This is not a proof. This is only advanced handwaving with a math notation cover to cloak it.
They propose a strange definition and model of consciousness and memory. And then, in that model the consciousness is (aparently) not computable. But they don't have any evidence that support their model of consciousness and memory. Moreover, the model is very unrealistic:
For example, from the article:
> What’s more, critics might point to other weaknesses in the formulation of this problem. For example, the proof that conscious experience is non-computable depends critically on the assumption that our memories are non-lossy.
They propose a strange definition and model of consciousness and memory. And then, in that model the consciousness is (aparently) not computable. But they don't have any evidence that support their model of consciousness and memory. Moreover, the model is very unrealistic:
For example, from the article:
> What’s more, critics might point to other weaknesses in the formulation of this problem. For example, the proof that conscious experience is non-computable depends critically on the assumption that our memories are non-lossy.