Abstract
At variance with previous understanding, a system 
with a
Hagedorn-like mass spectrum can sustain the unique temperature
T encoded in the spectrum itself. imposes the
same temperature to all emitted particles which are then in physical
and chemical equilibrium with and with each other.
Coexistence between hadronic and partonic phases is thus completely
characterized. This may explain the recurring constant physical and
chemical temperature observed in several experiments. The near
indifference of to fragmentation or coalescence makes this
approach relevant to heavy-ion and elementary-particle collisions
alike. The equation of state for a gas of systems has been
derived.