Alright, the conjecture looks to be true, in that I have not found any counterexample.
The 10th prime is 29 and the 10th perfect-square is 100
Now, how far out do I need to go to find the first and perhaps only 29 = p-sq_1 + or - p-sq_2
Well it happens that 225-196 = 29
However, 29 = 25 +4
So I need to discard the idea of 1-1 correspondence because it is somewhat clear that we have a sort of oscillation like a trig function of add or subtract. Perhaps even a harmonic oscillation.
Now it is clearly known that the primes are more abundant than the perfect-squares because we have to go to 100 for the 10th perfect square yet only 29 for the 10th prime. So I can forget or discard the idea of 1-1 correspondence of prime to perfect square.
Now the subtraction appears to give me many primes, but does the addition give me as many?