Inflation expectations in the Treasury market

In a post on another thread (Fed's estimate of current, forward inflation < 4.5%, roughly - #8 by WendyBG), there is discussion on inferring market expectations for future inflation from the Treasury bond market.

Actually, this statement is not correct:

Instead, the 5-year, 5-year forward rate measures expected inflation (on average) over the five-year period that begins five years from today.
Hence the word “forward.” Details of the correct formula are on the Fed’s site (5-Year, 5-Year Forward Inflation Expectation Rate (T5YIFR) | FRED | St. Louis Fed).

And the statement

is something different but related. It is the 5-Year Breakeven Inflation Rate (5-Year Breakeven Inflation Rate (T5YIE) | FRED | St. Louis Fed). It measures expected inflation (on average) over the five-year period that begins today (0 years forward).
Also, taking the difference in yields between nominal and TIPS bonds is typically used as an approximation to the exact formula (How to Calculate the Break-Even Interest Rate on Bonds | The Motley Fool).

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