Concerning/associated with the carbon migration.... here's something to think about, that I have come to realize over the years of doing it, is true.....
The best looking SS San-Mai, is a San-Mai with the poorest welding/forging technique. The opposite is also true.... the better the wedling/forging technique, the poorer the look. Sound crazy? Here's the explaination.... the best looking SS San-Mai has the biggest/boldest carbon migration "stripe" where the SS and the core/carbon steel meet. In order to achieve this, the billet must be soaked too long, at too high a temp, and there is significant carbon migration into the SS, which creates that bold stripe that everyone likes/have come to expect with a SS San-Mai. When good/proper welding/forging technique is used, the carbon migration is far less, and consequently, that "stripe" of carbon migration is far less (as in far more narrow).
The good news is... if you know this, and take it into consideration when building the billet, you still can come out with a great cutting, high quality blade. That is accomplished by leaving the core a bit thicker..... that way you make it past the depth of carbon migration when grinding/in the finished blade.