Non-Invertible Cosmological Inference: Operator Non-Commutativity, H₀ Drift, and the Geometric Origin of Dark Sector Phenomena

Modern cosmology relies on the inference of global dynamical properties from signals that have propagated across cosmological distances. Persistent tensions — including the Hubble constant discrepancy, the apparent need for dark matter, and the inference of cosmic acceleration — suggest that the standard interpretative framework may be structurally incomplete. In this work, we develop a unified operator-based formulation of cosmological observation, expressed as O = Π ∘ GX, where X denotes the underlying physical state, G represents the propagation of information through spacetime, and Π denotes the inferential operator mapping distributed signals into local observables. Within this framework, observation is intrinsically non-local and non-invertible. We show that this structure leads to a fundamental non-commutativity between dynamics and inference (Π ∘ G ≠ G ∘ Π), formally established in previous work (Prevedello 2026b, Theorem 5.1). This result implies that cosmological estimators are inherently operator-dependent. Building on a series of ten recent studies (Prevedello 2025a–b; 2026a–h), we demonstrate that: (i) the observed H₀ tension emerges as a consequence of window-dependent inference and estimator drift; (ii) apparent mass discrepancies in galactic systems arise from non-invariant estimators in low-acceleration regimes; (iii) the inference of accelerated expansion results from the Mirage Operator acting on underlying decelerating dynamics; and (iv) cosmic birefringence and effective axion-like phenomena can be interpreted as quantum-geometric signatures of the universal field Φ. These results, previously derived in distinct contexts, are shown here to be manifestations of a single structural principle: the non-invertibility of cosmological inference. The structural results are independent of the underlying cosmological model; the MRUV framework provides one explicit physical realization, but the operator-based non-commutativity and estimator non-equivalence follow from general properties of inference in finite observational domains. Six falsifiable predictions are presented for DESI, LiteBIRD, SKA, CMB-S4, and Euclid.