Superposition is the foundational technique of wave physics: express a complex field as a sum of simple basis waves, solve each component independently, and add the solutions. This works because the wave equation is linear. The technique is so fundamental that its failure modes are rarely examined.
The paper (arXiv:2602.20179, 2026) identifies one. In multilayered media with three or more interfaces, expanding fields in terms of evanescent and inhomogeneous waves — the standard basis for problems involving decay and tunneling — produces infinite series that diverge. The superposition fails. Not approximately, not numerically — mathematically. The series does not converge.
The cause is a normalization problem. Propagating waves carry finite energy per mode. Evanescent waves do not — they decay exponentially but their norm in the conventional basis diverges when the geometry allows multiple reflections. Each reflection amplifies the evanescent component, and the basis functions that were individually well-behaved become collectively divergent when summed across interfaces.
The fix is to change the basis. The authors construct power flux modes — basis functions that are orthonormal with respect to energy flux rather than field amplitude. In this basis, the scattering at each interface is unitary (energy-conserving), and the propagation eigenvalues are bounded. The series converges. The physics hasn't changed; the representation has.
The structural insight: the divergence is not a property of the physics. It is a property of the basis. The same physical system, described in different mathematical coordinates, is either convergent or divergent. The basis you choose determines whether your calculation works, not because of approximation or truncation but because the mathematical object you're constructing — an infinite series — either exists or doesn't depending on the space in which you define it. The right answer requires the right frame.