This paper presents a theoretical synthesis and mathematical consolidation of the Information-Topological Register Model, building upon the foundational frameworks established in Works 1 and 2. It provides a bridge between the discrete informational nature of the "register" and the smooth manifold of General Relativity, demonstrating that spacetime and mass are emergent properties of a 1D information-topological network. By unifying the geometric quantum of space (κ) and the informational mass coupling (μ), this supplement removes all arbitrary free parameters and provides rigorous proofs for the following milestones: Derivation of Mass-Energy Equivalence: We demonstrate that E = mc² is not a fundamental axiom, but a necessary geometric consequence of the topological field energy density within the register. Emergence of the Schwarzschild Metric: The model deterministically reconstructs the metric component g₀₀ = 1 - 2GM/rc². The holographic limit (S = 1) organically enforces the topological event horizon, allowing for a seamless transition to the physics of black holes without artificial saturation terms. Resolution of the Equivalence Paradox: The framework distinguishes between kinematic gravitation (following geodesics) and topological inertia. It introduces an exponential effective quasiparticle mass (Mₑff) for systems under extreme quantum coherence. Experimental Falsifiability via BECs: The theory offers a sharp, laboratory-testable prediction. We show that in a Bose-Einstein Condensate, the internal speed of sound (cₛ) must collapse exponentially as the degree of quantum entanglement (η) approaches its limit. This anomaly provides a direct empirical pathway to falsify the model using Bragg spectroscopy.
Nicolas Köllmer (2026) studied this question.