies intuition from smooth models. This is a rigorous mathematical proof, not a numerical experiment.

The practical benefits are already visible. Experimentalists creating quasicrystals in photonic lattices or ultracold atomic systems can now precisely predict in which finite system a desired topological index will appear. The spectral tree method opens the way to classifying other quasiperiodic models and to a rigorous proof of equivalence between the Komoto and Hofstadter butterflies. The question of a complete topological classification of all quasiperiodic operators remains open, but this work is an important step forward.