I asume OP is referring to:

population increases geometrically subsistence increases arithmetically

Let’s first remember that really simple functions can produce really complex patterns. We will use the Logistic map as an example, both because it is the simplest equation exhibiting chaotic properties, and because it can be used to model population. We can consider population growth to be dependent on population so that it increases geometrically: dN/dt=rN, can be integrated to N(t)=N^(rt) Where “N” is your population, and “r” is a proportionality constant.

You then add the condition that once there are no longer enough resources, population growth stops: dN/dt =0, when (N=K) Where “K” is the maximum sustainable population. One of the conditions when this will be true is if there is a term, (1-N/K). We then have: dN/dt= rN(1-N/K)

In this paper they used this equation to produce values of “N” and “K/N” and compare them with population and percapita income in different Chinese dynasties. They find strong correlation.

“K/N” is a measure of the aggregate per capita income, then, the term (1-N/K) implies reproductive success increases with wealth. This seems to have some historical basis as per the responses given in this other thread however as my post there states, this is not always the case. In this article they look at England and new France and conclude that this relationship is clearer when the society is reaching “equilibrium”, that is when it is close to its carrying capacity.

We can complicate the model by further dividing the society into different sub populations according to their economic behaviors. For example, in a pre-industrial society elites are of a predatory nature in which they extract their sustenance from the peasantry. We the have 2 sets of Malthusian equations one describing each. This can be further generalized into the Lotka-Volterra equations to include all sorts of relationships partaken by any number of distinct agents. Since this gets very complicated really fast, we shall stick with the most elementary form of predator prey dynamics. This book analyzes population and economic data of many different geographical places and time periods and makes a compelling case to explain them by using a predator prey model.

The expansionary parts of the cycle, where there are few and cohesive elites, a large economic and population surplus, would put a society in a position to conquer others, this was the topic of this other book. Which unfortunately I don’t have in digital format. I scanned one chapter about how you can model the Mongol invasion using an analogy with the population dynamics of lemmings or voles. We should keep in mind that many of the examples and formulas, I have used and referenced in this post are very simple, to better illustrate a point. But one can build very elaborate equations parting from these even the modeling of vole populations is a complicated affair.

Of course, there are other complications, in the Clark and Hamilton article linked above, the most significant predictor of poor reproductive success in medieval England was living in London. Since cities are often wealthier than the surrounding countryside this is a strong indication that urbanization runs according to non-Malthusian dynamics. There is after all evidence that some places in the world experienced a demographic transition before the industrial revolution, this might have been a consequence of the above mentioned urbanization dynamics, class competition, the dynamics of capital accumulation, cultural and other factors.

The question should not be is the above the only law of history but rather is it one of them? I hope to have made a compelling case for that position.