From chessboard of bipolarons of size 4a in cubic $$\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$$ to stripes of the same bipolarons in layered high $$T_c$$ cuprates

The compound [Formula: see text] exhibits a charge order (CO) state at [Formula: see text] and [Formula: see text], which recalls the CO state with a decrease in the temperature of the superconducting transition, [Formula: see text], observed in all cuprates at this doping value. Local excitations of lattice and magnetic origins measured in the two-dimensional metallic state of [Formula: see text] reveal the existence of bipolarons of size 4a resulting from structural and antiferromagnetic pairings of hole-rich orbital polarons of size 2a. They are intertwined with hole-poor domains in a disordered state at [Formula: see text] which become ordered on a chessboard organized in a 3D-order state of ferromagnetically paired polarons at [Formula: see text]. Applied to the [Formula: see text] planes of the cuprates of the "214" family, this model produces stripes of bipolarons intertwined with stripes of antiferromagnetically arranged spins, hole-poor, both of size 4a, leading to a spin density wave with a wave vector [Formula: see text], a charge density wave with [Formula: see text], the Yamada laws [Formula: see text] and [Formula: see text] and a decrease of [Formula: see text] at x=1/8. This work invokes relevance of a bipolaronic origin of high [Formula: see text] superconductivity, in which bipolarons of size 4a can play a major role.