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 Ξ(α) ≡ λγ(α)/dph(α), where λγ is the photon mean free path and dph is the GERT particle horizon. The boundary is set by Ξ = 1. We compute αem using two recombination treatments (Saha equilibrium and Peebles kinetics) and test robustness against the unknown Primordial Cauldron boundary αPC. Results: We obtain αem = −3.0±0.1, with uncertainty dominated by recombination kinetics (Saha vs. Peebles). Varying αPC over 25 orders of magnitude changes αem by less than 5×10−4, showing strong insensitivity to primordial microphysics. Together with Paper II (αcrit = 12.88 ± 0.12), the relativistic GERT domain spans 15.9 ±0.2 decades in α = log10(a). Conclusions: The relativistic ruler is an emergent operational regime, not an ontologically unlimited one. GERT nowprovides 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.