Object-constituting relations and the divine persons

Most versions of classical trinitarian theology recognize at least four properties in the trinity – and here I mean to use 'property' in the weakest possible sense as some recognizable characteristic of a thing, without meaning to entail any ontological claims about what such properties are. For example, I do not intend 'properties' to mean universals, tropes, or anything else typically associated with the term 'property'. I simply mean a recognizable characteristic of a thing.

In any case, I label these properties as follows:

(T1) There are four properties in the divinity like so:
e = the divine essence
p = the property 'being a father' (or paternity)
f = the property 'being a son' (or filiation)
s = the property 'being a spirit' (or spiration)

Additionally, classical theism holds that these properties are arranged in a particular way. Each divine person is made up of the divine essence and a personal property as follows:

(T2) Each divine person is made up of the properties from (T1) like this:
Father = e + p
Son = e + f
Holy Spirit = e + s

My question is: why are these properties arranged in this particular way? What explains this particular configuration? Answering this entails some discussion of the relations between all these properties.

To be sure, there are different kinds of relations we could talk about. For example, we could talk about identity and distinction relations here, and indeed this is what much of trinitarian theology invests its time in. But I'm not interested in those kinds of relations here. I'm interested in what I call object-constituting relations. An object-constituting relation is a relation which explains why its relata together constitute an object.

There are many sorts of object-constituting relations. For example, imagine if I could take a number of sand grains and throw them into a heap or pile. The grains would then be related to the whole as elements of a heap or a pile, and the sand pile would be an object (in the sense that heaps or piles are objects). The relations between the grains of sand are thus object-constituting relations, because they explain why the whole of them counts as an object.

But that can't be what we want here, because in heaps or piles, grouping any two elements together would be arbitrary. If the elements of our pile were the properties specified in (T1), I could group any two of them together arbitrarily. I could just as easily say the Father is p + s as I could say he is e + p. Thus, these divine properties cannot all just be 'there' in the divinity, in a heap or a pile, because then we don't get (T2) in any non-arbitrary sense.

What we need is some sort of relation between the divine essence and each of the personal properties which does not obtain between the personal properties themselves. In other words,

(T3) There is some object-constituting relation which obtains between
e and p,
e and f, and
e and s
which does not obtain between
p and f,
p and s, and
f and s.

Additionally, the kind of object-constituting relation we want here must be some kind of unity-making relation in the sense that it must explain the unity of the divine essence and a personal property in each divine person. Thus,

(T4) The constituents of each divine person are related by a unity-making object-constituting relation like so:
e is related to p such that e + p are one,
e is related to f such that e + f are one, and
e is related to s such that e + s are one.

Whatever we say about the object-constituting relation, it must be consistent with (T1) - (T4). So what is this object-constituting relation?

There are, of course, many strategies for answering this question. I'm not going to go into any of them here, mostly because I haven't worked them all out yet. Here I just wanted to put down the conditions (namely, (T1) - (T4)) for any answers to this question.