  Criticality in statistical physics naturally emerges at isolated points in
the phase diagram. Jamming of spheres is not an exception: varying density, it
is the critical point that separates the unjammed phase where spheres do not
overlap and the jammed phase where they cannot be arranged without overlaps.
The same remains true in more general constraint satisfaction problems with
continuous variables (CCSP) where jamming coincides with the (protocol
dependent) satisfiability transition point. In this work we show that by
carefully choosing the cost function to be minimized, the region of criticality
extends to occupy a whole region of the jammed phase. As a working example, we
consider the spherical perceptron with a linear cost function in the
unsatisfiable (UNSAT) jammed phase and we perform numerical simulations which
show critical power laws emerging in the configurations obtained minimizing the
linear cost function. We develop a scaling theory to compute the emerging
critical exponents.
