On what basis is the name “Ephraim” given the etymology of “ash heap”?

Both fruit and ash come from the two-letter gate פֶר .

Gates are occasionally used as words. They are primarily metaphoric concepts which are narrowed by the intersecting ideas of the letters which are added to them.

פר means young bull. As such it is the metaphor of the young bull. It is viril (fruitful) and it is the sacrifice (ash heap).

The aleph א represents separation (see the heavens and the earth separated by the firmament), so the separated bull is the ash heap. The yud י represents the initial spark of creation and when added to the bull becomes fruitfulness.

It is made plural with the suffix. The metaphoric meaning of doubling is that the thing happens in heaven and on earth.