Abstract This paper examines the cosmological-scale implications of the Precision Cascade framework for tests of gravitational physics. Building on the stabilised H₀ baseline established in Paper 1 and the structural covariance formalism introduced in Paper 3, the analysis explores how precision propagation across cosmological parameters constrains deviations from General Relativity (GR) at the largest observable scales. Using a multi-probe synthesis of cosmic microwave background anisotropies, baryon acoustic oscillations, weak lensing, galaxy clustering, and cluster-mass measurements, the framework predicts that cosmological observations should remain consistent with General Relativity at greater than six-sigma significance when complete survey datasets are analysed. Within this framework, spatial curvature is predicted to satisfy |Ωₖ| ≲ 0.0003, the growth index remains consistent with the GR prediction γ ≈ 0.545, and metric potentials Φ and Ψ remain observationally indistinguishable across cosmological redshifts. These predictions are conditional on validation of the stabilised H₀ baseline and will become directly testable with forthcoming datasets from Euclid, the Nancy Grace Roman Space Telescope, and next-generation gravitational-wave observatories. The results illustrate how cascade-propagated covariance structure within cosmological measurement ensembles constrains gravitational physics at the largest scales. Description This work presents the fifth paper in the Precision Cascade Cosmology Series, which investigates the role of structural covariance, provenance-aware inference, and precision propagation in modern cosmological analysis. Modern cosmology relies on synthesising heterogeneous observational probes—including cosmic microwave background measurements, baryon acoustic oscillations, weak gravitational lensing surveys, galaxy clustering observations, and galaxy cluster studies. These datasets are often combined under assumptions of statistical independence beyond survey-internal covariance matrices. However, many cosmological measurements share elements of their information architecture, including calibration pipelines, modelling assumptions, and observational systematics. The Precision Cascade framework provides a methodology for analysing how such shared structure influences multi-probe inference. Earlier papers in the series developed the reproducibility architecture (Paper 0), analysed structural correlations in Hubble constant measurements (Paper 1), examined independence assumptions in S₈ measurements (Paper S), and introduced a general theoretical framework for structural covariance and precision propagation (Paper 3). The present paper explores the implications of this framework for cosmological tests of gravitational physics. If the stabilised H₀ baseline and structural covariance framework are validated by independent datasets, the resulting precision propagation across cosmological parameters yields strong predictions for spatial curvature, structure growth, and metric perturbations under General Relativity. These predictions are explicitly falsifiable and will become testable as next-generation observational datasets become available