A Geometric-Arithmetic Framework for Critical Lines of L-Functions

We present a framework that bridges vesica piscis geometry with an arithmetic invariant to demonstrate the mathematical necessity of the critical line $\Re(s) = \alpha$ for L-functions encoding prime distribution. For the Riemann zeta function, $\alpha = \frac{1}{2}$. Our approach establishes $\alpha$ as a fundamental invariant derived from independent sources: the vesica piscis construction identifies the critical line $\Re(s)=\alpha$ geometrically (by setting scale $r=2\alpha$), while a generalized divisor function mapping $g_\alpha(n)=\frac{2\alpha}{d(n)}$ uniquely characterizes primes through the value $g_\alpha(p)=\alpha$. We demonstrate that the analytic structure arising from any function encoding prime distribution via an Euler product inherently necessitates that its non-trivial zeros respect this invariant by lying exclusively on the critical line $\Re(s)=\alpha$. The argument shows that internal structural contradictions arising from the simultaneous imposition of necessary geometric, arithmetic, and analytic constraints serve to demonstrate this necessity.While drawing on established principles, the framework's contribution lies in its synthesis: providing a unified explanatory structure that reveals why the value $\alpha$ (specifically $\frac{1}{2}$ for $\zeta(s)$) is mathematically necessary for non-trivial zeros, proving they cannot exist elsewhere. Our argument avoids asymptotic estimates or numerical bounds, relying instead on the structural rigidity imposed by the independent derivations of the critical line from geometry, arithmetic, and analysis.