  A fluctuating-valence impurity in a metal is quantum-critical unlike a Kondo
impurity which has the properties of a local Fermi-liquid. A systematic theory
for the fluctuating-valence lattice is constructed, based on the hybridization
and pairing of itinerant d-orbitals with localized f-orbitals both of which are
essential parts of the solution of the impurity problem. It also uses the fact
that the single-particle excitations at the Fermi-surface in any dimension can
be written as orthogonal Majoranas and those with linear departures from the
Fermi-surface as linear combination of bare particles and holes with the same
spin. The calculations on the lattice give four spin-degenerate one-particle
excitations of fractionalized fermions; two sets disperse across the chemical
potential and the other two have gaps. The former are shown to be dark to any
linear electro-magnetic probes of their charge and spin and observable only
through probes of their free-energy such as a Fermi-liquid specific heat and
magneto-oscillations characteristic of a Fermi-surface but without a Zeeman
splitting. The excitations with the gaps behave as in insulators but with
renormalized amplitudes. The superfluid density is zero. A magnetic field $H$
turns the insulator to a metal with a singularity in magnetization proportional
to $\sqrt{H - H_c}$, with $H_c$ related to the gap. Beyond $H_c$, the usual
Zeeman splitting appears in the magneto-oscillations. The properties and
predictions are compared to the momentous recent discoveries in
fluctuating-valence insulators. Similar excitations may be expected in
transition metal chalcogenide layers at fluctuating-valence, and quite likely
for Kagome lattices, and twisted multi-layer graphene near specific fillings.
