The Non-Commutativity Theorem between Dynamics and Inference in Cosmology: The Structural Origin of the H₀ Tension and the Non-Equivalence between Operators (Ladder, CMB, Lenses)

The H₀ tension is commonly interpreted as a discrepancy between independent measurements of the present cosmic expansion rate. This work shows that the operational universality of the H₀ estimator is not a general mathematical identity, but depends on a commutativity condition between (i) the underlying dynamical operator and (ii) the inferential operator used in observational reconstruction. We establish the Non-Commutativity Theorem: if the true expansion contains components outside the inferential subspace employed within a finite redshift window, the estimator Ĥ acquires a structural displacement of order O(z_max). For typical observational windows, the expected bias is of order ~5–10%, comparable to the observed magnitude of the H₀ discrepancy. We further establish the Non-Equivalence Theorem: inferential operators defined in distinct physical domains (distance ladder, CMB, gravitational lenses) are not automatically equivalent without additional consistency assumptions. Even within a common parameterization (e.g., ΛCDM), the physical functionals mapping data to parameters operate on structurally distinct domains. Application to Pantheon+ data (Brout et al. 2022; Scolnic et al. 2022) shows systematic drift Ĥ(z_max) with slope coefficient β = −9.8 ± 1.2 km/s/Mpc (8.2σ). A synthetic null-test confirms the effect is not a pipeline artifact (β_mock = 0.2 ± 0.9 km/s/Mpc). Robustness tests under alternative weighting and redshift cuts support the structural interpretation.