Topological materials are usually engineered from topological ingredients. You start with a system whose band structure has a nontrivial winding number, Chern number, or Z_2 invariant, and then you leverage the bulk-boundary correspondence to get protected edge states. The ingredients carry the topology; the assembly is just stacking.
Chen, Guan, Guo, Gao, Gu, and Zhu (arXiv 2602.18855, February 2026) build topology from nothing. Each layer in their system is individually trivial — zero winding number, no protected states, an ordinary gapped band structure with no topological character. But when you stack these layers with interlayer couplings that impose chiral symmetry on the composite, the system becomes topological. And whether it's topological in the standard sense or in a stranger sense depends on a single integer: whether the number of layers is odd or even.
For even layer count, the composite spectrum is gapped, and conventional in-gap edge states appear at the boundaries. This is the familiar picture — a topological insulator with exponentially localized edge modes sitting in the spectral gap. The winding number is nontrivial. The bulk-boundary correspondence works as expected.
For odd layer count, the spectrum is gapless — the authors prove this as a theorem. The chiral symmetry operator, when acting on an odd-dimensional space, necessarily has an unbalanced number of +1 and -1 eigenvalues. This forces at least one band to cross zero energy, closing the gap. But despite the gaplessness, topological boundary states still exist. They sit inside the continuum of delocalized bulk modes — their energies overlap with extended states — yet they remain localized at the boundaries. These are bound states in the continuum, protected not by an energy gap but by symmetry alone.
The mechanism is chiral symmetry emergence. A single layer doesn't have chiral symmetry. The interlayer coupling is engineered so that the N-layer composite acquires it: the Hamiltonian anticommutes with a chiral operator that acts uniformly on each layer's sublattice degree of freedom. This places the system in symmetry class AIII of the tenfold classification, which supports a winding number invariant in one dimension. But whether that invariant manifests as gapped edge states or gapless bound states in the continuum depends entirely on whether N is odd or even.
The parity dependence is not a quantitative shift — more layers giving stronger topology or weaker topology. It's a qualitative phase transition driven by a discrete parameter. Adding one layer to an even stack closes the gap and converts in-gap edge states to bound states in the continuum. Adding one more reopens the gap and restores the conventional picture. The topology oscillates with layer count, switching between two fundamentally different manifestations with each additional layer.
The authors verify this experimentally in 3D-printed acoustic lattices — mechanical structures where the lattice constant is 6 centimeters and the couplings are set by cavity geometry. They measure the acoustic density of states, confirming gapped spectra for two and four layers, gapless spectra for three and five. They image the pressure field distributions, showing boundary-localized modes for both in-gap states and bound states in the continuum, versus delocalized profiles for bulk modes.
The result generalizes across platforms. The mechanism doesn't require exotic materials or fine parameter tuning — it requires chiral symmetry in the coupling and a count of how many layers you stacked. Photonic metamaterials, mechanical lattices, electronic circuits, and acoustic systems can all implement it. The topology lives in the symmetry and the integer, not in the material.
An integer — odd or even — determines whether a stack of trivial layers is topological, and if so, what kind.