Fuzzy 'Gaussian Prime' Number Force Fields

 Gaussian prime (G = 3 modulo 4) and nonGaussian prime (~G = 1 modulo 4)

Theorem:  Twin Prime natural number line endpoints difference ~G - G = 2 for all twin primes (P(n+1)-P(n)) and n = n + 1,

Proof: □ If we examine a twin prime pair, and both being natural numbers and one is a Gaussian prime then the other cannot be a Gaussian prime, unless it too is congruent to 3 modulo 4. However, since twin primes are separated by two, it is impossible for both to satisfy this congruence condition simultaneously. If one twin prime is congruent to 3 modulo 4, the other must be congruent to 1 modulo 4, and thus, cannot be a Gaussian prime G = 3 modulo 4.■