Yes, some things have been clarified. Generally, our models of reality have to deal with both substance and form as fundamental issues. Reductionism does this by talking about local materials which have inherent properties, while the holistic or systems view of reality talks about local degrees of freedom in interaction with global bounding constraints.
So a panpsychist is reductionist in seeking an explanation to something "higher order" like complexity or mind in terms of the micro-scale properties of the material realm, while the equivalent systems project is pansemiosis - the search for the ultimately simple, yet essentially scalefree, description of the "localised degrees of freedom in interaction with global constraints" relationship.
Penrose starts out acknowledging the importance of formal cause - global organisation - but his thinking quickly collapses into the search for some panpsychic property of matter.
His "non-computability" is of course the same as Peirce's abduction in talking about the ability of minds to think holistically about causality. Computability is again just the world according to SOV logic - modelling in terms of efficient causality, simple deterministic cause-and-effect. And the "non-computable" part of thought is the abductive jump to general principles, such as axioms, which then can be tested against reality for their pragmatic value. Humans can cope with vagueness or indeterminacy as a starting point for forming a systems view of what is going on. Turing machines can't.
Penrose - a card-carrying Platonist - does try to make some kind of systems sense of the issue with his "three worlds/three mysteries" model of metaphysics. He creates a self-closing circle of the three realms of form, material and mind. He says each arises from some small part of the prior and then fully encompasses the latter. So mind arises from a small part of total material possibility, form arises from a small part of total mental possibility, and materiality arises from a small part of total formal possibility. You go round in a circle with each realm having a restricted starting point that then unfolds into a new species of causal action.
So this is beyond dualism, and is a triadic story (like Popper, and of course Peirce/hierarchy theory).
As a grand metaphysical view, it has the interesting ring of truth to it perhaps. It seems superficially attractive.
But consider what is actally going on. First it fails completely as a causal
model. It is not telling us in what way each realm creates the next - in either a constructive or constraining fashion.
And then it only works at all by confounding the human creation of formal concepts (ie: epistemology, the modelling relation) with the actual existence of formal cause (ie: an ontological acceptance of the downward causality due to constraints). And also of course by accepting an ontological dualism of the mental and the material.
So (as befits a topologist
) he posits three ontic realms - formal, material, mental - then glues them into a circle by an illegal splicing of the formal realm. The mind's epistemic generation of mathematical models gets discretely twisted into the ontic concept of formal causality so that the connection can be made back to the material realm.
Confused? Penrose certainly is.
The Peircean or systems view does it differently. All arises out of the one-ness (firstness) of vagueness, then via dichotomistic separation (secondness) becomes the triadic causal relationship of a hierarchy (thirdness).
So it is a developmental view rather than the circular or Ouroboros logic Penrose uses. And it is a causal view because you end up with global constraints in interaction with local degrees of freedom (as the formal and material "realms"). And then you get reality itself as that which arises due to the action of this causality on a ground of raw potential, or vagueness. So reality is just whatever crisply exists within the constrasting limits of upwards and downwards causes. And this reality spans the gamut from the simple to the complex.
Something further is then required to explain this spectrum. Which is where some kind of thermodynamics must come in. Complexity is tied to the dissipation of gradients. Negentropy is the partner of entropification.
Which gets us back to pansemiosis - the story of how constraints get constructed. This is something that happens over all scales, from the simple to the complex. And the "realm" of mind is semiosis at its most negentropic, at its most complex. The reductionist question that Penrose wants to ask then comes down to an understanding of semiotic mechanism - what is the "least" form it takes. When the material world is being organised by downwards constraint, what is the simplest possible example of this kind of interaction?
As said, reality has gone through some phase transitions so far as semiosis is concerned. You have a step from genomic to memetic semiosis. And an even bigger one from a-biotic to biotic semiosis (as in the step from non-living dissipative structures such as gyres to living ones such as cells). The project for pansemiosis is then to define the essential causal mechanism in a way so general that it can encompass all these complicating transitions.
Panpsychism? Well that has only ever proved to be a cul-de-sac of metaphysical thought. An easy and tempting path for the reductionist to head down. But it is a blind alley, leading nowhere.
Penrose tries to suggest there is a magic door out of this cul-de-sac - his topology cut-and-splice trick with epistemology/ontology which rotates you through human modelling and back out into formal cause behind the concealing cloak of a "third Platonic realm".
, a theatrical wave of the wand, and you are stumbling blinking again into the street marked Materialism, ready for your next go-around of his Ouroboros coil.
A systems theorists instead says there are no "local properties", only a top-down restriction on degrees of freedom that thus creates degrees of freedom of some definite kind. The material "realm" is just as much an emergent aspect of reality as the global constraints which constitute "Plato's Heaven".
Which is not a bad thing, because all definite things are emergent in the systems view.