Where Geometry Meets Number Theory: A Constructive Framework

This work presents a series of interconnected mathematical \emph{constructions} that take the zeros of the Riemann zeta function as primordial elements. Rather than seeking a conventional proof of the Riemann Hypothesis, we investigate: what kind of mathematical reality emerges when we \emph{postulate} that these zeros form the spectrum of an operator within a specific geometric arena? Our constructions reveal a remarkable chain of coherence, linking geometry (minimal surfaces), topology (M\"obius bands), statistics (GUE), and fundamental physical constants. Within the constructed framework, the critical line $\Re(s)=1/2$ appears as a \emph{necessary condition}, GUE statistics as an intrinsic geometric property, and relations between the first four zeros encode the fine structure constant $\alpha^{-1} = 137.035999084\ldots$ to experimental precision \cite{CODATA2018}. We present these constructions not as final theorems, but as substantive \emph{insights} from a perspective that treats the zeta function not merely as an object of analysis, but as a potential organizational principle of mathematical reality.