In my original chained CPI article I noted that one reason deficit hawks like this method of cutting Social Security so much is that it's also a tax increase!
That's because Social Security checks aren't the only thing indexed to inflation, tax bracket thresholds are as well. So when the federal government switches to the C-CPI-U, households will "creep" upward into higher brackets a bit faster than they otherwise would. This raises about $60 billion in the relevant scoring window, but the White House doesn't want to count that $60 billion in the math perhaps because they don't want to draw attention to the fact that this—yes—violates their much-repeated pledge to never raise a dime of tax revenue for the non-rich. I've been saying that was a dumb pledge for years, so I don't mind. But it is what it is.