Cosmological Background and Linear-Growth Validation of the Entropic No-Boundary Hypothesis: A Symmetric Extension of Cosmology with Implications for the Edge Problem, Cosmic Topology and Dark Sector Dynamics

The Entropic No-Boundary (ENB) hypothesis proposes that the Universe possesses no spatial boundary when spacetime is extended by a real-valued entropic coordinate 𝑆. Enforcing smoothness, analyticity, and integrability across this extended manifold eliminates the possibility of spatial edges and generates an additional geometric stress–energy contribution 𝑇𝜇𝜈 (𝑆). This entropic term acts as an intrinsic source of curvature, producing dark-matter-like and darkenergy-like phenomena without invoking new particles or external fields. The ENB framework therefore offers a unified geometric origin for the dark sector. The first companion paper demonstrated that the entropic geometric contribution naturally produces the additional centripetal acceleration required to explain galactic rotation curves. A second validation paper extended this to a full population-level empirical test using 175 galaxies from the SPARC database, including deterministic fits, hierarchical Bayesian modelling, and weaklensing-based predictions. Together, these results established that the entropic term 𝑇 𝜇𝜈 (𝑆) consistently reproduces dark-matter-like behaviour across a broad range of galactic environments. The purpose of this third validation paper is to examine the implications of the ENB framework at cosmological scales. Specifically, we analyse how the entropic stress–energy affects: 1. the background expansion history 𝐻(𝑧), 2. the deceleration parameter 𝑞(𝑧), 3. the linear matter growth factor 𝐷(𝑎), and 4. the observable growth rate 𝑓𝜎8(𝑧). These quantities represent the minimal set of cosmology-level diagnostics required to evaluate any alternative to the standard ΛCDM model. We perform the analysis in two stages. First, we examine the Tier-I minimal ENB configuration, corresponding to the special case in which the entropic sector behaves like a curvature-like contribution with equation-of-state parameter 𝑤𝑆 = - 1 3 . This configuration is analytically simple and was chosen in the main ENB theory paper to emphasise the geometric origin of the entropic term. Second, we extend the analysis to the Tier-II generalised entropic sector, where 𝑤𝑆 is treated as a free parameter. This allows the entropic contribution to mimic a wide range of dark-sector behaviours, including full recovery of ΛCDM in the limit 𝑤𝑆 = -1. The central goals of this paper are therefore: • to determine whether the minimal ENB cosmology (Tier-I) is consistent with the observed expansion history and structure growth; • to identify the region in the generalised entropic parameter space (Tier-II) that yields cosmologies compatible with current observations; and • to assess, in combination with the galaxy-scale results of the companion papers, whether the ENB hypothesis remains a viable unified geometric alternative to particle dark matter and vacuum dark energy. The results show that the curvature-like Tier-I configuration is inconsistent with the required late-time acceleration, while the generalised Tier-II entropic sector naturally includes a physically viable accelerating regime and reproduces the ΛCDM limit exactly. This provides a coherent cosmological extension of the ENB framework and reinforces its interpretation as a unified geometric theory of the dark sector.