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Abstract
This paper introduces the formal theoretical framework of Zero-Entropy Background Topological Dynamics (ZEBTS), establishing $\chi$-ontology as a primary fundamental layer from which spacetime metrics, quantum state vectors, thermodynamic entropy, and inertial particle masses emerge. We construct the rigorous mathematical apparatus of $\chi$-structures, defining differential operators over topological spaces $(X, \tau_\chi)$ and modeling non-local correlations via dynamical entropic cobordisms. Quantization in curved $\chi$-geometry is achieved through path integrals incorporating non-minimal curvature coupling and topological class summations. We mathematically prove the Zero Background Entropy Principle ($S_{\chi,\text{background}} = 0$), demonstrating that it constitutes the unique boundary condition required for ground-state non-degeneracy, structural minimality, and directional entropy flow ($t_\chi = S_\chi$). From this principle, we derive the Universal $\chi$-Equation, yielding exact analytic solutions for scalar topological solitons and cosmological expansion attractors while proving direct low-energy projections to Schrödinger, Einstein, and Hubble dynamics. To establish empirical falsifiability, we present the complete setup, system Hamiltonian, and Lindblad master equation for the $\chi$-Interferometric Fluctuation Test ($\chi$-IFT), using a SQUID interferometer coupled to a tunable qubit entropic array at sub-10 mK temperatures. Quantitative confirmation criteria—including entropy-gradient-dependent coherence time $T_2 = T_2^{(0)}(1 + \beta \vert{}\nabla S_\chi\vert{})$, topological charge step-quantization, and non-local cross-correlations at $>5\sigma$ significance—are explicitly formulated alongside definitive falsification protocols.
