An electron dives into a crystal, changing its energy as it goes. In an ordinary lattice, it encounters a sequence of allowed bands and forbidden gaps — this is the ABC of semiconductors. But in a quasicrystal, the pattern of atoms never exactly repeats. The spectrum is also riddled with gaps, and within any gap the density of electronic states is constant. What numbers can it take? The answer, it turns out, is written into the topology of the pattern itself — much like a genetic code determines the shape of a protein.

Forty years ago, Jean Bellissard showed that the key lies in the K-theory of C*-algebras (operator algebras that generalize complex numbers) and the homology of the moduli space of a groupoid (a generalization of a group where multiplication is defined only for certain pairs). But an exact algebraic description remained elusive. New work by Hiroki Matui offers a unified method — sequencing the aperiodic genome. He studies dynamical systems — actions of poly-Z groups on Cantor sets, classic specimens from topology. Instead of directly computing K-groups, Matui builds bridges to homology via exact sequences (chains of maps where the image matches the kernel).

For poly-Z groups with Hirsch length up to 4, the method yields an almost complete picture. In one- and two-dimensional cases, the HK conjecture holds exactly: K-theory is isomorphic to a sum of homologies, and gap-labelling works out in integers. For a three-dimensional action of Z³, the integral HK conjecture is rigorously proved for the first time. For more complex systems, the factor ½ appears, and the obstruction always has order 2 — an echo of transpositions in the topological full group. The ideas of Poincaré (duality), Hilbert (abstract spaces), and von Neumann (operator algebras) permeate the whole construction.

The method also reveals why rational versions of the conjectures hold widely while integral versions do not: the obstructions have torsion of order 2. To eliminate the pesky ½, one may need the de la Harpe–Skandalis determinant or a more refined spectral sequence. Next steps: testing the HK conjecture for groups of Hirsch length 5 and extending the technique to groupoids with torsion. The long-term goal is to classify aperiodic patterns themselves by their spectral 'genome'. The genome of the aperiodic crystal is almost read; only a few typos in the codons remain to be fixed.