I can't help with your question, but maybe with some definitions:
Smelting ist not melting.
While smelting is a pyrometallurgical (chemical) process, often used to prepare ore for further processing or to gain the metallic form by the means of redox processes,
melting is the physical process to bring solids to their liquid aggregate state by the means of heat.
On the forum I read, that the continious remelting of gold alloys will reduce the quality of their physical proberties.
Since you have to expect that there isn't only one kind of 10K yellow gold but a thousand different alloys of it - not to mention all the ugly solders used, that will alloy with the rest of your "high-karat-gold", it might be hard to obtain a consistent qualtity or any appreciable quality at all.
That is why there are refiners.
The word "inquartation" is, as far as I know, only used for the process to alloy high-karat alloy with silver or copper in order to dissolve silver and basemetals by the means of nitric acid before further processing of the gold.