Newtonian statics, gravitational radiation, and the equivalence principle for non-topological solitons in elastic media

Proposals in which spacetime is an elastic medium, matter a localized field excitation, and gravity the medium's long-range strain response recur from the nineteenth century to the present. We show that for the simplest such model — Q-balls coupled to the dilatation of an isotropic elastic continuum — the standard gravitational requirements are not independent constraints but a chain of mutually forced failures. (i) In linear elasticity the static force between separated solitons vanishes identically (the field-theoretic Bitter–Crum theorem): there is no Newtonian limit. (ii) The only repair within isotropic elasticity is the fine-tuned "floppy" point λ + 2μ = 0 with strain-gradient stabilization (Kleinert's world-crystal elasticity), which restores an exact 1/r law but forces a negative bulk modulus (an unstable homogeneous mode) and quartic compression-wave dispersion, so Newtonian statics and luminal gravitational radiation cannot share the compression sector. The dispersive sector also has zero Landau critical velocity: a uniformly moving soliton Cherenkov-radiates at any speed, feeling a derived and independently verified v² drag, so the medium rest frame is locally detectable and orbits decay with a velocity scaling incompatible with the binary-pulsar-confirmed quadrupole law. (iii) Free fall is not universal: amplitude coupling gives an order-unity Eötvös parameter across the stable soliton branch, while energy-density coupling recovers the equivalence principle only up to a computed, non-cancelling elastic self-energy fraction — the elastic counterpart of the Hui–Nicolis theorem for scalar forces. The chain closes every branch of this recurring class of proposals within classical isotropic elasticity. Four Python scripts reproducing every numerical result are included as ancillary files.