Recursive Geometric Closure as a Universal Fixed-Point Principle

We introduce a general mathematical framework based on the principle of recursive geometric closure, defined on scale-free projective configuration spaces. We show that a broad class of nonlinear closure operators admits a unique, globally attractive fixed point, independent of auxiliary choices such as kernels, norms, or nonlinear response functions. Furthermore, we demonstrate the existence of universal critical geometric invariants associated with the closure process, stable under iteration and coarse-graining. Explicit realizations in Banach spaces are provided, establishing that the framework is mathematically well-defined and non-trivial. These results furnish a rigorous foundation for emergent approaches in which physical constants and effective laws arise as geometric fixed-point invariants rather than free parameters.