  The suppression of density fluctuations at different length scales is the
hallmark of hyperuniformity. However, its existence and significance in jammed
solids is still a matter of debate. We explore the presence of this hidden
order in a manybody interacting model known to exhibit a rigidity transition,
and find that in contrary to exisiting speculations, density fluctuations in
the rigid phase are only suppressed up to a finite lengthscale. This length
scale grows and diverges at the critical point of the rigidity transition, such
that the system is hyperuniform in the fluid phase. This suggests that
hyperuniformity is a feature generically absent in jammed solids. Surprisingly,
corresponding fluctuations in geometrical properties of the model are found to
be strongly suppressed over an even greater but still finite lengthscale,
indicating that the system self organizes in preference to suppress geometrical
fluctuations at the expense of incurring density fluctuations.
