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 Σ(z). ER introduces a two-frame kinematic structure, gμν=e2ψĝμν, in which null geodesics are described by a pattern metric ĝμν while massive clocks accumulate proper time dtphys=eψdt̂. The framework is tested by an overconstrained closure protocol: Σ(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 Σ(z)=Ωₘ(z)^2αeff/3 (for the benchmark κ=1/2) with an automatic late-time turn-on. Using a representative late-time suppression Σ₀ 0.9, ER predicts an O(5%) rate inflation H₀ᵖʰʸˢ/Ĥ₀Σ₀-1/2 and a correlated suppression of lensing and growth under the no-slip restriction μ=Σ. Within the fixed-background closure implementation used here, a decoupled modification (Σ<1 with μ=1) is strongly disfavored by RSD growth constraints, while current cosmic-chronometer precision leaves κ effectively unconstrained over a broad range. We also report a diagnostic environment-dependent correlation in supernova calibrator residuals whose sign is consistent with the entropy-production interpretation, but whose amplitude remains sensitive to analysis choices. As an external (non-cosmological) consistency check, we compare ER’s quasi-static transition scale to the SPARC radial-acceleration relation (RAR). The thermally-completed interpolation (empirically equivalent to the McGaugh form) is strongly preferred over the simplest mean-field algebraic form (equal parameter count), with 93 at fiducial stellar mass-to-light ratios and preference >0 across a broad Υ grid. This yields an order-unity KMS susceptibility C_χ ≡ 2π (a₀/c H₀)=O(1), sharpening the “a_* ~ cH₀” 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_* ~ cH₀ 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.