A cross-epoch endpoint-consistency test of a single effective scaling from dark energy to inflation

Abstract

A cross-epoch endpoint-consistency test is formulated for a single power-law effective scaling, $M_U(\mu)=\beta^{1/4}\,M_P(\mu/M_P)^\gamma$, connecting late-time cosmic acceleration to the inflationary energy scale. The normalization is anchored at $\mu=H_0$ by the late-time dark energy closure relations of the density-responsive gravity (DRG) framework and extrapolated to the inflationary Hubble rate $\mu=H_*$. Comparison with the CMB-normalized Starobinsky plateau defines the residual matching factor $C_{\rm infl}\equiv V_0^{A_s}/V_0^{\rm RG}$. The novelty is the formulation of the inflation - dark-energy connection as a quantitative endpoint test between two empirically anchored energy scales. The lever arm $H_*/H_0\sim 10^{55}$ compresses endpoint uncertainties into $\delta\gamma\propto [p\ln(H_*/H_0)]^{-1}\delta\ln X$, so requiring $C_{\rm infl}=\mathcal{O}(1)$ selects a narrow consistency band. For the benchmark $V_0\propto M_U^4$ with $c_4=\mathcal{O}(1)$, matching requires $\gamma\simeq 0.491$ and $\beta\simeq 0.68$, both $\mathcal{O}(1)$. Varying $p\in[2,8]$ shifts $\gamma$ only mildly (approximately $0.45$--$0.52$), with natural matching favoring $p\simeq 3$--$4$. We present the full uncertainty budget, including cumulative threshold effects ($\Xi$) and observational errors on the late-time anchor ($w_0$, $A_s$, $H_*$). A secondary consequence is that the effective vacuum sector dilutes as $\rho_\Phi\propto a^{-6\gamma}$ at late times, faster than spatial curvature for $\gamma>1/3$. In a closed universe, this permits a cosmological turnaround at $|\Omega_K|\sim\mathcal{O}(10^{-4})$, testable with Stage-IV surveys and qualitatively distinct from $\Lambda$CDM.