Critical Geometric Thresholds and Emergent Dimensionless Invariants in Recursive Closure Frameworks

Can geometry alone force the existence of non-adjustable dimensionless quantities, the kind of robust, scale-free constants that any serious physical theory eventually has to confront? This paper shows that the answer is yes, and gives the precise mechanism. Working entirely within a mathematical framework of recursive geometric closure on scale-free projective configuration spaces, we introduce a class of geometric densification functionals and prove that their evolution under an admissible closure operator exhibits a sharp critical threshold: a unique, non-tunable value determined entirely by the fixed-point geometry of the operator. This threshold is stable under iteration and coarse-graining and cannot be adjusted without changing the closure structure itself. Beyond the threshold, we show that the full hierarchy of invariants {Ip(g∗)}p≥2 forms an injective curve on the unit hyperbola x2−y2=1, with position and curvature determined entirely by the operator, nothing is imposed from outside. All spectral quantities are algebraic for algebraic fixed points. To confirm that the framework is non-vacuous, explicit realizations are constructed in standard functional spaces and the invariants are computed directly. No physical constants are identified, no dynamics are assumed, and no phenomenological interpretation is made. The work establishes a rigorous intermediate layer between abstract fixed-point principles and any future identification of geometric invariants with physical quantities.