Significance The Cu extended X-ray absorption fine structure of YSr ₂ Cu _2.75 Mo _0.25 O _7.54 (with superconducting critical temperature, T _c , = 84 K) and Sr ₂ CuO _3.3 (T _c = 95 K) through their superconducting transitions demonstrates that the common factor in superconductivity in cuprates, including those prepared by high-pressure oxygenation, is an internal quantum tunneling polaron in its dynamical structure. In addition, Sr ₂ CuO _3.3 is the first material to show a concomitant transformation in this structure involving atom displacements >1 Å that would be expected to modify its Fermi surface, which would complicate the transition beyond a purely electronic one consisting of the pairing of electrons of opposite momentum across fixed electronic states.