Searching with a X2 mage (no crosspaths) using a remote search spell for a given number of turns, then searching manually by walking and using the search order on a new province every other turn for another given number of turns.

C: cost of the remote
r: number of provinces searched with a remote
N: number of provinces to cover with the mage
T: number of turns considered total (T gotta be larger than 2N here)
G: the expected gem income from the remote spell, g the gem income from the mage search (per turn value)

The expected number of gems received since starting the remote searching is:

-C·r ——————————- cost of the r remotes
+ r·(r+1)/2·G ——————— gems found by the remoted target provinces during the r turns of remote searching
+ (N-r)·(N-r+1)/2·g·2 ————– gems found by the remoted target provinces during the 2·(N-r) turns of walking search
+ 2·(N-r)·r·G ———————- gems found by the N-r provinces searched manually during the 2·(N-r) turns of walking search
+ (T-(r+2·(N-r))·(r·G+(N-r)·g) —– gems found by all the provinces during the T-(x+2·(N-x) turns left after everything has been searched

to be compared with N·(N+1)/2·g·2 when casting 0 remotes .

Here is the difference for 15 provinces and T=2x15 turns of horizon (that's a lot) as a function of the number r of remotes used: https://www.desmos.com/calculator/hlhzskmuvr

The optimum way of searching those 15 provinces with a single mage if you look at your gem income in 30 turns (ie when the mage has finished walking in the no remote scenario) seems to be 5-8 casts of remotes (less than a gem of difference). (assuming no error in the modeling)