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I have a problem with your Possibilist TI that I also had with original TI, and with almost every ontological interpretation except for Bohmian mechanics - I can't figure out what the ontology is; nor even what the mathematical object is, that represents reality in the theory.
If Einstein had had his way, reality would have been described by classical fields on a manifold. Mathematically the universe would be represented by some particular exact solution of the equations of motion. Even given that, we could still ask the ontological questions like, what is a property, what is a causal relation and why does it necessitate anything, and so on; but at least the mathematics would be clear.
Quantum mechanics also has a certain clarity, if you resolutely regard it as not ontological, but just as an algorithm for making predictions. The observables are what's real, but they are an incomplete description of reality, and wavefunctions etc are a recipe for making predictions, whose reasons for working are unknown and remain to be discovered.
A peculiar laxity regarding the notion of reality, and regarding what counts as an adequate specification of an ontological theory, entered physics when people started trying to regard quantum mechanics as a complete theory of reality, rather than an incomplete one; and many ontological interpretations have inherited some of these lax attitudes, even as they try to restore objectivity to physical ontology. At least, this is how I explain to myself the oddities that I keep encountering in the literature on quantum foundations.
I will give another example of an ontological interpretation whose mathematical basis I think is clear - and it's a "back-and-forth-in-time" theory like TI: Mark Hadley's idea that QM arises from subatomic time loops. Hadley's ontology is like Einstein's, fields on a manifold, but the difference is that the manifold is non-orientable, it's full of little time loops, and quantum mechanics is supposed to arise from the global consistency constraints imposed by the coexistence of innumerable coexisting causal loops. The idea may or may not work, but at least the mathematical starting point is clear.
One more example of non-clarity before I turn to TI: MWI. MWI says that reality consists of one big wavefunction or state vector - OK, that much is clear. The non-clarity in this case comes when you ask, what parts of the wavefunction or state vector correspond to observable reality? Are the "worlds" the components of the wavefunction, when decomposed in a special basis? Or do all possible basis decompositions produce another, equally real set of worlds? Etc., lots of questions which have been raised many times on this site.
Now to TI. Let me give an example of an ontological claim that might have been made about TI, which would have provided a clear starting point. It could have been claimed that what exists are particles and fields. The particles trace out world-lines, the fields do their thing. And then the TI claim could have been, that the fields can be decomposed, in some specific way, into a particular set of advanced waves and retarded waves, which can be arranged into the "pseudo-time sequence" making up a "transaction".
That sounds like a clear starting point to me. And then the challenge would be to go into the details - describe how the decomposition works, and explain why the quantum formalism is the appropriate and correct way to compute probabilities in this world where influences are going back and forth in time "simultaneously".
That is not what I found in John Cramer. Instead, his only visible mathematical foundation is just, the usual quantum formalism. Then he has a few specific physical setups, where he tries to communicate the gist of the TI way of thinking. Also, as I recall, there is a path integral formalism in which advanced and retarded waves appear.
At this point, as a "philosophy of QM", TI appears structurally very similar to CI. The math is still just the same quantum formalism, perhaps amended to include advanced waves in the path integral. There is no clear mathematical description of the ontological part of the theory. Instead, there is just a way of thinking and a way of talking about the traditional quantum formalism. In CI, it's Bohr going on about complementarity and the uncertainty principle, in TI, it's Cramer going on about pseudotime sequences.
I have not yet seen your book, but so far, I don't find, in Possibilist TI, an improvement on this situation. Instead, there seems to be just an extra layer to the talking, in which "possibilities" are ascribed an important role. It's a little odd that something nonexistent should matter so much for the understanding of that which exists, but I can let that go, it's not my main concern. My main concern is just - what is the mathematical object, that corresponds to reality? I've already given two examples of theories where there is no mystery at all about what that is - fields on a manifold, and fields on a nonorientable manifold. I've also given a clear example of a theory that does not attempt to be ontologically complete, namely, QM with observables regarded as real, and wavefunctions regarded as not real.
What I would like to know is just, what sort of mathematical object describes the actual part of Possibilist TI ontology? Is it a definite history of particles and fields, which then gets ontologically analyzed in a certain way (and perhaps that is where the "possibilities" come in)? If I open your arxiv paper, I see kets, propagators, quantum fields, squared amplitudes, and a whole pile of stuff which just looks like standard quantum formalism. So it looks like you have produced just another way of talking about the quantum formalism, rather than a clear ontology whose objects can be specified with mathematical exactness. Please prove me wrong, and show me the part where you just say "These are the entities that exist, and these are the states they can have." :-)
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After some puzzlement (because it is so unlike what I expected), I think I now understand your interpretation. Possibilist TI is essentially a growing block universe which consists of a set of state vectors with a timelike partial order (a little like this), and the growth is a stochastic feeling out of immediate future extensions of this poset, via potential transactions.
For various reasons I can't believe in that as a final ontology, but I can imagine that it would have heuristic value, and maybe even practical value, for people trying to understand the nature of time and causal dependency in a universe containing backward as well as forward causality.
Thanks Mitchell -- it's only at the nonrelativistic limit that there is a timelike partial ordering in this sense, and that emerges stochastically from the relativistic level. I.e., there is no temporal causal relationship in the basic field propagation. So my picture isn't quite captured by the formulation in this paper (which also doesn't appear to address wf collapse and the possible relation of collapse to an emergent spacetime). But in any case, thanks again for your interest and I hope you will take a look at the book. The main dividend you get from the TI picture is a robust solution to the measurement problem, in contrast to the 'FAPP' quasi-solution obtainable from decoherence approaches. In particular, decoherence never gives true irreversibility, since you never get real collapse with decoherence alone. In PTI you get true collapse, which also sheds light on macroscopic irreversibilty. I discuss this in my book as well.