This seems to suffer from a finite-size effect. Wolfram's machines have a tiny state space (s ≤ 4, k ≤ 3). For some class of NP problems, this will be insufficient to encode complex algorithms and is low dimensional enough that it is unlikely to be able to encode hard instances ("worst case") of the problem class. The solution space simply cannot support them.

In this regime, hard problem classes only have easy solutions, think random k-SAT below the satisfiability threshold, where algorithms like FIX (Coja-Oghlan) approximate the decision problem in polynomial time. In random k-SAT, the "hardness" cannot emerge away from the phase transition and by analogy (watch my hand wave in the wind so free) I can imagine that they would not exist at small scales. Almost like the opposite of the overlap gap property.

Wolfram's implicit counter-claim seems to be that the density of irreducibility among small machines approximates the density in the infinite limit (...or something? Via his "Principle of Computational Equivalence"), but I'm not following that argument. I am sure someone has brought this up to him! I just don't understand his response. Is there some way of characterizing / capturing the complexity floor of a given problem (For an NP-hard Problem P the reduced space needs to be at least as big as S to, WHP, describe a few hard instances)?

I appreciated the great explanation of space complexity and it eludicated why some scientific authors don't include it in their analysis of algorithms. However, Wolfram found that "by successively investigating both larger inputs and longer runtimes, one can develop reasonable confidence that—at least most of the time—one is correctly identifying both cases that lead to halting, and ones that do not." There are exceptions like Machine 600720 that have exceptionally long runtimes but I gain a much better understanding about an algorithm if I'm provided the space complexity. It's still an open question in pure theory but it could be understood from empirical results.

"But what if one were to look at the question empirically, say in effect just by enumerating possible programs and explicitly seeing how fast they are, etc.?"

It is absolutely rammed with m dashes, which is not conclusive. For me, a bit of a clanger is that the writer might have decided to instruct the beastie to go fast and loose with grammar "norms". So, we have loads and loads of sentences starting off with a conjunction (and, but).

It just gets worse. The article is huge - it's over 17,000 words. I've skimmed it and its awful.

Please don't do this.