Entropic Relativity: Horizon Equilibration and Metrological Time

Two late--time puzzles have long invited a common origin. One is the appearance of a constant term in the Friedmann equation; the other is the systematic departure of galactic dynamics from Newtonian expectation below an acceleration scale $a_0\sim 10^{-10}\,\mathrm{m\,s^{-2}}$. The numerical proximity $a_0\sim cH_0\sim c\sqrt{\Lambda/3}$ is too persistent to be dismissed as a mere coincidence without asking whether both phenomena reflect the same underlying limitation: a universe with a causal boundary of finite information capacity. Entropic Relativity (ER) is an operational attempt to take this limitation seriously. ER distinguishes a geometry reconstructed from null observables from the physical geometry read by material clocks and rods. Physical time is defined operationally by the accumulation of distinguishable microstates, implying that clock rates are fixed by horizon-referenced entropy production rather than by coordinate choice. The universality of coupling to a single physical metric then follows as a consequence of metrological consistency, yielding the equivalence principles as structural outcomes rather than independent postulates. From the holographic entropy bound we identify a maximum-entropy (de~Sitter) scale $H_{\min}$ and hence $\Lambda=3H_{\min}^2$. Within an additive two-source bookkeeping (matter plus an irreducible horizon floor) we show that closure fixes the clocked expansion law uniquely as $H'^2=H^2+H_{\min}^2$, leaving no freedom for mixed ``bridge'' terms. Under minimal near-equilibrium assumptions, fluctuation--dissipation structure fixes the clock--response exponent to $\kappa=\tfrac12$ in the operational map $d\tau'=\Sigma^\kappa d\tau$, where $\Sigma$ is an entropy-production \emph{response} rather than an entropy stock. Since the background identity $\Sigma_{\rm bg}=H^2/H'^2=\Omega_m$ cannot represent perturbative Weyl response, the lensing/growth sector is described by a distinct susceptibility-controlled function $\Sigma_{\rm pert}(k,z)$ with KMS-normalized DC scale $\chi_0=C_\chi/(2\pi)$, where $C_\chi=\mathcal O(1)$ expresses the remaining microphysical normalization. In the quasi-static (Rindler) limit, defining the local channel as \emph{excess distinguishability} above the de~Sitter bath and imposing channel additivity yields $\Sigma_{\rm qs}=a_N/(a_N+a_\dagger)$ with $a_\dagger=cH_{\min}$, and therefore the deep-MOND scaling $a_{\rm obs}\to\sqrt{a_N a_\dagger}$ as a theorem. Including KMS thermal fluctuations with geometric-mean coupling produces a finite-temperature completion,\[\nu(x)=\bigl[1-e^{-\sqrt{x}}\bigr]^{-1},\]algebraically identical to the empirically preferred McGaugh form. On the SPARC compilation (175 galaxies; 3391 points) this thermal completion is strongly favoured over a zero-temperature mean-field alternative ($\Delta\mathrm{AIC}=93$). The associated acceleration scale implies $\chi_0^{\rm SPARC}\equiv a_0/a_\dagger=\mathcal O(0.1)$, with dominant uncertainty from stellar mass-to-light systematics, consistent with KMS-scale normalization ($C_\chi=\mathcal O(1)$) without parameter tuning. At the level of response, ER admits a Schwinger--Keldysh description of the horizon as an open system constrained by metrological covariance, passivity, and causality. Within this setting we obtain an arrow statement: bounded monotone horizon entropy implies that clock-defined time has a lower bound---a metrological onset of time. Imposing \emph{clockability} (path-independent synchronizability of local modular flows across overlapping causal diamonds) forces local Weyl suppression at this onset, aligning Penrose's Weyl-curvature hypothesis with metrological consistency. We also outline a constrained early-universe extension in which primordial perturbations arise from a state-selected, scale-localized horizon bath, preserving a GR window through BBN and recombination. Cosmologically, ER predicts a hierarchy between galaxy weak-lensing shear and CMB lensing suppression fixed by redshift and scale weighting: shear may exhibit few-percent suppression while CMB lensing remains nearly unchanged when the coherence scale is intermediate. Using published CMB-lensing reconstructions as an orthogonal check, we obtain an upper bound on large-scale suppression that disfavors large-coherence realizations and yields a multi-probe consistency wedge in $(\chi_0,\zeta)$ that naturally accommodates the KMS-scale expectation $\chi_0\sim 1/(2\pi)$. A precision cosmological determination of $\chi_0$---and the decisive cross-domain closure test $\chi_0^{\rm cosmo}\simeq\chi_0^{\rm SPARC}$---requires a dedicated tomographic weak-lensing likelihood inference under the full $\Sigma_{\rm pert}(k,z)$ kernel with nuisance refitting and scale cuts; we state this as the principal falsification target.