The statement "for all elements, the smallest particle that can exist independently is an atom" is not true. While atoms are indeed the smallest units of matter that retain the properties of an element, there are subatomic particles that are even smaller and can exist independently. These subatomic particles include electrons, protons, and neutrons, which are the building blocks of atoms. Additionally, in certain cases, even smaller particles such as quarks and gluons can exist independently, although they are not considered to be elements in the traditional sense.