A Scale-Dependent Unified Framework for Quantum Mechanics and Gravitation via Functorial Operators

We propose a scale-dependent unified framework for quantum mechanics and general relativity based on a single variational principle acting on a joint functional U [ψ, g], where ψ is a quantum state in a Hilbert bundle and g is a Lorentzian spacetime metric. The central ingredient of the construction is a scale-dependent linear operator Ôλ that smoothly interpolates between quantum-dominated and gravity-dominated regimes through a monotonic transition function f (λ). This operator implements a continuous crossover between ultraviolet quantum dynamics and infrared classical gravity, avoiding sharp quantization boundaries and treating the quantum–classical transition as intrinsically scale dependent. The resulting variational principle yields coupled equations of motion for the quantum state and the spacetime geometry. In the appropriate limits, the framework consistently reduces to quantum field theory on curved spacetime and to the semiclassical Einstein equations with quantum backreaction. The interpolation structure admits a natural interpretation in terms of weighted superposition and deformation of quantum and gravitational dynamics, and can be formulated categorically as a natural transformation between quantum and gravitational functors. As a concrete application, we implement the framework in a spatially flat FLRW cosmology, deriving modified Friedmann equations with scale-dependent quantum corrections. The numerical evolution demonstrates that quantum effects can influence early-universe dynamics while standard cosmological expansion is recovered at late times. This approach provides a mathematically consistent and physically transparent pathway toward quantum gravity, bridging abstract formalism and phenomenological implications.