No, the claim is that an explanatory framework limited to physical terms is not complete enough to encompass mental events.

I’m not sure if I’m reading you right. You seem to be saying some mental events don’t have physical causes, or maybe that some mental properties are not encoded physically.

In which case it sounds like there should be some mental events or experiences that are impervious to physical interference. So for example there is no conceivable physical intervention that could affect them. Do you believe there may be mental phenomena that are not causally related to physical processes in that way? It sounds like this should be testable.

"You seem to be saying some mental events don’t have physical causes, or maybe that some mental properties are not encoded physically."

I think he is only saying that an empirical language alone will not be enough to relate mental processes to physical ones. You have to define what a mental event really is before you can properly test anything about it, and depending on what definition you choose it might be inherently un-testable. If you define a mental process as being composed of physical processes from the start then of course you will conclude that all mental processes are influenced by physical ones. There are other possible definitions though such as physical processes being composed of multiple mental processes (the opposite of the last) in which case no physical test could be performed to verify a "change" occurred in the mental subject (but a "mental test" of sorts might still be able to tell us)

The existing terminology may be inadequate. But when we discovered atomic forces, and it turned out that Newtonian physics could not accurately describe them, we did not split the natural world at that boundary and claim atomic forces could not be explained by "appeal to fundamental laws of nature". No, we developed our understanding, phenomena that occurred inside nature came to be considered natural. Our understanding of what the physical world consisted of grew, it did not fragment.

the claim is not that they are not explained by laws of nature, but that they are not explained by physics.

It's like saying you can't model a football game in terms of the behavior of proteins. Sure there are proteins, and there is no magic anywhere, but the conceptual gap is just too vast.

Right, so they are meta-phenomena. They are still entirely physical. The actual process occurring is physical, but there is information being processed and meaning in what’s happening that can be analysed independently of the physical substrate.

It would be better to claim cannot be complete enough, and AFAIK that's still an open question. But people on HN don't seem to think it's a question at all, which shouldn't be the case.

They don't have to be supernatural because they could still host hypotheses that are observable and testable, and don't have to contradict other natural phenomena; but it could be that we would never be able to resolve in-practice how they connect to other phenomena with clarity.

No, the claim is that not everything in the world is physical, even if it all supervenes (is constituted by) the physical. See David Chalmers on property dualism and natural versus logical supervenience. Also, there is panpsychism and other versions of dualism. None of that needs to be supernatural. It's just a question of what the natural world is made up of.