Falsified by the global economic growth in the period between Adam Smith and Karl Marx.
Once we reach the end of the road for further invention and improvements (even if that is in the form of our own capacity and theoretically there's more), that's when capitalism becomes zero sum. Until than point, capitalism is a way of distributing more resources to people who are better at finding those improvements.
"Distributing" doesn't require "evenly". Amazon has distribution centres, they don't send stuff to all customers equally regardless of what is ordered.
The entire foundational assumption of (Smith's) capitalism is the idea that people who are better at making a profit are exactly who should be given more money to work with, and that this benefits all of society — and while I will agree that the phrase "trickle down economics" doesn't fit reality of the behaviour of billionaires (who act more like aristocrats), that's where that phrase comes from, and it seems to often work up to deca-millionaires at least.
The flaw with this (even in the case of millionaires) is it presumes no parasites. As it happens, both Smith and Marx noticed this, but as history shows, the proponents extolling the virtues of each were not very effective at preventing economic parasites.
If I read an article that titled "this NGO is distributing food and medical aid to the refugees of the conflict" and then on the body of the article I find out that a single guy got all the supplies and the rest got nothing, I would consider the title very misleading.
But the distribution for wealth under capitalism isn't "one" person, and capitalism wouldn't work if it was, because nobody else could buy anything with the money, and therefore everyone else would invent a new currency or barter, and then all the money which that one person has would be worthless.
The more capital you have, the more purchasing power you have relative to the other players in the capitalist game. There is a fixed amount of purchasing power (the total sum of capital). Increasing currency in circulation does not increase the number of things that can be purchased with that currency, it just changes the distribution of purchasing power.
The US Federal Reserve data indicates that the top 10% of households own about 67% of the nation's wealth, while the bottom 50% hold less than 4%
Also let us not forget other tools that have significantly helped capitalist nations: Slavery, Colonialism, Imperialism, Sanctions, Wars, Coup d'etat's... these have all contributed to what capitalism is today. So yes Capitalism in the grand scheme of things is entirely zero sum.
Nothing of what you said has anything to do with the concept of "zero sum". It has to do with if the resources in a system are fixed or if they are changing.
"zero sum: relating to or denoting a situation in which whatever is gained by one side is lost by the other."
The data from the Fed show great inequality which can be expected from a zero-sum game. Another example check how under develop DR Congo is in the electronic age, while hardware (and to an extend software) companies are some of the most valuable companies, they all source raw materials from Congo.
Its like chess in that white has an advantage because they get to play first. The rich get richer because you need capital to efficiently participate in capitalism and it has a compounding effect. It than becomes a zero-sum game where the poor are trying to play catch up with the rich. For example if you start an innovative company the rich can simply buy it or start a competing company that is hard to compete against because of their resources.
