GETT Correspondence Paper 4.3: Empirical Recovery of General Relativity in the Einstein-Domain

Description This work presents Paper 4.3 of the General Expanse Tension Theory (GETT) Correspondence Series, a structured programme aimed at reconstructing established physical frameworks as domain-limited emergent descriptions of an underlying scalar-field system. The Correspondence Series proceeds in stages: establishing the special relativistic limit (Paper 3), demonstrating the emergence of metric structure (Paper 4.1), recovering Einstein-domain field dynamics (Paper 4.2), and, in the present work, completing the programme at the empirical level. Objective The objective of this paper is to demonstrate that: Within a clearly defined Einstein-domain, the General Expanse Tension Theory reproduces the full set of experimentally verified weak-field predictions of General Relativity (GR). The Einstein-domain is defined as the regime in which: the scalar field Φ is smooth in space, temporal evolution is adiabatic, the coupling function is effectively constant (mid-density plateau), and the system lies within the weak-field limit. Methodology A structured and non-selective validation framework is applied across a comprehensive set of observables. Each is treated using a consistent four-stage methodology: Observable → GR Prediction → GETT Reduction → Equivalence For each observable, equivalence is required at: the level of functional form, the level of parameter dependence, and the level of numerical agreement with experimental observation. All derivations are carried out within the Einstein-domain, with no parameter fitting or case-specific adjustment. Full technical derivations are provided in dedicated appendices to ensure transparency and reproducibility. Content The paper evaluates nine canonical tests of General Relativity, spanning all major experimentally accessible weak-field phenomena: Perihelion precession (Mercury) Gravitational redshift Light deflection (weak-field lensing) Shapiro time delay Gravitational time dilation (including GPS systems) Gravitational-wave propagation speed Geodetic precession Universality of free fall (Einstein-domain limit) Frame dragging (Lense–Thirring effect) Each observable is independently derived and compared to its GR counterpart, with full derivational detail provided in Appendices A–I. Results The analysis demonstrates that: All nine observables are recovered exactly within the Einstein-domain. Agreement is achieved without parameter tuning. The correspondence holds across: orbital dynamics, relativistic timing, signal propagation, precession effects, and fundamental principles such as free fall. The results establish that: GETT is empirically indistinguishable from General Relativity across all currently tested weak-field regimes. Interpretation Within the GETT framework: gravitational phenomena arise from the behaviour of an all-pervading scalar field ΦΦΦ, local measurement processes (time, space, signal propagation) are governed by response functions, and the spacetime metric of GR emerges as a coarse-grained representation of these responses. The geometric description of GR is therefore recovered as an effective, domain-limited representation of underlying scalar-field dynamics. Domain of Validity The correspondence established in this work is explicitly domain-limited. It applies only within the Einstein-domain. Outside this regime—particularly in: ultra-low-density environments (e.g. galactic outskirts, cosmic voids), ultra-high-density systems (e.g. compact objects), and density-transition regions, GETT predicts systematic departures from GR behaviour due to density-dependent coupling and non-uniform scalar-field dynamics. These regimes are not treated in this paper and form the basis of subsequent work in the series. Conclusion This paper establishes a complete empirical correspondence between GETT and General Relativity within the Einstein-domain. By demonstrating that all tested weak-field gravitational observables are reproduced exactly and without parameter adjustment, the work provides: a unified underlying mechanism for relativistic phenomena, and a clear, testable framework for exploring deviations beyond the regimes in which GR has been experimentally validated. Series Context This paper forms a central component of the GETT Correspondence Series, completing the transition from: theoretical construction → metric emergence → field equation recovery → empirical validation Future work (Paper 4.4) will address behaviour beyond the Einstein-domain and the observational implications of density-dependent coupling.