Model of charge triplets for high-T$_c$ cuprates

Starting with a minimal model for the CuO$_2$ planes with the on-site Hilbert space reduced to a charge triplet of the three effective valence centers [CuO$_4$]$^{7-,6-,5-}$ (nominally Cu$^{1+,2+,3+}$) with different conventional spin, different orbital symmetry, and different local lattice configuration, we develop a unified non-BCS spin-pseudospin model to describe the main phase states of doped cuprates. We argue that antiferromagnetic insulating, charge ordered, superconducting, and Fermi-liquid phases are possible phase states of a model parent cuprate, while typical phase state of a doped cuprate, in particular mysterious pseudogap phase, is a result of a phase separation. Superconductivity of cuprates is not a consequence of pairing of doped holes, but the result of quantum transport of on-site composite hole bosons, whereas main peculiarities of normal state can be related to an electron-hole interplay for unusual Fermi-liquid phase and features of the phase separation. Puzzlingly, but it is the electron-lattice interaction, which in the BCS model determines $s$-wave pairing, in the model of local composite bosons gives $d_{x^2-y^2}$-symmetry of the superconducting order parameter, thus showing once again a substantial involvement of the lattice in the cuprate's HTSC.