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The investigation of the phase state of dense matter is hindered by complications of first-principle nonperturbative quantum chromodynamics. By performing the first consistent general-relativistic calculations of tidal-excited g-mode of neutron stars with a first-order strong interaction phase transition in the high-density core, we demonstrate that the gravitational-wave signal during binary neutron star inspiral probes their innermost hadron-quark transition and provides potent constraints from present and future gravitational-wave detectors.