From Hyperbolic Spacetime Geometry to Cosmic RLC Resonators: A Unified Top-Down Synthesis of Space Curvature, Casimir Dynamics, and Quantum Circuit Dualities

This paper synthesizes a continuous conceptual and mathematical thread bridging the geometric foundations ofnon-Euclidean hyperbolic space to the quantum architectures of theoretical physics. Beginning with Nikolai Lobachevsky and JanosBolyai’s constant negative curvature parameter k, we trace its manifestation into Albert Einstein’s General Theory of Relativity and Anti-deSitter (AdS) space. We introduce an original electrodynamic analog mapping the macro-cosmos and micro-cosmos to a passive, resonantResistor-Inductor-Capacitor (RLC) network, where matter boundaries behave as capacitor plates and spatial flux acts as inductive loops.We extend this synthesis to higher-dimensional Kaluza-Klein and braneworld cosmologies, demonstrating how the higher-dimensionalCasimir effect acts as a regularizing quantum spring preventing cosmic collapse. Finally, we review nanoscale experimental metrologyefforts utilizing iso-electronic isotope shifting to detect extra dimensions via non-Newtonian deviations in the real-world Casimir force.