N-site-lattice analogues of V(x)=ix3

► Elementary non-Hermitian quantum-lattice Hamiltonians H(a,z) with real spectra proposed. ► Exceptional points a=α calculated. ► All topologically nonequivalent stability-loss patterns shown reachable via “exponent” z. ► “Critical exponents” of PT-symmetry breakdowns assigned Fibonacci-related classification and calculated. ► Hermitization matrices Θ shown obtainable (and samples constructed) via symbolic manipulations and extrapolations in N.