Note: This gives a broader causal relation which automatically satisfies "if x C y and x P z then z C y" as well as "if x C z and y P z then x C y", loosely "anything which is caused by a part is caused by the whole" and "anything which causes the whole, causes the part". So we don't need to state those as extra premises.
This will lead to a problem.
Consider assumption A3:
A3. Let S be any endless chain in E; then there is some z in E such that every x in S is a proper part of z.
Consider any endless chain consisting of at minimum two elements. Consider two elements in that chain, x and y, such that x C y. x and y are both proper parts of z. Therefore, x C z, and z C y. But then we have a longer chain; using x C z C y in place of x C y. Each element of that longer chain must then, by A3, be a proper part of a larger entity, z2. But then, similarly, we can construct the chain using x C z C z2 C* y. There are therefore an infinite number of entities z, z2, z3, z4... and so on, each including the one before it as a proper part (and nothing that is not part of the one before it).
Furthermore, anything which is a proper part of anything else is a part of such an infinitely recursive loop by default.
This leads to trouble in the proof of theorem 3.
Consider any endless chain consisting of at minimum two elements. Consider two elements in that chain, x and y, such that x C y. x and y are both proper parts of z. Therefore, x C z, and z C* y.
It follows that z C y but it does not follow that x C z or that y C z.. The "whole" z may be a cause of its parts, without in turn being caused by its parts. Note that by construction of C it is true that if x is a cause of y and x is a part of z, then z C y. However, it is not generally true that if x is a cause of y and z is a part of x then z C y. ...
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.