One God-Being, three God-Persons (part 2)

In my previous post, I suggested that "the God-Person known as the Father sometimes speaks on behalf of, and represents, the group (or family) known as the God-Being." However, on reflection I think this is problematic for two reasons. Firstly, it means that each of the God-Persons constitutes only one third of the God-Being which raises the question of whether or not any of them possesses the fullness of the divine nature. Secondly, The import of biblical passages which reference the Father as the one, or the only, true God seems weakened by the notion that they really mean that He merely represents the one true God.
Since I believe (and stated in my previous post) that the Father is the source of the divine nature of the Son and Holy Spirit, it seems appropriate to say that the Father Himself is the God-Being (as well as being one of the God-Persons) while the other two God-Persons, since they derive their divine nature from the Father, can be said to participate in His Being.
To use a rather outlandish example, imagine that the entire human race has been wiped out in some freak nuclear accident apart from one survivor - a genius scientist named Tom. Tom has two best friends called Dick and Harry and although he couldn't completely save them, being a scientific genius he did manage to preserve each of their brains which he now keeps in two jars of formaldehyde. Again, being a scientific genius, Tom has managed to wire each of his two friends' brains up to his own body so that each of them is now able to see through Tom's eyes, to speak through his mouth and even to walk around, pick things up etc, using Tom's feet, hands and limbs. Fortunately, Tom Dick and Harry have similar tastes in terms of where they want to go and what they want to do, so there are no conflicts over how to make use of Tom's body. In this bizarre scenario, I think it would be appropriate to say that there are three human persons - Tom, Dick and Harry - but only one human being, namely Tom, although the other two persons participate in Tom's being through having access to his human physical attributes. In other words they derive the non-personal aspects of their human nature from Tom. Likewise, in the case of the Holy Trinity we can say that there are three God-Persons and one God-Being, the Father, from whom the other two God-Persons derive their divine nature and attributes.
The scripture passages which speak of the Father as the one true God do so because he is the only true God-Being. However, all three God-Persons, in equal measure, possess and partake of the nature and attributes of that Being just as, in my analogy, the three human persons, Tom, Dick and Harry, share in the one human nature which is derived from the one surviving human being, Tom.