The Recursive Universe: A Joint Fixed-Point Framework for Quantum States, Effective Laws, and Quantum Gravity

This paper develops a consistency-first formulation of the recursive-universe hypothesis. Rather than beginning with a state-dependent unitary and treating stationarity as self-reference, we represent a physical theory by two coupled objects: a quantum state and a set of effective laws or couplings. A recursive universe is defined as a joint fixed point of a state-update map and a law-update map. This separation resolves an ambiguity in the condition ∣Ψ⟩=U[Ψ]∣Ψ⟩, which by itself is only a nonlinear stationarity equation and is insensitive to the distinction between state dynamics, computability-theoretic recursion, and renormalization-group fixed points. In finite-dimensional truncations, we prove fixed-point existence for continuous admissible maps on compact convex state-law spaces, uniqueness and iterative stability for contractions, and perturbative persistence of non-degenerate fixed points under weak recursive deformation. A self-consistent two-state benchmark exhibits a pitchfork bifurcation: above a critical feedback strength, the symmetric state becomes unstable and two stable symmetry-broken states emerge. We then apply the framework to renormalization-group theory space. If an undeformed beta function possesses a non-degenerate fixed point, a small recursive deformation shifts its coordinates and critical exponents by computable first-order expressions. This establishes a controlled bridge to asymptotic-safety research without claiming numerical quantum-gravity results that have not been calculated. We also formulate action-level and covariant implementation criteria, explain why deterministic subsystem-level nonlinear quantum dynamics raises no-signalling and ensemble-dependence problems, and replace a universal recursive coupling with normalized, model-specific parameters. A Ramsey-interferometry benchmark yields an explicit conversion from phase sensitivity to a coupling bound, while gravitational-wave and cosmological tests are expressed as template-based programmes requiring a specified recursive kernel. The framework does not yet constitute a UV-complete theory of quantum gravity or a solution to the cosmological-constant problem. Its contribution is a mathematically explicit architecture that separates proven fixed-point results, illustrative models, and conjectural physical applications, thereby turning recursion from a metaphor into a disciplined research programme.