Deriving the Weak-Field GR Limit from the Minimal Core of R-Theory

This paper provides a derivation-focused bridge from a minimal dynamical core of R-Theory to the standard weak-field predictions of General Relativity (GR) at first post-Newtonian order (1PN). The strategy is intentionally conservative: we assume (A1) a single dimensionless scalar coherence perturbation χ sourced by mass density, (A2) Poisson correspondence in the static, long-range limit, (A3) universal coupling of both matter and radiation to one effective metric g_{\mathrm{eff}}(\chi), and (A4) an isotropic weak-field expansion. Under these explicit postulates, the isotropic 1PN metric is not merely fit to GR—it is forced into the standard PPN structure, yielding \gamma = 1. With a minimal exponential completion of the time–time potential, \beta = 1 follows as well. The paper then connects this “GR gate” directly to classical observables: once the single-metric postulate holds, Maxwell waves propagate on the null cone of g_{\mathrm{eff}}, recovering the Cassini-class Shapiro time delay and the leading-order light-deflection regime as consequences of the same weak-field scalar χ. The Newtonian potential is identified via \chi=\Phi/c^{2}, calibrated by the empirical value of G in the Poisson limit. Beyond producing a clean baseline, the paper clarifies what remains open: the derived GR gate rests on explicit postulates (especially universal coupling), which future R-Theory development must derive from deeper resonance dynamics. The appendices relate the minimal core to the R-Theory Lagrangian (v2) and discuss how electromagnetic coupling must be specified at the micro-level to reproduce the universal single-metric behavior at the effective level. This establishes a reusable “gate” that any serious R-Theory variant should satisfy before introducing genuinely discriminant, testable deviations (as proposed in Paper 2). Keywords: weak-field gravity; post-Newtonian limit; PPN; effective metric; Poisson limit; coherence perturbation; equivalence principle; Maxwell propagation; Shapiro delay; Cassini radio science; theory reconstruction.