CDUFD Supplementary Material VII: Emergence of the Born Rule from Topological Instrument Dynamics

The Born rule---the probability of obtaining measurement outcome aᵢ given by P (aᵢ) =|aᵢ|²---is the last independent postulate of quantum mechanics not yet reduced to dynamical principles. Existing decoherence programs explain why measurement outcomes appear classical, but cannot derive why the probabilities are precisely ||². Gleason's theorem establishes the Born rule as the unique consistent probability assignment on Hilbert space, but it presupposes the Hilbert space framework itself. In this paper, we provide a complete derivation of the Born rule from the axioms of the Constraint-Dispersion Unified Framework (CDUFD). The central innovation is replacing statistical randomness with topological determinism: using A4 topological defects as natural instruments, their discrete topological charges provide a discrete spectrum of measurement outcomes; topological pairing constraints ensure that the relaxation of the system-instrument composite is deterministically directed---the instrument does not create probability, it merely ``reads out'' the topological sector of the system; the equilibrium distribution of the system in the critical regime is rigorously determined to be ||² by the uniqueness of the Fokker-Planck steady state. The Born rule emerges as a direct corollary of the equilibrium statistical mechanics of the critical regime. The derivation strictly follows the CDUFD methodology of solution-space construction and constraint-driven reduction, with all steps anchored to axioms A1--A5. Together with Supplementary Material I (Madelung transformation) and Supplementary Material VI (Wallstrom condition), this work completes the full emergence of quantum mechanics from CDUFD axioms.