Context-Free Fixed Points and Complete Classification of Orbits in Picard Iteration for Guarded Power Language Operators

We study the language-theoretic structure of fixed points and finite Picard iterates for guarded q-power language operators. For the general operator, we prove that a context-free seeded language deter- mines a unique context-free fixed point and give an effective construction of a context-free grammar generating this fixed point, independently of the initial language.We then consider a marked single-guard special case in which two new symbols separate the recursive contribution from the the contribution of the seed. In this setting, the initial language can be traced and recovered exactly from every finite Picard iterate by means of a regular slice and fixed-word quotients. This yields injectivity of the finite-time maps and an abstract finite-time class-preservation prin- ciple. As a consequence, we obtain a classification of the finite Picard iterates according to the exact position of the initial language in the Chomsky hierarchy. Thus the exact language-theoretic complexity may persist at every finite stage, while all Picard orbits converge to the same context-free fixed point.