Chalmers' zombie argument, best presented in The Conscious Mind, concerns the ontological status of phenomenal consciousness in relation to physics. Here I'll present a somewhat more general analysis framework based on the zombie argument.

Assume some notion of the physical trajectory of the universe. This would consist of "states" and "physical entities" distributed somehow, e.g. in spacetime. I don't want to bake in too many restrictive notions of space or time, e.g. I don't want to rule out relativity theory or quantum mechanics. In any case, there should be some notion of future states proceeding from previous states. This procession can be deterministic or stochastic; stochastic would mean "truly random" dynamics.

There is a decision to be made on the reality of causality. Under a block universe theory, the universe's trajectory consists of data specifying a procession of states across time. There are no additional physical facts of some states causing other states. Instead of saying previous states cause future states, we say that every time-adjacent pair of states satisfies a set of laws. A block universe is simpler to define and analyze if the laws are deterministic: in that case only one next state is compatible with the previous state. Cellular automata such as Conway's Game of Life have such laws. The block universe theory is well-presented in Gary Drescher's Good and Real.

As an alternative to a block universe, we could consider causal relationships between physical states to be real. This would mean there is an additional fact of whether X causes Y, even if it is already known that Y follows X always in our universe. Pearl's Causality specifies counterfactual tests for causality: for X to cause Y, it isn't enough for Y to always follow X, it also has to be the case that Y would not have happened if not for X, or something similar to that. Pearl shows that there are multiple causal networks corresponding to a single Bayesian network; simply knowing the joint distribution over variables is not enough to infer the causal relationships. We could imagine a Turing machine as an example of a causal universe; it is well-defined what will be computed later if a state is flipped mid-way through.

These two alternatives, block universe theory and causal realism, give different notions of the domain of physics. I'm noting the distinction mainly to make it clearer what facts could potentially be considered physical.

The set of physical facts could be written down as statements in some sort of axiomatic system. We would now like to examine a new set of statements, S. For example, these could be statements about high-level objects like tables, phenomenal consciousness, or morality. We can consider different ways S could relate to the axiomatic system and the set of physical facts:

1. S-statements are not well-formed statements of the axiomatic system.
2. S-statements could in general be logically inferrable from physical facts. For example, S-statements could be about high-level particle configurations; even if facts about the configurations are not base-level physical facts, they logically follow from them.
3. S-statements could be well-formed, but not logically inferrable from physical facts.

In case 2, we would say that S-statements are logically supervenient on physical facts. Knowing all physical facts implies knowing all S-facts, assuming enough time for logical inference. Chalmers gives tables as an example: there does not seem to be more to asserting a table exists at a given space-time position than to assert a complex statement about particle configurations and so on.

In case 3, we can't infer S-facts from physical facts. Through Gödel's completeness theorem, we can show the existence of models of the axiomatic system and physical facts in which the S-statements take on different truth values. These different models are in some sense "conceivable" and logically consistent. S-facts would then be "further facts"; more axioms would need to be added to determine the truth values of the S-statements.

So far, this is logically straightforward. Where it gets trickier is considering S-statements to refer to philosophical entities such as consciousness and morality.

Suppose S consists of statements like "The animal body at these space-time coordinates has a corresponding consciousness that is seeing red". If these statements are well-formed, then it is possible to ask whether they do or do not logically supervene on the physical facts. If they do, then there is a reductionist definition of mental entities like consciousness: to say someone is conscious is just to make a statement about particle positions and so on. If they don't, then the S-statements may take on different truth values in different models compatible with the same set of physical facts.

This could be roughly stated as, "It is logically conceivable that this animal has phenomenal consciousness of red, or not". There is much controversy over the "conceivability" concept, but I hope my formulation is relatively unambiguous. Chalmers argues that we have strong reason to think phenomenal consciousness is real, that we don't have a reductionistic definition of it, and that it is hard to imagine what such a definition would even look like; accordingly, he concludes that facts about phenomenal consciousness are not logically supervenient on physical facts, implying they are non-physical facts, showing physicalism (as the claim that there are no further facts beyond physical facts) to be false. (I'll skip direct evaluation of this argument; the purpose is more to present a general analysis framework.)

Suppose S-statements are not logically supervenient on physical facts. They might still follow with some additional "metaphysical" axioms. I will not go into much detail on this possibility, but will note Kant's Critique of Pure Reason as an example of an argument for the existence of metaphysical entities such as the a priori synthetic. Chalmers also notes Kripke as making metaphysical supervenience arguments in Naming and Necessity, although I haven't looked into this. Metaphysical supervenience would challenge "conceiveability" claims by claiming that possible worlds must satisfy additional metaphysical axioms to really be conceivable.

Suppose S-statements are not logically or metaphysically supervenient on physical facts. Then they may or may not be naturally supervenient. What it means for them to be naturally supervenient is that, across some "realistic set" of possible worlds, S-statements never take on different truth values for the same settings of physical facts.

The "realistic set" is not entirely clear here. What natural supervenience is meant to capture is that a functional relation between physical facts and S-facts always holds "in the real world". For example, perhaps all animals in our universe with a given brain state have the same phenomenal conscious state. There would be some properties of our universe, similar to physical laws, which constrain the relationships between mental and physical entities. This gets somewhat tricky in that, arguably, only one set of assignments of truth values to physical statements and S-statements corresponds to "the real world"; thus, natural supervenience would be trivial. Accordingly, I am considering a somewhat broader set of assignments of truth values to physical statements and S-statements, the realistic set. This set may capture, for example, hypothetical universes with the same physics as ours but different initial conditions, some notions of quantum multiversal branches, and so on. This would allow considering supervenience across universes much like our own, even if the exact details are different. (Rather than considering "realistic worlds", one could instead consider a "locality condition" by which e.g. natural supervenience requires that phenomenal entities at a given space-time location are only determined by "nearby" physical entities, as an alternative way of dodging triviality; however, this seems more complex, so I won't focus on it.)

Chalmers argues that, in the case of S-facts being those about phenomenal consciousness, natural supervenience is likely. This is because of various thought experiments such as the "fading qualia" thought experiment. Briefly, the fading qualia thought experiment imagines that, if there are some physical entities (such as brains) that are conscious, and others (such as computers) that are not, while having the same causal input/output properties, then it is possible to imagine a gradual transformation from one to the other. For example, perhaps a brain's neurons are progressively transformed into simulated ones running on a computer. The argument proceeds by noting that, under these assumptions, qualia must fade through the transformation, either gradually or suddenly. Gradual fading would be strange, because behavior would stay the same despite diminished consciousness; it would not be possible for the person with fading consciousness to express this in any way, despite them supposedly experiencing this. Sudden fading would be counter-intuitive due to an unclear reason to posit any discrete threshold at which qualia stop.

One general objection to natural supervenience is epiphenomenalism. This argument suggests that, since physics is causally closed, if the S-facts naturally supervene on physical facts, then they are caused by physics, but do not cause physics. Accordingly, they do not have explanatory value; physics already explains behavior. So Occam's razor suggests that these statements/entities should not be posited. (Yudkowsky's "Zombies? Zombies!" presents this sort of argument.)

Here we can branch between block universe theory and causal realism. According to the block universe theory, physics simply makes no statement as to whether some events cause others. So the notion that physics is causally closed is making an extra-physical claim. This is a potential obstacle for the epiphenomenalism objection. However, there may be a way to modify the objection to claim that S-facts lack explanatory value, even without making assumptions about physical causality; I'll note this as an open question for now.

According to causal realism, physics does specify that physical states cause other physical states. Accordingly, the epiphenomenalism objection holds water. However, causal realism opens the possibility of epistemic skepticism about causality. Possibly, physical events do not cause each other, but rather are caused by some other events (N-events); N-events cause each other and cause physical events. There is not an effective way to tell the difference, if the scientific process can only observe physical events.

This possibility is somewhat obscure, so it might help to give more motivation for the idea. According to neutral monism, there is one underlying substance, which is neither fundamentally mental nor physical. Mental and physical entities are "aspects" of this single substance; the mental "lens" yields some set of entities, while the physical "lens" yields a different set of entities. The scientific process is somewhat limited in what it can observe (by requirements such as theories being replicable and about shared observations), such that it can only effectively study the physical aspect. Spinoza's Ethics is an example of a neutral monist theory.

Rather than explain more details of neutral monism, I'll instead re-emphasize that the epiphenomenalism objection must be analyzed differently for block universe theory vs. causal realism. These different notions of the physical imply different ontological status (non-physical vs. physical) for causality.

To summarize, when considering a new set of statements (S-statements), we can run them through a flowchart:

1. Are the statements logically ill-formed in the theory? Then they can be discarded as meaningless.
2. Are the statements well-formed and logically supervenient on physical facts? Then they have reductionist definitions.
3. Are the statements well-formed and metaphysically but not logically supervenient on physical facts? Then, while there are multiple logically possible states of S-affairs given all physical facts, only one is metaphysically possible.
4. Are the statements well-formed and neither logically nor metaphysically supervenient on physical facts, but always take on the same settings given physical facts as long as those physical facts are "realistic"? Then they are naturally supervenient; physical facts imply them in all realistic universes, but there are metaphysically possible though un-realistic universes where they take on different values.
5. Are the statements well-formed and neither logically nor metaphysically nor naturally supervenient on physical facts? Then they are "further facts"; there are multiple realistic, metaphysically possible universes with the same physical facts but different S-facts.

This set of questions is likely to help clarify what sort of statements, entities, events, and so on are being posited, and serve as a branch point for further analysis. The overall framework is general enough to cover not just statements about phenomenal consciousness, but also morality, decision-theoretic considerations, anthropics, and so on.