Intrinsic 1-D Embedding of Riemann Zeta Zeros: Deterministic Computational and Thermodynamic Validation of Umbrella Topology, Building on Geometric Axial Alignment

We propose a 1-D embedding model of the Riemann Zeta function zeta(s), forming what we term the "Umbrella Topology," which conceptually collapses the complex plane into a single axis analogous to closing an open umbrella, with the tip marking the first non-trivial zero. The axis carries a nested 3D geometry that reflects factor interactions and recursive behavior derived from the natural numbers. Deterministic computational experiments, designed to probe potential deviations from the critical line, are described with explicit consideration for intrinsic versus extrinsic coordinate distinctions and symmetry properties. The framework builds directly on prior work (Perrine, 2025), providing a reproducible foundation for validating the topological hypothesis.