Entropic Relativity: A Metrological Closure Framework for Late-Time Cosmology

We present Entropic Relativity (ER), a metrological closure framework in which late-time cosmological observables separate into operational classes (null-geometry distances, clock-based rates, lensing, and growth) whose mutual consistency is governed by a single horizon response function $\Sigma(z)$. ER introduces a two-frame kinematic structure, $g_{\mu\nu}=e^{2\psi}\hat{g}_{\mu\nu}$, in which null geodesics are described by a pattern metric $\hat{g}_{\mu\nu}$ while massive clocks accumulate proper time $dt_{\mathrm{phys}}=e^{\psi}d\hat{t}$. The framework is tested by an overconstrained closure protocol: $\Sigma(z)$ inferred from lensing predicts both clock-rate deformations and the growth history through fixed maps. We sketch a thermodynamic ansatz, the Local Entropic Extremum (LEE), which yields a simple power-law envelope $\Sigma(z)=\Omega_m(z)^{2\alpha_{\mathrm{eff}}/3}$ (for the benchmark $\kappa=\tfrac{1}{2}$) with an automatic late-time turn-on. Using a representative late-time suppression $\Sigma_0\simeq 0.9$, ER predicts an $\mathcal{O}(5\%)$ rate inflation $H_0^{\mathrm{phys}}/\hat{H}_0\simeq\Sigma_0^{-1/2}$ and a correlated suppression of lensing and growth under the no-slip restriction $\mu=\Sigma$. Within the fixed-background closure implementation used here, a decoupled modification ($\Sigma0$ across a broad $\Upsilon$ grid. This yields an order-unity KMS susceptibility $C_\chi \equiv 2\pi (a_0/c\hat H_0)=\mathcal{O}(1)$, sharpening the “$a_* \sim cH_0$” target into a quantitative calibration. ER reduces the number of independent late-time dark-sector response functions by treating lensing, growth, and clock-rate discrepancies as different operational projections of a single horizon response, while identifying the galactic RAR scale $a_* \sim cH_0$ as an external consistency target. The primary contribution of this work is methodological: a sign-audited, overconstrained closure protocol that can be applied to present and future datasets to falsify (or support) a one-function horizon-response hypothesis.