Considerable efforts have been devoted to modifying gravity, which aim to elucidate the possible existence or nature of dark matter and dark energy, achieve a better description of observation data, verify theoretical restrictions in the strong curvature regime and to formulate quantum gravity. In addition, despite the enormous success of the quantum field theory, the framework requires the so-called renormalization techniques and breaks down at high energies. Recently, the Planck Legacy 2018 release has confirmed the presence of an enhanced lensing amplitude in the cosmic microwave background power spectra, which prefers a positively curved early Universe with a confidence level greater than 99%. This study considers the implied positive curvature of the early Universe as the curvature of ‘the background or the 4D conformal bulk’ and distinguishes it from the localized curvature that is induced into the bulk by the presence of comparably smaller celestial objects that are regarded as ‘4D relativistic cloud-worlds’. To consider the interaction between the bulk and cloud-worlds, this paper presents interaction field equations in terms of the brane-world modified gravity counting for the bulk conformal curvature and the boundary terms, which could remove the singularities and satisfy a conformal invariance theory. Similarly, the quantum fields and clouds are regarded as ‘4D relativistic branes’ embedded in vacuum energy of a field strength reliant on the background curvature due to gravity. A visualization of the evolution of the 4D relativistic cloud-worlds over the conformal space-time of the 4D bulk is presented.