Various physicists, famously John Wheeler, have asserted that physical information is the central object of study in physics, in the sense that an object or concept is "physically meaningful" if it makes a difference to the information that can be extracted from (actual or hypothetical) physical experiments. For example, gauge transformations are "unphysical" in this sense.
Some have gone further and proposed that the Universe is "computational in essence": that, at the most fundamental level, it consists of the processing of information according to fixed rules; see e.g. the Stanford Encyclopedia of Philosophy's article "Computation in Physical Systems" and Wikipedia's article on digital physics.
I am trying to understand the possible interpretation and coherence of this proposal.
Robin Gandy, in "Church's Thesis and principles for mechanisms", formulated a broad model for parallel computation subject to "locality" constraints. The model subsumes Turing machines and cellular automata, and is given in terms of hereditarily finite sets, although John Byrnes and Wilfried Sieg have recast it in terms of locally finite labelled graphs, in "A graphical presentation of Gandy's parallel machines".
Since the information processing in a Gandy machine is local, it seems plausible to me that this formulation is already consistent with relativity; if not, I can't imagine it being difficult to modify the definitions to make them so.
The "pancomputationalist" proposal might then be formalized as claiming that the Universe is (equivalent to) a "Gandy-like" machine. My question is: Is it possible for this proposal to be consistent with current physical observations? More precisely, are there any known obstacles to the possibility that some Gandy-like machine, with certain initial conditions, could exhibit "large-scale" behaviour consistent with current physical models? I am aware of several possible objections:
The Church-Turing thesis may be false, in that there could be physical systems ("hypercomputers") capable of performing computational supertasks. An example is given by Oron Shagrir and Itamar Pitowsky in "Physical Hypercomputation and the Church-Turing Thesis". Whether such a device is physically possible is obviously unknown, but if so it would refute the proposal.
Bell's theorem essentially* rules out local hidden variables, so the proposed "locality" of computation may not correspond simply to locality in spacetime (*another possibility is to reject counterfactual definiteness; this position is known as superdeterminism).
Even the state of a simple quantum mechanical system resides in a Hilbert space and so contains an infinite amount of classical information. That said, the amount of information which may actually be extracted from a system (and is hence "physically meaningful") is finite. In particular, there is a bound due to Jacob Bekenstein which arises when considering thermodynamics together with general relativity. It states that the entropy (or, to my understanding, "classical information") enclosed in a region of space is bounded from above by the surface area of the region.
I have tagged this question as soft since it may not have a definite answer; nevertheless I would find relevant literature and discussion very useful.
Some may contend that this question belongs on physics.stackexchange rather than here; however, my perspective is that the question - at least, the more precise version of it stated above - is ultimately a mathematical one.