On the Uniqueness of Prime Solutions to the Equation $a^n+b^n=n^c+n^d$

We investigate the structure of solutions to the self-referential exponential Diophantine equation$a^n+b^n=n^c+n^d,\qquad a,b,c,d,n\in\mathbb{N}$under the constraint that all variables are prime numbers. Through elementary number-theoretic analysis and modular constraints, we prove that if $a,b,c,d,n$ are all prime and satisfy the equation, then necessarily $a=b=c=d=n$. This result demonstrates that the primality constraint uniformly projects the solution space---spanning both algebraically generated parametric families and arithmetically escaping resonance families---onto the diagonal form $(p,p,p,p,p)$. Computational verification reveals a parity bifurcation in the integer solution space: solutions are dense for $n=2$, exhibit a double-exponentially sparse distribution for $n=2^k$, and are strictly empty for odd $n\ge 3$. By comparing self-referential hierarchies and analyzing structural duality, we elucidate how the exponent-base coupling acts as an arithmetic filter that reshapes exponential solution spaces.