Finite-Observation Conditional Guarded Language Operators: Kannan Contractions Beyond Banach Contractivity

Abstract

We introduce finite-observation conditional guarded language operators on the complete length-based ultrametric space of formal languages over a finite alphabet. A finite observation map selects a guarded branch according to the membership of prescribed test words. Although each branch is Banach-contractive, branch switching may destroy global Banach contractivity. We derive depth–residual conditions yielding a max-type Kannan inequality with an explicit constant α<1/2. Consequently, the operator has a unique fixed language, and the Picard iteration converges from every initial language. We also construct multi-branch operators that are Kannan-contractive but not Banach contractive and obtain residual error estimates, explicit convergence bounds, finite-depth certification, and eventual stabilization of the selected branch.