Quasicrystalline Signatures in Gravitational-Wave Propagation: A Falsifiable Effective Framework for Anisotropic Dispersion, Polarization Mixing, and Spectral Sidebands in the LISA Band

We formulate a conservative and explicitly falsifiable framework for gravitational-wave

propagation beyond standard General Relativity, motivated by structured-spacetime scenarios

with direction-dependent effective transport. Rather than postulating a complete microscopic

theory, we work at the level of an effective propagation sector and identify a correlated

signature class consisting of: (i) anisotropic dispersion, (ii) polarization mixing beyond pure

tensor propagation, and (iii) quasi-discrete spectral sidebands. The central claim is not that

any single one of these effects would be unique, but that their correlated appearance in the

milliHertz band would define a nontrivial phenomenological fingerprint testable with LISA. A

minimal geometric correction to the wave equation is introduced, an anisotropy estimator is

defined, and order-of-magnitude benchmark targets are specified for scalar/vector admixture

and group-velocity deformation. The framework reduces continuously to General Relativity in

the isotropic limit and provides a direct falsification route through null tests on polarization

content, frequency-dependent delays, directional modulation, and residual sideband structure.