A fixed point of a function f is a value x such that f(x) = x — the function, applied to x, returns x unchanged.
Fixed-point theorems guarantee the existence of fixed points under various conditions. The Knaster-Tarski theorem states that every monotone function on a complete lattice has a least fixed point (and a greatest fixed point). The Banach fixed-point theorem guarantees a unique fixed point for contractive functions on complete metric spaces.
Fixed points appear in the relationality derivation in two roles. At step 10, a Fixed Point is a term whose evaluation yields itself — self-application finding stable configuration. At step 18 (the grand closure), the entire derivation is a fixed point: profiles reconstruct the full derivation, and the full derivation produces profiles. The structure applied to itself yields itself.