The Universal Saturation Principle

We prove that the hyperbolic saturation form f(x) = x/(x + K) emerges universally in

any system satisfying three axioms: finite capacity, reversible competition, and steady-state

equilibrium. This unifies the Universal Saturation Law (observer agreement), Michaelis-

Menten kinetics (enzyme catalysis), Langmuir isotherms (surface adsorption), Monod growth

(bacterial dynamics), Hill equations (cooperative binding), and information channel capacity

into a single mathematical framework. We demonstrate that the dimensional term 1/dim

in the USL corresponds to the “geometric concentration” of disagreement states in highdimensional

spaces, providing a first-principles derivation of observer agreement from concentration

of measure. The unified principle yields a master equation from which all specific

saturation laws can be derived as special cases.