No, this is showing a counterexample, which would render the original theory moot. If we find a planet tomorrow that pushes you away rather than attracting you, then Newton’s theory of gravitation is (probably) no longer a valid model of the real world, or would have to be revised. That is just how science works.

Thanks for the detailed explanation, and sorry for the late response. Mine was just a simple counterexample to show that the tendency doesn’t always apply. You’re right that the c2 I used is wrong, and it should be s1+v1+c1, although that would still not change the result. My example was in the case where one producer wants to compete with another with a lower price, so chooses to trade a lower s for a bigger market share, so I wasn’t really getting into improved productivity, I was just addressing your initial statement of “competition forces prices down”.

In a real economy this chain would be much more complicated with way more steps and even backpropagation of some of the values. If we have a rate of decline of profit for company 1 called R1 and a rate R2, the overall R would only decline if R2 > R1, otherwise it would increase. So to prove a general declining rate of profit you would have to prove that the decline propagates fast enough through the entire chain.

Also, I fail to see how c/v (organic composition of capital) necessarily increases. If prices lower (due to competition, or productivity as you have said), then c will also decrease for the companies using those products (as I have shown in my example) as the cost of machines and input lowers (a computer in 2025 costs way less than the same one in 2000). To prove that c/v increases you would have to prove that dc > dv (derivatives), which is not at all clear since, while they both decrease, they can decrease at varying rates which are not predictable.

You didn’t address any of my concerns, nor was I talking about productivity. Let’s try again for the the first one with a simple example:

Company 1 makes a product (let’s say timber) at 50 surplus value. That 50 is a cost for company 2 that uses the product as an input material (it makes wooden chairs). We can calculate the total rate of profit of both companies. Now company 1 is forced to lower the price to 40 because of competition. We calculate the total rate of profit again and the total rate of profit has actually increased.

Thus, it does not follow that lowering prices/profits leads to a decrease in the overall rate of profit

Competition forces prices down, and rates of profit with it

This is not true in the general case. If prices for input materials are down, profits rise for the company using them. One company’s profit loss is another’s gain. That is even with the shaky assumption that competition can exist long term in a free market. Imperialism, as defined by Lenin, results in concentration of capital and the removal of competition.

this process can be struggled against by expanding markets or finding new industries

There are counteracting forces for it, but expanding is not one of them. Expanding does not change the rate of profit (profit/capital invested); at most, it changes the total profit.

There is no tendency of the rate of profit to fall. The theory is inconclusive, as is empirical research. If TRPF were true, then growing a company would, in fact, accelerate the process.