The Onset of the Relativistic Ruler: Metric Emergence and the Pre-Relativistic Boundary of the GERT Universe

Background: The Gibbs Energy Redistribution Theory (GERT) program established a thermodynamic ontology for cosmology (Paper~I) and later identified the post-relativistic dissolution boundary of the relativistic ruler in the Hyperdilute Regime (Paper~II). The complementary open question is the onset of relativistic metric legibility in the early Universe. Objective: To determine, within GERT, the emergence boundary of the relativistic metric ruler and define the lower limit of validity of the effective relativistic regime. Methods: We define the metric-emergence parameter $\Xi(\alpha)\equiv\lambda_\gamma(\alpha)/d_{\mathrm{ph}}(\alpha)$, where $\lambda_\gamma$ is the photon mean free path and $d_{\mathrm{ph}}$ is the GERT particle horizon. The boundary is set by $\Xi=1$. We compute $\alpha_{\mathrm{em}}$ using two recombination treatments (Saha equilibrium and Peebles kinetics) and test robustness against the unknown Primordial Cauldron boundary $\alpha_{\mathrm{PC}}$. Results: We obtain $\alpha_{\mathrm{em}}=-3.0\pm0.1$, with uncertainty dominated by recombination kinetics (Saha vs.~Peebles). Varying $\alpha_{\mathrm{PC}}$ over 25 orders of magnitude changes $\alpha_{\mathrm{em}}$ by less than $5\times10^{-4}$, showing strong insensitivity to primordial microphysics. Together with Paper~II ($\alpha_{\mathrm{crit}}=12.88\pm0.12$), the relativistic GERT domain spans $15.9\pm0.2$ decades in $\alpha=\log_{10}(a)$. Conclusions: The relativistic ruler is an emergent operational regime, not an ontologically unlimited one. GERT now provides a complete domain map with pre-relativistic, relativistic, and post-relativistic sectors. The onset and dissolution boundaries are thermodynamically controlled, giving a symmetric validity structure for Layer 3.