From Particle Horizon to Solution Space Selection of the Wheeler-DeWitt Equation: A Mathematical Physics Proof Based on Quantum Mechanics Postulates and Cosmological Boundary Conditions

The Wheeler-DeWitt equation, as the central equation of quantum gravity, suffers from an excessively large solution space due to the lack of well-defined boundary conditions, leading to a loss of predictive power. Traditional approaches introduce artificial boundary conditions or topological constraints, but they lack support from first principles. This paper demonstrates that, by combining the fundamental postulates of quantum mechanics with the cosmological particle horizon, the physically relevant solutions of the Wheeler-DeWitt equation must belong to the space of bandlimited functions. This selection is entirely based on the principle of observability and requires no presupposition of a transcendent external observer.