Quantum Mechanics from Finite Observation: Born Rule, Interference, and Uncertainty as Consequences of Markov Blanket Geometry

Paper 6 of the Metric Bundle Programme. We prove that the core axioms of quantum mechanics---the Born rule, superposition with interference, the uncertainty principle, and spectral quantisation---arise as necessary consequences when a finite agent with Fisher--Rao information geometry observes an infinite-dimensional geometric structure through a finite Markov blanket. The central result (thm:born-uniqueness) shows that among all power-law probability rules p_i \propto |\psi_i|^\alpha, only the Born rule \alpha = 2 yields a total Fisher information that is independent of the quantum state when summed over a complete set of mutually unbiased bases. This is verified analytically for qubits and numerically for higher dimensions. Combined with the metric bundle framework's derivation of gauge theory from \Met(M^4), these results suggest that both the Standard Model and quantum mechanics emerge from the geometry of the space of metrics, observed by agents of finite capacity. This paper proves that the core axioms of quantum mechanics — the Born rule, superposition with interference, the uncertainty principle, and spectral quantisation — arise as necessary consequences when a finite agent with Fisher–Rao information geometry observes an infinite-dimensional geometric structure through a finite Markov blanket. The central result (Theorem 4.1) shows that among all power-law probability rules, only the Born rule yields a total Fisher information that is independent of the quantum state when summed over complementary measurements. v2 changes (March 2026): Restricted interference theorem to piecewise smooth functions (was false as stated). Fixed off-diagonal cancellation claim. Added Gleason (1957) and Masanes-Mueller (2011) citations. Fixed author name (Vatan). Rewrote Appendix A. Tempered abstract.