  We explore universality and phases of matter in hybrid quantum dynamics
combining chaotic time evolution and projective measurements. We develop a
unitary representation of measurements based on the Stinespring Theorem, which
we crucially identify with the time evolution of the system and measurement
apparatus, affording significant technical advantages and conceptual insight
into hybrid dynamics. We diagnose spectral properties in the presence of
measurements for the first time, along with standard, experimentally tractable
probes of phase structure, finding no nontrivial effects due to measurements in
the absence of feedback. We also establish that nonlinearity in the density
matrix is neither sufficient nor necessary to see a transition, and instead
identify utilization of the measurement outcomes (i.e., ``feedback'') as the
crucial ingredient. After reviewing the definition of a phase of matter, we
identify nontrivial orders in adaptive hybrid dynamics -- in which measurement
outcomes determine future unitary gates -- finding a genuine
measurement-induced absorbing-state phase transition in an adaptive quantum
East model. In general, we find that only deterministic and constrained
Haar-random dynamics with active feedback and without continuous symmetries can
realize genuine, measurement-induced phases of matter.
