From Primordial Curvature Perturbation to the Minimal Difference Theorem On the inevitability of gravitational quantum noise from two postulates

If both quantum mechanics and general relativity hold without exception in their respectively confirmed domains, what does their logical intersection strictly enforce? Starting from two postulates — the universality of quantum mechanics and the macroscopic validity of Einstein's field equations — and with a single explicitly designated default presupposition, we prove, under the stated postulates, that the relative momentum variation rate between adjacent freely falling particles (the primordial curvature perturbation) is an objective physical quantity already exposed within classical general relativity, determined directly by the difference of the energy-momentum tensor at two points. Under Postulate I, this difference is itself a quantum operator, so the primordial curvature perturbation possesses ineliminable quantum fluctuations. By invoking the information conservation inherent in unitary quantum evolution, we further prove that these fluctuations must take the irreducible minimum set by the material vacuum fluctuations, thereby establishing the minimal difference principle as a deductive theorem. Based on this theorem, we establish a unique mapping from the quantum noise of matter to fluctuations of spacetime geometry, and derive a rigid scaling law for gravitationally induced quantum decoherence. The scaling law contains no free parameters, being fixed solely by the Planck scale together with the system mass, size, and superposition separation, and constitutes a quantitative prediction testable in principle by foreseeable experiments. If confirmed, the prediction provides positive evidence for the direct compatibility of quantum mechanics and general relativity without additional hypotheses; if excluded, this compatibility is strictly refuted.