In order to get the desired result that the 3D and 4D views are equivalent, the authors need “sums-at-times” to satisfy two requisites: (a) sums-at-times are acceptable for endurantists, i.e. they are not additions to the endurantist ontology, they are nothing over and above the enduring particles that the endurantist already accepts (b) Sums-at-times are “timebound”, i.e. they exist at only one time. For any two different times t and t’ in which an object O exists, (O, t) is numerically distinct from (O, t'). (Because of problems with the blogger, I use brackets instead of > and < to represent sums-at-times...In my notation, (O, t)represents the sum of particles that constitute O at t).
The second requisite is necessary for the translation scheme they propose to work. If sums-at-times are not timebound, then something is true of them that is not true of temporal parts (namely, that they exist or may exist at more than one time). This is why, I think, the authors hasten to emphasize that
(O, t) [the sum of particles that constitute O at t] may be understood as a 3D object which exists only at time t and no other time. […] The upshot of this is that the intertranslatability of 3D and 4D descriptions rests ultimately upon entities which can be described indifferently as “instantaneous 4D temporal parts”, or “3D objects which exist at one time only”. (p. 574)
But in ensuring that sums-at-times satisfy (b), the authors compromise (a). Understood as entities that exist at only one time, sums-at-times are genuine additions to the endurantist ontology. And this is so independently of how ontologically promiscuous the endurantist decides to be about other issues (i.e. whether she accepts coincidence, arbitrary composition, etc) while still being endurantist.
Take an example. Suppose that there are two times t and t’ such that Tibbles does not change in its constituent particles from t to t’. Then the set of particles that constitute Tibbles at t is the same set that constitutes it at t’. However, given (b), (Tibbles, t) is not identical to (Tibbles, t’). They are two different entities, one existing only at t and the other only at t’. But why should the endurantist accept the existence of these two numerally distinct things, (Tibbles, t) and (Tibbles, t’)? She accepts the existence of Tibbles, the existence of times, and the existence of enduring particles that constitute Tibbles at different times. Let us assume that she will also accept the existence of sums of these particles. So she will accept the existence of a sum of particles that constitute Tibbles at t, and a sum that constitutes Tibbles at t’. But why should she say that these are two numerically distinct things? After all, they are composed of exactly the same enduring particles. Nothing in the endurantist’s position commits her with the existence of two things here. In fact, the endurantist position can be understood precisely as the negation of the existence of two distinct things in a case like this. So understood, the endurantist view is that there are sums-at-times, but not as many as the perdurantist think there are. Notice that the endurantist can have this view even if she accepts unrestricted mereological composition. The existence of two different sums-at-times in the example above does not follow from accepting arbitrary composition. It would follow from accepting arbitrary decomposition. But this is precisely the doctrine that the endurantist refuses to accept, and what makes her position non-equivalent to perdurantism. .