I experimented with this last year by pulling the ice which was around two inches thick in my buckets over top of the remaining two to six inches of unfrozen sap and saving both.
The unfrozen sap was high in sugar. If I remember correctly, it was about 4 or 5% compared to my normal 1.6% and boiled down nicely to a light color.
The ice was then melted and the sap that it made did not register on my hydrometer which is not very accurate at that low of a content.
I proceeded to boil it down to almost nothing and got nothing.
Before that, I saved the ice and boiled it all. Now I throw it all away since I have an insulated milk tank in the shade and I do not let my sap sit for very long.
Like the previous poster said, there has to come a point when too much freezing starts capturing sugar and then discarding the ice would be throwing away syrup.
2 x 6 A&A raised flue w/ preheater, OFA & UFA
600 gal Zero tank
125 taps on vacuum w/homemade releaser, 70 on gravity, 75 on buckets + a partner with 100 more buckets
7" short stack Wesfab press
Homemade water jacketed canner
Polaris Ranger sap hauler
Two English Setters to keep watch on things