Nothing more than a modest proposal to spur discussion...!
In some civilization collapse scenarios that I am vaguely familiar with, they may have been proceeded by a dark age (like post bronze age collapse or post Roman empire collapse). But the world is connected in these days in ways that make that scenario seem less likely. Knowledge on things like fertilizer production are very wide spread and not central to one specific culture.
1. Industrial activity will continue to require nonzero human participation, even in the future.
2. Culture has no impact on the fertility rates of citizens in developed economies, they are incorrigible homo sovieticus, entirely shaped by material conditions and quality of life. Populations immigrating from other regions quickly reach the same quality of life, and follow suit.
3. There exists some level of economic contraction that would lead to present-day Western countries' inability to feed themselves. In one extreme case, if productive activity in the OECD went to zero, nobody would do anything at all, not even work knowledge-sector jobs; and therefore couldn't even afford to import fertilizer, food, etc. from other parts of the world. By a sort of intermediate-value theorem, assume the level in question exists, and lies between 0 growth and total cessation of industrial activity.
4. Starvation constitutes poor quality of life.
If the level of contraction (3) happened, I advance that one of two cases would happen.
1. People starve. The population does not stabilise. The marginal fertility response of humans in developed countries to this particular decline in living standards is zero or negative. The population continues to shrink. In the long run, if this happens again and again, the working-age population repeatedly falls below some number required to maintain successive level of industrial production, until there is fewer than 1 working-age human left. (This could take several hundred years.) By the first assumption, we need at least one person to continue all industrial production in the OECD countries, so at least one such minimum number exists; and so in that person's absence, we recover a situation with zero industrial activity. The remaining n non-working citizens cannot eat and quickly die.
2. People starve. The population does stabilise at a new equilibrium. This requires, by definition, at least replacement fertility in response to declining living standards, and constitutes a non-cultural solution as sought by the GP comment.
If at any point case (2) happens, then a level of industrial contraction of some degree solves the GP's request without requiring cultural factors. Otherwise, case (1) will happen again and again unconditionally, and we will go extinct in the long run.
It's a very unpleasant solution, but it does exist. I don't really believe that humans' behavioural responses are so non-smooth that the last two people on Earth would choose to have a single child, and then send that child off to learn the Haber-Bosch process -- it's only a limiting case.
As the most trivial example off the top of my head, I cannot imagine any situation remotely close to what you are suggesting without war. I don't think the cost of the precursors to fertilizer will stop Western countries from taking it.
At any rate, the idea that the nail in the coffin of our society will be our inability to produce fertilizer seems an unlikely and remote possibility. I'm not saying that there is no path to that outcome (we could probably spend all day making up scenarios where that is a possible outcome), it just seems rather unlikely.
The 20th century west made an all-in bet that those things might be correlated, and coaxed (and sometimes coerced) others into joining, but it's not clear that the bet is paying off or that it will.
The environment is the familiar one, but also at the expense of health safety (PFAS/lead/etc), and at the expense of our societal makeup. Perhaps the optimal distribution of labor for food production isn't 0.00001%, but would be good for growth and other intangibles if it were say 0.05%.
And worrying about 'what will happen if human population actually drops to zero' is like worrying about commuter traffic congestion on Mars.
Even in a worst-case projection scenario, like, say, China (Currently at 1.16 births/woman) in 2070:
https://www.populationpyramid.net/china/2070/
43.7% of the population is still working age (25-65).
And in 2100, it would be 41.2%.
As of 2020, it is 58.2%.
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Now, you tell me, in 80 years of technological progress and economic development, do you think China will be able to become 40% more productive? And manage to do ~the same amount of work with 30% fewer workers?
(PS. China's per-capita GDP grew 40 times in the past 40 years. It's full of smart people, I think it's going to figure something out.)
Why not? I don't think it will happen, but at the same time, I don't want to say anything about human behaviour without stating assumptions. If you think (as do I) that there is always some point above zero population at which birth rates do stabilise, then it's valuable to examine what exactly that would entail.
It sounds like we agree the point exists somewhere - we could extend your example to the whole Earth's population becoming 2 and the population becoming (8 bn / 2)% more productive. Why can't we extend it to the population becoming 1?
It's a completely different situation, and none of the shit we are arguing about applies to it. There's a fundamental difference between running and meeting the long-term needs of a human society of 10 billion people and a human society of 10.
The mathematics you're using is the same reduction to absurdity that leads a manager to believe that nine women can deliver one baby in one month.
Oh no, that would be terrible.