SeashellGull congrats on a well-reasoned argument delivered via a well-written essay. I have sometimes thought about what I like to call the 'paradox of maintenance': as every homeowner is keenly aware, any constructed, artificial structure needs near-constant upkeep not to succumb to decay; while on the other hand, the weeds in the garden need cutting down to be kept from taking over. What's built crumbles, what grows flourishes, if left to its own device.
It is an intriguing idea to pose this as the unifying principle behind life. While it's not without its precursors (e.g. Jeremy England's theory that more organized assemblies of molecules are better at dissipating waste heat than random conglomerates), you link it to a notion of memory: a continuation of form in the face of the thermodynamic tendency towards dissipation. This puts the focus less on the structures, and more on the dynamics they are subject to. The same structure in a cellular automaton with a 'forgetful' rule might dissipate, while flourishing within an environment that allows it to retain traces of its earlier form. Of course, there might be a degree of interplay here: one could imagine a complex structure within a forgetful CA that has some sub-part that effectively acts as a memory, such as the tape of von Neumann's universal replicator or even a Langton loop (as opposed to something like a glider in GoL).
But I've got a few questions. One is that maybe mere persistence isn't quite enough for life. Consider the aforementioned glider: it succeeds in self-preservation, but effectively only because it lives in a 'sterile' environment: there is no outside influence on it, which would immediately disrupt its continuity. Life, on the other hand, is always (and necessarily) in contact with the environment, and persists despite this constant Brownian bombardment. It would be interesting to see if your idea of non-Markovian replication might actually be helpful, here: modify your thought experiment to a CA which includes a little bit of random noise (i.e. sometimes cells just randomly transitioning between states), and monitor to what degree your self-replicating structure preserve coherence across time.
Also, I'd be interested to see if the quantum version has some quantitative (or perhaps even qualitative) advantage over its classical analogue. I could very well imagine that within a quantum random walk, you can exceed measures of coherence between successive states beyond what is classically possible, e.g. as in a Leggett-Garg inequality. This would be an intriguing hint towards the indispensability of quantum mechanics for life: perhaps there are settings such that only this added coherence is sufficient to guarantee a minimal degree of persistence.
But anyway, all in all, I thoroughly enjoyed your essay, and hope it will do well in the contest!
