  A background-independent quantization of the Universe near its Big Bang
singularity is considered using a drastically simplified toy model. Several
conceptual issues are addressed.
  (1) The observable spatial-geometry characteristics of our empty-space
expanding Universe is sampled by the time-dependent operator $Q=Q(t)$ of the
distance between two space-attached observers (``Alice and Bob'').
  (2) For any pre-selected guess of the simple, non-covariant time-dependent
observable $Q(t)$ one of the Kato's exceptional points (viz., $t=\tau_{(EP)}$)
is postulated {\em real-valued}. This enables us to treat it as the time of Big
Bang.
  (3) During our ``Eon'' (i.e., at all $t>\tau_{(EP)}$) the observability
status of operator $Q(t)$ is mathematically guaranteed by its self-adjoint
nature with respect to an {\em ad hoc} Hilbert-space metric $\Theta(t) \neq I$.
  (4) In adiabatic approximation (i.e., in Heisenberg picture) the passage of
the Universe through its $t=\tau_{(EP)}$ singularity is interpreted as a
quantum phase transition between the preceding and the present Eon.
  It is worth adding that in our model the widely accepted ``Big Bounce''
regularization of the classical Big Bang singularity after quantization gets
replaced by the full-fledged quantum degeneracy. At the
quantum-phase-transition singularity, operator $Q(\tau_{(EP)})$ becomes
unobservable and acquires a non-diagonalizable Jordan-block structure.
