GERT and the Internal Thermodynamic Anatomy of the Relativistic Window — Cohesive and Entropic Peaks in the Gibbs Dance

Background: The first three papers of the Gibbs Energy Redistribution Theory (GERT) programme established the thermo-relativistic ontology of the expanding Universe, bounded the regime of metric legibility by two operational limits — the primordial emergence of the relativistic ruler at α_em = -3.0 ± 0.1 and its hyperdilute future dissolution at α_crit = 12.88 ± 0.12 — and validated the framework against supernovae, CMB, and BAO data (χ2_ν ≈ 0.99, H0 ≈ 72.5 km/s/Mpc). The relativistic window defined by those boundaries spans 15.9 ± 0.2 decades in the expansion parameter α and has hitherto been treated as a homogeneous interval of effective validity. Objective: This paper investigates the internal thermodynamic organization of that window. We argue that the relativistic domain of GERT is not featureless between its boundaries: it possesses an internal anatomy, structured by smooth background transitions and localized episodes of reorganizational thermodynamic Work encoded in the Gaussian peaks of the GERT dynamical functions. Methods: Using the fixed parameters of Paper I [1] (log ρ_c = -17.41, f_m,peak = 0.37; log ρ_L2 = -23.93, f_L,peak = 4.62), the late-time boundary from Paper II [2], and the emergence condition ξ(α_em) = 1 of Paper III [3], we develop the interpretation of the logistic-Gaussian functional architecture through the lens of macroscopic phase transition theory. We introduce the concept of integrated thermodynamic impulse, compute the derivative structure of the GERT functions, and map each feature of the fitted model onto a specific epoch of cosmic reorganization. Results: We identify four major thermodynamic milestones within the relativistic window: (1) the cohesive inauguration at log ρ_c = -17.41, coinciding with recombination understood as a triple transition — chemical, optical, and metric; (2) the liquid stabilization at log ρ_M = -20.30, marking the passage from builder to maintainer; (3) the entropic reversal at log ρ_L2 = -23.93, the concentrated outward Work that transfers thermodynamic primacy from cohesion to expansion; and (4) the long dissolution from entropic trigger through gas-dominated asymptotic future. Conclusions: The Gaussian peaks of GERT are not phenomenological fitting artifacts but the exact mathematical forms demanded by macroscopic phase transition theory: the cosmological analogues of specific-heat peaks at critical thresholds. The relativistic window is a finite thermodynamic organism: born by recombination, stabilized by cohesive relaxation, reversed by an entropic surge, and dissolved by gaseous expansion. Relativity, in GERT, has not only a domain of validity, but a thermodynamic biography.