We have very clearly just found the lower limit on unemployment which will then produce inflation. We're never going to hit a limit of 0%, there will always be people who are getting fired, and some people who are pretty unemployable. The limit will also change based on conditions and the lower limit from decades ago or decades in the future may be different from today. There's no known good algorithm to tell you what that magic number is.
I'd also argue that "inflation death cycle" packs a lot of negative political assumptions into it. And the "death spiral" economists were typically worried about decades ago was the deflationary death spiral of savings gluts causing deflation causing more savings.
And my assertion is that "the Fed conquering inflation" cannot be read off of the CPI numbers for the past 6 months. We're still at what the Fed would call "full employment" and inflation is lurking and all we've seen is a short term abatement in inflation. If the economy revvs up again, then in the short term the curve will be the same and inflation will reappear.
For the Fed to consider the job done the curve needs to move so that the economy can rev up again without inflation reappearing. That will only happen if there's more slack in unemployment numbers. If we are at 6-8% unemployment then it will take time for the unemployment rate to fall back down to 3.5% and for inflation to reappear. By creating more unemployment you shift the curve so that the economy can run hot for awhile and grow while keeping wage inflation down. So shifting that curve is what the Fed's actual goal is, they're looking beyond what last months inflation numbers were (and they look not only beyond the y-o-y headline inflation numbers but they look beyond the m-o-m inflation numbers). They want to shift the curve, and the way to shift the curve is to have more slack in the jobs markets, which is economist-speak for a higher unemployment rate.
They also will very likely overshoot whatever target they think they have, they don't have a magic wand to control the economy. They mostly just detonate a few bombs in the economy and they don't know how much dry powder is waiting to go off in secondary explosions. Or the way this is usually phrased, "when the tide goes out we find out who was swimming naked".
And there's a tension here in HN between the oft stated belief that a decade of low rates has led to all kinds of malinvestment, and the belief that we're due for a soft-landing kind of recession. One of those I think has to be incorrect. And looking at the vacancy rates in downtown cores, etc I'm pretty sure the soft-landing story is going to be the wrong one. There are a lot of entities surviving on persistently low short rates, and they're going to go under once their loans adjust. The strict capitalists will celebrate this as bad investments going under and getting flushed, but that comes with pain, just like it was painful in 2008.