The 10000 individual sets of clues can be uniquely solved with the additional constraint that every number from to 9999 appears exactly once. The final grid is available here.
As suggested by the individual puzzles, it is useful to check which entries are triangular numbers. There are 141 of them, and this yields the following grid:
XXXXX X X XXXXX X X
X X X X XX X
X X X X XXX X
X X X X X X X X
X X X X X X XXX
X X X X X X XX
X X X XXXXX X X
XXX XXX XXXXX X X XXXX
X X X X X XX XX X
X X X X X X X X X
XXX XXX X X X X XXX
X X X X X X X
X X X X X X X
X X X XXXXX X X XXXX
Doing the same thing for the 409 twin primes which appear in the grid (205 pairs, but 5 appears in two pairs) yields:
XXXXXXX XXXX XXXXX XXXXX
XXXXXXX XXXX XXXXXX XXXXXXX
XX XX XX XX XXX XX XX
XX XX XX XX XX XX
XX XX XX XXX XX XX
XXXX XX XXXXXXX XX XX
XXXX XX XXXXXX XX XX
XX XX XX XXX XX XX
XX XX XX XX XX XX
XX XX XX XXX XX XX
XX XXXX XXXXXX XXXXXXX
XX XXXX XXXXX XXXXX
XX XX XXX XXXXX XXXXX XXXX
XXX XX XXXXX XXXXX XXXXX XXXX
XXXX XX XX XX XX XX XX
XXXXX XX XX XX XX XX XX
XX XXXXX XX XX XX XX XX
XX XXXX XXXXXXX XX XX XX
XX XXX XXXXXXX XX XX XX
XX XX XX XX XX XX XX
XX XX XX XX XX XX XX
XX XX XX XX XX XX XX
XX XX XX XX XXXXX XXXXX XXXX
XX XX XX XX XXXXX XXXXX XXXX
There are 20 Fibonacci numbers in the grid, taking f(0) = 0. Doing the same produces a square:
XXXXXX
X X
X X
X X
X X
XXXXXX
Finally, taking the 100 square numbers which appear produces the answer:
XX X X XXX XXX
X X X X X X X
XX XXX X X XXX XX
X X X X X X X X X
X X X X XXX X X XXX
X X X X X XXX X X
X X X X X X X XXX
X X XXX X X XX X
X X X X X X X
X X X XXX XXX XXX X
The final answer is RHONE VALLEY.