Now wealth income can easily be spent on large amounts of things that are typically seen as CPI items. At any moment the top 10 billionaires can choose to spend their wealth on an extravagant amount of labor, drastically rising labor incomes, but they choose not to as it lowers their long-term utility.
The fed's mandate is to create a system where they don't mix. It must always be the case for wealth assets like fancy homes and stocks to on average rise in value more than CPI items. If at any point the reverse is true, all of the wealthy will liquidate their stockpiles of stocks(crashing the stock market in the process) and instantly transfer their wealth into the thing that brings the best returns. If the best returns is GPU's, gamers won't get to game. If the best returns is on beans and rice, people will starve.
One of the best ways that the fed does this is by intentionally creating recessions whenever CPI inflation kicks up. Whenever oil money starts targeting water money, the fed kicks up interest rates above the 30 year[0], triggering a yield curve inversion, which triggers a recession. The recession forces employers to let employees go, which drives down the demand on CPI items, and that reduces inflation. At the same time, they put interest rates below what it was when it started, and the wealthy move their money back to oil.
(Note: I'm not an economist, just a programmer with an interest in economics and an eye for details. I'm mostly posting here for feedback, and I know I'm not exactly correct, but every post is one more chance for clarity.)