I detect a kind of "straw-man" argument against emergence here. My understanding of emergence comes out of Kolmogorov Complexity

https://en.wikipedia.org/wiki/Kolmogorov_complexity

One of the underlying principles of Kolmogorov complexity is the notion of a Universal Turing Machine. This concept illustrates the difference between reductionism and emergence quite nicely. Going back to Turing's original paper, the reductionist can claim that the Universal Turing Machine is reducible to the table of states that appear in the paper. But that is like having the pile of Legos that make up the Millenium Falcon set and saying that this pile is the model of the Falcon. It is not. The model emerges from the pile. Likewise, the list of states that make up the Universal Turing Machine is just a list. The identification of this list as a Universal Turing Machine emerges from the list, but the reduction of the Universal Turing Machine to the list misses the point. Eventually, the reductionist has to (maybe through clenched teeth) say that this list of states that they have in front of them is an example of a Universal Turing Machine. That statement is the admission that there is an emergent property that comes from that list of states. That property is different from the reduction.

As to Kolomogorov Complexity itself, a fundamental result actually formalizes the difference between reduction and emergence: the notion of an incompressible string. Basically, this string is not reducible to something simpler. It is what it is: it is emergent. Note that this notion of emergence has a profound relationship to the notion of randomness, as shown by Martin-Lof. In a fundamental sense, randomness is not as much a measure of our ignorance (which implies that if we knew more, we could reduce a seemingly random event into its constituents) but that randomness arises from the emergent nature of information.

To apply this distinction between reductionism and emergence to physics, there is a fundamental problem with the statement: "there is no evidence for the existence of any phenomena that falls outside of what reductionism is capable of explaining." A phenomenon is anything that can be observed. An explanation is a reason or justification for that phenomenon. But reductionism alone does not explain emergent phenomena.

To reduce a proton to its components - the quarks and the forces binding them - does not make the concept of the proton go away. The proton is emergent from the parts it is reduced to. But the proton is not just the sum of its parts. It is a phenomenon that is more than the sum of its parts. To explain a proton by reducing it to its parts is not really an explanation. In a very fundamental sense, a proton qua proton cannot be explained. It is the physical equivalent of an incompressible string. It just is: a brute fact.

https://en.wikipedia.org/wiki/Brute_fact

Another way of putting this distinction more formally is to look at what reductionism claims. To reduce a property to simpler elements is to describe that property, or in some formal sense "explain" it. Let us assume that the description or explanation can be expressed logically, or mathematically. In that case, the reduction provides an effective procedure for describing or explaining the property. But in that case, you have expressed the property in a simpler form - that of the effective procedure. Although this is possible for many properties, Kolmogorov complexity states that there are just as many properties that are incompressible - you just cannot make this simplification.

The advocates of emergence point out that evolution is an example of emergence. Evolution is an example of Solomonoff's theory of Inductive Inference, which is a version on Kolmogorov Complexity.

https://en.wikipedia.org/wiki/Solomonoff%27s_theory_of_inductive_inference

Kolmogorov himself credited Solomonoff with coming up with this notion first.