"If it's not physical, then what is it? Is it an algorithm? Well, for one, there is no precedent. Nothing else in the material world is purely an algorithm...except maybe a computer running an algorithm..."

I don't see the distinction you're making. I think I want to say "ok, so nothing is purely an algorithm". A computer relies on the rest of the universe just like your brain and I guess it's therefore a "physical phenomenon"?

In order to draw a distinction, perhaps one could say that a "pure algorithm" is something that can run on "many" substrates, so the difference between a "pure algorithm" and a "physical phenomenon" is about the variance in the substrates maybe? But everything is physical. Statements about abstractions are statements about patterns of physical things.