In this paper, we review how the “cusp” predicted in the nuclear symmetry energy generated by a topology change at density n1/2 ≳ 2n0 can have a surprising consequence, so far unrecognized in nuclear physics and astrophysics communities, on the structure of dense compact-star matter. The topology change translated into nuclear EFT with “effective” QCD degrees of freedom encoded in hidden local and scale symmetries predicts an EoS that is soft below and stiff above n ≳ n1/2, and yields the properties of neutron stars with no tension with all the astrophysical observations available up to date. Furthermore it describes the interior core of the massive stars populated by fractionally charged quasi-fermions that are neither baryonic nor quarkonic. It is argued that the cusp “buried” in the symmetry energy resulting from strong correlations with hidden heavy degrees of freedom leads, at n ≳ n1/2, to a “pseudo-conformal” sound speed, vpcs2/c2 ≈ 1/3, converged from below at n1/2. It is not conformal since the trace of energy–momentum tensor is not zero even in the chiral limit. It reflects an emergent scale symmetry. This observation with the topology change implies that the quantities accurately measured at ∼ n0 cannot give a qualitatively stringent constraint for what takes place at the core density of compact stars ∼ (3–7)n0. This is because there intervenes a change of degrees of freedom in the effective field theory. We discuss the implication of this on the recent PREX-II “dilemma” in the measured skin thickness of 208Pb.