A Geometric-Arithmetic Framework for the Critical Line of the Riemann Zeta Function III

We present a framework that bridges vesica piscis geometry with an arithmetic invariant to explain the critical line of the Riemann zeta function. Our approach establishes 1/2 as a fundamental invariant from two independent sources: the vesica piscis construction identifies the critical line ℜ(s) = 1/2 geometrically, while the divisor function mapping g(n) = 1/d(n) uniquely characterizes primes through the same value.We demonstrate that any analytic function encoding prime distribution must respect this invariant, and verify through analytic arguments that zeros off the critical line would contradict established bounds on prime distribution.While drawing on established principles, the novelty of this framework lies in its synthesis: providing a unified explanatory structure that explains why the value 1/2 is mathematically necessary for non-trivial zeros, not just proving that it must be.