This is a straw man, as I didn't claim that something "suddenly appears" at the nth step of induction. (See my words in Italics at the end of paragraph below, or re-read carefully what I said before.)
Ergodicity, pressure, temperature, chemical equilibrium, planetary formation, ecosystems, degenerative syndromes, differential cell development, protein synthesis, chaperonin-regulated protein function, embyonic development, feedback mechanisms, viral population dynamics, animal behaviour, population equilibria... and all that. IOW, everything except the raw summary of the fundamental laws of Nature. None of these things can be seen in the raw equations of physics. They appear somewhere along the huge buildup of complexity from the elementary particle to swarms of billions and billions of them.
But it's not like: "Now!, the adding of one particle has made it because..." It's gradual, rather.
OK. I didn't claim to have answers for everything. Although it's not really so much that I have no answer for it. I never have answers for questions I do not understand.
Here I have to put my foot down. Either we agree on what you mean by random, or we stop talking about this. Otherwise I might be talking about an elephant, and you be talking about a mouse --in a manner of speaking. Non-random is just a limit in a sequence of probability distributions of decreasing entropy. A probability distribution with entropy equal to the natural logarithm of the number of states is very random. A probability distribution of 10-100 entropy (just a small fraction of unity) is almost deterministic. Only at zero entropy we are at the non-random (deterministic) realm. So again, what do you mean by random?
Living things have somehow "managed" to exploit regions of very, very low entropy (very non-random in that sense).
Ok.
If you wan to open that can of worms, it's ok. Only be aware it is a can of worms. Scenarios in which constants of physics may be changing in a much, much wider context could make a universe in which life, consciousness, etc can arise actually in an inevitable way. The possibilities are endless. Remember Haldane:
IOW: What meta-conditions could make what we see as a formidable coincidence actually inevitable? This should give you pause. One man's coincidence is another (better informed) man's inevitability.
If there is no mind (as a thing separate from matter) certainly there can be no mind using my mind. As Schopenhauer said, man can do what he wants, but he cannot want what he wants. Can you choose what mind you will have tomorrow at 10:30? Nah. It doesn't make sense. Mind must arise from something physical. There is enough mathematical leeway for me to think that mind is something that arises physically. The simple-minded mechanistic view of the universe is long dead and gone.
Modern science does not claim full understanding. I'm assuming by "materialism" you mean that. "Pattens of behaviour in matter" is vague enough that it can include pretty much everything, so I cannot see how it could be weak. Not precise is OK, but not weak.
Let me put it this way: Life (and mind, as a consequence) finds its way by following its grove. Only it is a much more intricate grove than the one found by planets and asteroids. Impossible to see by just solving an equation from any simple statement of principles.