  The antisoliton-soliton {\cal G}={\cal S_-}\!\vee{\cal S_+} neutral state is
a proper geon in a family of stable squashed-S^3\times\IR pp-wave electrovacua
along a primordial Q/r^2 field. With {\cal S_-} propagating backwards in time,
the dominant EM field is that of an effective electric-dipole moment {\sf p}.
If disjointed, the {\cal S_\pm} carry \pm Q charge (on a round-S^2 {\em
physical} singularity of radius r_o) as non-singular alternatives to the
Kerr-Newman solution. {\cal G} has three scales (gravitational \kappa, metric
scale, NUT-charge \kappa Q=2r_o) in a full 4-scale hierarchy without
supersymmetry. A particular {\cal G} with effective mass and a near-zero {\sf
p} is proposed as dark-matter particle. A gas of such {\cal G}s would
`freeze-out' before the electroweak era as CDM, whose present mean density is
predicted by this model (via Casimir-effect data on earth) as, roughly, 100{\rm
Mev}/{\rm cm}^3.
