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That's what I thought. If your J-nat is such an aggressive cutter to step up from 4k it couldn't do the polishing job of a 30k+ stone :) I know fast polishers exist but really now, who jumps from...

You don't say what type of stone your Nakayama is, Asagi ,Suita, Namito ??
Just as an example I wouldn't dream of taking a 4k edge to my Nak Asagi.
As a polisher it is a relatively fast stone but...