Obviously, when you are taxed, you lose wealth (or at least a potential claim on future wealth if you pay in cash). But that's different from the statement that, "Everytime you purchase something you give up wealth". If one properly understands trade, and assuming the trade is voluntary, the proper statement would be, "Every time you purchase something you gain wealth" since if you weren't better off from the trade you wouldn't make it in the first place.
If you purchase a stock that goes down, your actual wealth remains the same, just as it would if the stock went up. Your potential wealth has changed, or at least it is likely to once you sell.
Here's where we are different.
I do not believe that the purchase of a speculative investment, such as purchasing stock, is what you call 'actual wealth'. It has a component of risk associated with it, so it could be partially or totally lost. How can you call that 'actual wealth'?
Wealth is a more tangible asset than that. Although you appear not to agree that money is actual wealth, a savings or checking account to me is actual wealth as it stands little risk of not being there when needed.
I also see you added an adjective of 'potential' to the word wealth. I do not agree with you that your actual wealth remains the same when purchasing a stock that goes down.
I also do not agree with the statement
"Every time you purchase something you gain wealth" since if you weren't better off from the trade you wouldn't make it in the first place.. At times one makes purchases not to increase wealth but to cover necessities. At other times one makes purchases to speculate for
potentially increasing wealth. Those latter amounts are subject to risk once again and could result in a
decrease in wealth.
Maybe the word wealth is where all the problems are here and the adjectives we give it.