Fluctuation as a Consistency Condition of Finiteness: A Unified Derivation of Minimum Length, Maximum Curvature, and the Cosmological Constant

This study demonstrates that three fundamental finiteness conditions of space-time— (1) finiteness of information, (2) finiteness of curvature, and (3) finiteness of Fisher information— cannot simultaneously hold unless space-time possesses an intrinsic fluctuation structure. This fluctuation is not introduced as an assumption; rather, it emerges as a consistency requirement ensuring that the three finiteness conditions do not contradict one another. Within this fluctuation geometry, a parameter-free minimum length α\* is uniquely determined. The existence of this minimum length immediately implies a maximum curvature Rmax⁡=1/α\*2, which in turn fixes the cosmological constant Λ uniquely, since vacuum space-times with bounded curvature must be constant-curvature manifolds. Thus, α\* → Rmax⁡ → Λ forms a single, continuous chain of implications requiring no external parameters. The resulting value of Λ is consistent with observational data from the accelerating universe. These results show that the minimum length, maximum curvature, and cosmological constant are not independent quantities but arise from a single underlying structure dictated by the finiteness of space-time. The framework provides a principled resolution to two foundational issues in general relativity: (1) the allowance of infinite curvature, and (2) the externality of the cosmological constant. It also offers a natural foundation for quantum gravity based on the intrinsic finiteness and fluctuation geometry of space-time.