SDRIS: The Geometric Origin of Probability - Deriving the Born Rule and Wave-Particle Duality from the Dimensional Reduction of Recursive Amplitudes

Paper 102 of the SDRIS Research Series Following the geometric derivation of forces (Paper 100) and non-locality (Paper 101), this work addresses the origin of quantum probability. Abstract: The Born Rule, which states that the probability of a quantum measurement is proportional to the square of the wavefunction's amplitude ($P = |\psi|^2$), is one of the most fundamental yet least understood axioms of quantum mechanics. Standard theory treats it as an empirical postulate without a rigorous geometric derivation. In this paper, we derive the Born Rule as a necessary consequence of the Static-Dynamic Recursive Information Space (SDRIS) framework. We demonstrate that the "amplitude" $|\psi|$ corresponds to the linear Euclidean length of the information vector in the 5-dimensional bulk, while the "probability" $P$ corresponds to the interaction flux through the 3-dimensional projection interface. Since the information flux through a holographic boundary scales with the surface area ($A \propto r^2$) rather than the linear radius, the quadratic relationship is geometrically inevitable. Furthermore, we show that Wave-Particle Duality is resolved as an observational artifact: The "Wave" is the continuous rotation of the vector in the recursive time dimension ($\tau$), while the "Particle" is the discrete intersection of this vector with the static metric slice ($t$). Thus, quantum randomness is identified not as intrinsic indeterminism, but as the geometric cross-section of a higher-dimensional determinism. Includes Computational Verification: This publication includes a Python simulation (SDRIS_Born_Rule_Simulation_Code_v1.0.ipynb) that numerically validates the flux-projection mechanism, confirming that the measurement of a deterministic 5D-vector yields a statistical distribution strictly following the Born Rule.