You can make up all the fancy particles you like, but they do not exist in reality. There are only so many sub-atomic particles, and we know what they are. There are sub-sub-atomic particles and such, but they can not form anything comparable to atoms, just sub-atomic particles.
You can not have half a particle because half a particle results in either a fairly large explosion or simply in the pieces floating off. Matter is only stable in very rigid and specific patterns. And, the sub-atomic particles are made of 3 smaller particles anyways, so 1/2 is not even a valid fraction.
But joking aside, those smaller particles can not form any larger particles other than the ones we know about already. There are only so many ways matter can combine, and we have seen or modeled them all.
I am not sure what you mean by quadruple cores, but it sounds like wishful thinking to me. The only possibilities for the core of an atom are protons and neutrons (except for antimatter, which has identical particles except that they have opposing charges and spins, so they are essentially the same anyways). What gives a piece of matter almost all of its properties is the number of protons in the nucleus. Change that, you get entirely different matter.
IF (and that is a big if) gravity is found to be particle like in nature (in addition to wave-like), like light is (photons), then these "gravitons" would be no more able to form new types of matter than photons are (which they are not). If they exist, they would have a similar interaction with electrons to that of photons with electrons. I am not sure what precisely this interaction would be, but it most certainly would not be to form new types of matter with the electrons. They would have to be immensely smaller than photons anyways, as we have never been able to detect or see the evidence of any such gravitons.