A Narrow Analytic Framework for the Primitive Exponential Integer Equation A^x + B^y = C^z

We apply a Tijdeman-type variable transformation to the equation $A^{x}+B^{y}=C^{z}$ and embed the resulting logarithmic deviation into a Euclidean Pythagorean framework. This continuous normalization yields a geometric inequality determining when $C^{z}$ remains compatible with integer solutions. Combined with the affine decomposition $Qn+x$, the method provides a structural criterion distinguishing primes from composite pseudoprimes such as Carmichael numbers.