Joint State-Law Fixed Points in Quantum Theory: Recursive Dynamics, Bifurcations, and Renormalization-Group Deformations

This article develops a joint state-law fixed-point method for self-consistent quantum models. A physical model is represented by a density operator and a finite-dimensional vector of effective law parameters; admissible dynamics consist of a state-update channel and a law-update map. A recursive solution is their joint fixed point. This formulation separates stationarity of a state-dependent Hamiltonian from genuine co-determination of states and laws and preserves the option of completely positive trace-preserving subsystem dynamics. Three general results are obtained: existence for continuous maps on compact convex state-law domains, uniqueness and iterative stability under a contraction condition, and persistence of non-degenerate fixed points under weak deformation. An exactly solvable two-state model displays a supercritical pitchfork bifurcation, critical exponent one-half, and a divergent zero-field susceptibility at the feedback threshold. In renormalization-group theory space, the same perturbative result yields first-order shifts of fixed-point coordinates and critical exponents. An action-level construction shows how recursive consistency operators can enter the standard functional flow without adding ad hoc terms to the Wetterich equation. Covariant gravity and laboratory phase-shift formulas are presented as conditional templates whose predictions require a specified kernel and normalization. The framework therefore contributes a broadly applicable method linking quantum channels, nonlinear dynamics, fixed-point theory, and renormalization-group deformations, while making no claim to a completed theory of quantum gravity.