The latest core CPI data came out today and is up on FRED. I thought I'd post a picture of the model results without smoothing as I did for the PCE numbers that came out at the beginning of the month. However I noticed a pretty obvious lag in the CPI data relative to the model (there is no obvious lag in the PCE inflation data ), so I decided to try and do a four-parameter fit with a lag y0 (along with α, m0 and γ -- see paper).
Overall, it works pretty well (model = blue, data = green and 1-sigma error bands) with the lag being y0 = 1.2 years:
And here's a zoom to the last 15 years:
If the lag is correct, then the model gives the future values of core CPI (past the vertical line) using data available today.
Although less obvious, the lag has the opposite sign for Japan (y0 = -0.9):
 Here is the graph for reference:
One interesting thing is that maybe PCE inflation measures CPI inflation y0 = 1.2 years in the future -- which might (partially) explain why PCE inflation comes in about 0.3 percentage points lower (per Scott Sumner).