Recursive Spectral Cosmology and Higher-Order Admissibility Flow A Higher-Order Effective Geometric Cosmology from Bures–Wasserstein Admissibility Dynamics

We develop a nonlinear operator-geometric framework for cosmology based on metric-gradient flow on the positive trace-class cone equipped with a Bures–Wasserstein information geometry. The framework is constructed as a higher-order effective geometric theory in which ultraviolet regularization emerges dynamically from admissibility-induced spectral stiffness rather than explicit topological dimension reduction. The central dynamical object is a nonlinear admissibility functional defined on the positive trace-class Banach space of quantum states. The nonlinear sector is governed by a conformally covariant fourth-order Paneitz-type operator acting as an interior-point barrier on the admissible state domain. The resulting dynamics define a globally well-posed m-accretive Crandall–Liggett semigroup on the trace-class cone. In the ultraviolet boundary-dominant regime, the principal symbol of the effective generator becomes asymptotically biharmonic. This modifies the effective spectral scaling from standard Laplacian Weyl growth to fourth-order Weyl behavior, yielding heat-kernel scaling consistent with apparent spectral dimension running from d_s = 4 in the infrared to d_s → 2 in the ultraviolet. Importantly, this phenomenon arises from operator-order doubling rather than literal coordinate-topology reduction. Applying the framework to perturbed FLRW cosmology generates scale-dependent dissipative corrections to primordial tensor propagation. High-frequency gravitational-wave modes acquire Bures–Wasserstein damping corrections while scalar perturbations develop anisotropic stress signatures characterized by gauge-invariant potential splitting Φ ≠ Ψ. The framework predicts stretched-exponential tensor transfer suppression, modified polarization asymmetries, evolving effective equation-of-state corrections, and constrained early-universe radiation injection signatures testable by next-generation CMB-S4, DESI, Euclid, LISA, LiteBIRD, DECIGO, and Einstein Telescope observations. The work is explicitly formulated as a higher-order effective geometric cosmology and does not claim a complete ultraviolet completion of quantum gravity. Instead, it establishes a mathematically disciplined operator-theoretic cosmological framework with concrete observational predictions.