The Planck Length as a Consistency Condition of Quantum Localization, Gravitational Collapse, and Causal Structure: A Parameter-Free Derivation of Singularity Avoidance

This work demonstrates that the Planck length lp=ℏG/c3 is not an externally introduced assumption nor a dimensional-analysis artifact, but the unique consistency solution required for the simultaneous validity of three independent finite constraints: finite quantum localization, finite gravitational collapse, and finite causal structure. Quantum localization demands increasingly small Δx, while gravitational collapse imposes a lower bound through the Schwarzschild radius, and causal structure requires the preservation of light-cone openness. Analyzing the consistency condition at the intersection of these constraints yields a single physically admissible length scale, Δx=lp, without introducing any free parameters. The existence of this minimum length implies a maximum curvature Rmax⁡∼1/lp2, rendering all classical spacetime singularities—such as the Schwarzschild center, the FRW initial singularity, and the Kerr ring singularity—physically unreachable. Singularity avoidance emerges not from modified field equations or additional hypotheses, but as a necessary consequence of the finite structure of quantum mechanics, gravity, and causality. This result provides a unified and parameter-free foundation for the compatibility of quantum theory and general relativity, offering a new paradigm in which the Planck length arises as the structural closure point of existing physics rather than a speculative quantum-gravity postulate.